Found problems: 72
EMCC Accuracy Rounds, 2021
[b]p1.[/b] Evaluate $1^2 - 2^2 + 3^2 - 4^2 + ...+ 19^2 - 20^2 + 21^2$.
[b]p2.[/b] Kevin is playing in a table-tennis championship against Vincent. Kevin wins the championship if he wins two matches against Vincent, while Vincent must win three matches to win the championship. Given that both players have a $50\%$ chance of winning each match and there are no ties, the probability that Vincent loses the championship can be written in the form $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$.
[b]p3.[/b] For how many positive integers $n$ less than $2000$ is $n^{3n}$ a perfect fourth power?
[b]p4.[/b] Given that a coin of radius $\sqrt{3}$ cm is tossed randomly onto a plane tiled by regular hexagons of side length $14$ cm, the chance that it lands strictly inside of a hexagon can be written in the form $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
[b]p5.[/b] Given that $A,C,E,I, P,$ and $M$ are distinct nonzero digits such that $$EPIC + EMCC + AMC = PEACE,$$ what is the least possible value of $PEACE$?
[b]p6.[/b] A palindrome is a number that reads the same forwards and backwards. Call a number palindrome-ish if it is not a palindrome but we can make it a palindrome by changing one digit (we cannot change the first digit to zero). For instance, $4009$ is palindrome-ish because we can change the $4$ to a $9$. How many palindrome-ish four-digit numbers are there?
[b]p7.[/b] Given that the heights of triangle $ABC$ have lengths $\frac{15}{7}$ , $5$, and $3$, what is the square of the area of $ABC$?
[b]p8.[/b] Suppose that cubic polynomial $P(x)$ has leading coecient $1$ and three distinct real roots in the interval $[-20, 2]$. Given that the equation $P\left(x + \frac{1}{x} \right) = 0$ has exactly two distinct real solutions, the range of values that $P(3)$ can take is the open interval $(a, b)$. Compute $b - a$.
[b]p9.[/b] Vincent the Bug has $17$ students in his class lined up in a row. Every day, starting on January $1$, $2021$, he performs the same series of swaps between adjacent students. One example of a series of swaps is: swap the $4$th and the $5$th students, then swap the $2$nd and the $3$rd, then the $3$rd and the $4$th. He repeats this series of swaps every day until the students are in the same arrangement as on January $1$. What is the greatest number of days this process could take?
[b]p10.[/b] The summation $$\sum^{18}_{i=1}\frac{1}{i}$$ can be written in the form $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Compute the number of divisors of $b$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Speed Rounds, 2013
[i]20 problems for 20 minutes.[/i]
[b]p1.[/b] Determine how many digits the number $10^{10}$ has.
[b]p2.[/b] Let $ABC$ be a triangle with $\angle ABC = 60^o$ and $\angle BCA = 70^o$. Compute $\angle CAB$ in degrees.
[b]p3.[/b] Given that $x : y = 2012 : 2$ and $y : z = 1 : 2013$, compute $x : z$. Express your answer as a common fraction.
[b]p4.[/b] Determine the smallest perfect square greater than $2400$.
[b]p5.[/b] At $12:34$ and $12:43$, the time contains four consecutive digits. Find the next time after 12:43 that the time contains four consecutive digits on a 24-hour digital clock.
[b]p6.[/b] Given that $ \sqrt{3^a \cdot 9^a \cdot 3^a} = 81^2$, compute $a$.
[b]p7.[/b] Find the number of positive integers less than $8888$ that have a tens digit of $4$ and a units digit of $2$.
[b]p8.[/b] Find the sum of the distinct prime divisors of $1 + 2012 + 2013 + 2011 \cdot 2013$.
[b]p9.[/b] Albert wants to make $2\times 3$ wallet sized prints for his grandmother. Find the maximum possible number of prints Albert can make using one $4 \times 7$ sheet of paper.
[b]p10.[/b] Let $ABC$ be an equilateral triangle, and let $D$ be a point inside $ABC$. Let $E$ be a point such that $ADE$ is an equilateral triangle and suppose that segments $DE$ and $AB$ intersect at point $F$. Given that $\angle CAD = 15^o$, compute $\angle DFB$ in degrees.
[b]p11.[/b] A palindrome is a number that reads the same forwards and backwards; for example, $1221$ is a palindrome. An almost-palindrome is a number that is not a palindrome but whose first and last digits are equal; for example, $1231$ and $1311$ are an almost-palindromes, but $1221$ is not. Compute the number of $4$-digit almost-palindromes.
[b]p12.[/b] Determine the smallest positive integer $n$ such that the sum of the digits of $11^n$ is not $2^n$.
[b]p13.[/b] Determine the minimum number of breaks needed to divide an $8\times 4$ bar of chocolate into $1\times 1 $pieces. (When a bar is broken into pieces, it is permitted to rotate some of the pieces, stack some of the pieces, and break any set of pieces along a vertical plane simultaneously.)
[b]p14.[/b] A particle starts moving on the number line at a time $t = 0$. Its position on the number line, as a function of time, is $$x = (t-2012)^2 -2012(t-2012)-2013.$$ Find the number of positive integer values of $t$ at which time the particle lies in the negative half of the number line (strictly to the left of $0$).
[b]p15.[/b] Let $A$ be a vertex of a unit cube and let $B$,$C$, and $D$ be the vertices adjacent to A. The tetrahedron $ABCD$ is cut off the cube. Determine the surface area of the remaining solid.
[b]p16.[/b] In equilateral triangle $ABC$, points $P$ and $R$ lie on segment $AB$, points $I$ and $M$ lie on segment $BC$, and points $E$ and $S$ lie on segment $CA$ such that $PRIMES$ is a equiangular hexagon. Given that $AB = 11$, $PS = 2$, $RI = 3$, and $ME = 5$, compute the area of hexagon $PRIMES$.
[b]p17.[/b] Find the smallest odd positive integer with an odd number of positive integer factors, an odd number of distinct prime factors, and an odd number of perfect square factors.
[b]p18.[/b] Fresh Mann thinks that the expressions $2\sqrt{x^2 -4} $and $2(\sqrt{x^2} -\sqrt4)$ are equivalent to each other, but the two expressions are not equal to each other for most real numbers $x$. Find all real numbers $x$ such that $2\sqrt{x^2 -4} = 2(\sqrt{x^2} -\sqrt4)$.
[b]p19.[/b] Let $m$ be the positive integer such that a $3 \times 3$ chessboard can be tiled by at most $m$ pairwise incongruent rectangles with integer side lengths. If rotations and reflections of tilings are considered distinct, suppose that there are $n$ ways to tile the chessboard with $m$ pairwise incongruent rectangles with integer side lengths. Find the product $mn$.
[b]p20.[/b] Let $ABC$ be a triangle with $AB = 4$, $BC = 5$, and $CA = 6$. A triangle $XY Z$ is said to be friendly if it intersects triangle $ABC$ and it is a translation of triangle $ABC$. Let $S$ be the set of points in the plane that are inside some friendly triangle. Compute the ratio of the area of $S$ to the area of triangle $ABC$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Guts Rounds, 2023
[u]Round 1[/u]
[b]p1. [/b] What is the sum of the digits in the binary representation of $2023$?
[b]p2.[/b] Jack is buying fruits at the EMCCmart. Three apples and two bananas cost $\$11.00$. Five apples and four bananas cost $\$19.00$. In cents, how much more does an apple cost than a banana?
[b]p3.[/b] Define $a \sim b$ as $a! - ab$. What is $(4 \sim 5) \sim (5 \sim (3 \sim 1))$?
[u] Round 2[/u]
[b]p4.[/b] Alan has $24$ socks in his drawer. Of these socks, $4$ are red, $8$ are blue, and $12$ are green. Alan takes out socks one at a time from his drawer at random. What is the minimum number of socks he must pull out to guarantee that the number of green socks is at least twice the number of red socks?
[b]p5.[/b] What is the remainder when the square of the $24$th smallest prime number is divided by $24$?
[b]p6.[/b] A cube and a sphere have the same volume. If $k$ is the ratio of the length of the longest diagonal of the cube to the diameter of the sphere, find $k^6$.
[u]Round 3[/u]
[b]p7.[/b] Equilateral triangle $ABC$ has side length $3\sqrt3$. Point $D$ is drawn such that $BD$ is tangent to the circumcircle of triangle $ABC$ and $BD = 4$. Find the distance from the circumcenter of triangle $ABC$ to $D$.
[b]p8.[/b] If $\frac{2023!}{2^k}$ is an odd integer for an integer $k$, what is the value of $k$?
[b]p9.[/b] Let $S$ be a set of 6 distinct positive integers. If the sum of the three smallest elements of $S$ is $8$, and the sum of the three largest elements of $S$ is $19$, find the product of the elements in $S$.
[u]Round 4[/u]
[b]p10.[/b] For some integers $b$, the number $1 + 2b + 3b^2 + 4b^3 + 5b^4$ is divisible by $b + 1$. Find the largest possible value of $b$.
[b]p11.[/b] Let $a, b, c$ be the roots of cubic equation $x^3 + 7x^2 + 8x + 1$. Find $a^2 + b^2 + c^2 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$
[b]p12.[/b] Let $C$ be the set of real numbers $c$ such that there are exactly two integers n satisfying $2c < n < 3c$. Find the expected value of a number chosen uniformly at random from $C$.
PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3131590p28370327]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Guts Rounds, 2021
[u]Round 1[/u]
[b]p1.[/b] What is the remainder when $2021$ is divided by $102$?
[b]p2.[/b] Brian has $2$ left shoes and $2$ right shoes. Given that he randomly picks $2$ of the $4$ shoes, the probability he will get a left shoe and a right shoe is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p + q$.
[b]p3.[/b] In how many ways can $59$ be written as a sum of two perfect squares? (The order of the two perfect squares does not matter.)
[u]Round 2 [/u]
[b]p4.[/b] Two positive integers have a sum of $60$. Their least common multiple is $273$. What is the positive diffeerence between the two numbers?
[b]p5.[/b] How many ways are there to distribute $13$ identical apples among $4$ identical boxes so that no two boxes receive the same number of apples? A box may receive zero apples.
[b]p6.[/b] In square $ABCD$ with side length $5$, $P$ lies on segment $AB$ so that $AP = 3$ and $Q$ lies on segment $AD$ so that $AQ = 4$. Given that the area of triangle $CPQ$ is $x$, compute $2x$.
[u]Round 3 [/u]
[b]p7.[/b] Find the number of ordered triples $(a, b, c)$ of nonnegative integers such that $2a+3b+5c = 15$.
[b]p8.[/b] What is the greatest integer $n \le 15$ such that $n + 1$ and $n^2 + 3$ are both prime?
[b]p9.[/b] For positive integers $a, b$, and $c$, suppose that $gcd \,\,(a, b) = 21$, $gcd \,\,(a, c) = 10$, and $gcd \,\,(b,c) = 11$. Find $\frac{abc}{lcm \,\,(a,b,c)}$ . (Note: $gcd$ is the greatest common divisor function and $lcm$ is the least common multiple function.)
[u]Round 4[/u]
[b]p10.[/b] The vertices of a square in the coordinate plane are at $(0, 0)$, $(0, 6)$, $(6, 0)$, and $(6, 6)$. Line $\ell$ intersects the square at exactly two lattice points (that is, points with integer coordinates). How many such lines $\ell$ are there that divide the square into two regions, one of them having an area of $12$?
[b]p11.[/b] Let $f(n)$ be defined as follows for positive integers $n$: $f(1) = 0$, $f(n) = 1$ if $n$ is prime, and $f(n) = f(n - 1) + 1$ otherwise. What is the maximum value of $f(n)$ for $n \le 120$?
[b]p12.[/b] The graph of the equation $y = x^3 + ax^2 + bx + c$ passes through the points $(2,4)$, $(3, 9)$, and $(4, 16)$. What is $b$?
PS. You should use hide for answers. Rounds 5- 8 have been posted [url=https://artofproblemsolving.com/community/c3h2949415p26408227]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Accuracy Rounds, 2011
[b]p1.[/b] What is the maximum number of points of intersection between a square and a triangle, assuming that no side of the triangle is parallel to any side of the square?
[b]p2.[/b] Two angles of an isosceles triangle measure $80^o$ and $x^o$. What is the sum of all the possible values of $x$?
[b]p3.[/b] Let $p$ and $q$ be prime numbers such that $p + q$ and p + $7q$ are both perfect squares. Find the value of $pq$.
[b]p4.[/b] Anna, Betty, Carly, and Danielle are four pit bulls, each of which is either wearing or not wearing lipstick. The following three facts are true:
(1) Anna is wearing lipstick if Betty is wearing lipstick.
(2) Betty is wearing lipstick only if Carly is also wearing lipstick.
(3) Carly is wearing lipstick if and only if Danielle is wearing lipstick
The following five statements are each assigned a certain number of points:
(a) Danielle is wearing lipstick if and only if Carly is wearing lipstick. (This statement is assigned $1$ point.)
(b) If Anna is wearing lipstick, then Betty is wearing lipstick. (This statement is assigned $6$ points.)
(c) If Betty is wearing lipstick, then both Anna and Danielle must be wearing lipstick. (This statement is assigned $10$ points.)
(d) If Danielle is wearing lipstick, then Anna is wearing lipstick. (This statement is assigned $12$ points.)
(e) If Betty is wearing lipstick, then Danielle is wearing lipstick. (This statement is assigned $14$ points.)
What is the sum of the points assigned to the statements that must be true? (For example, if only statements (a) and (d) are true, then the answer would be $1 + 12 = 13$.)
[b]p5.[/b] Let $f(x)$ and $g(x)$ be functions such that $f(x) = 4x + 3$ and $g(x) = \frac{x + 1}{4}$. Evaluate $g(f(g(f(42))))$.
[b]p6.[/b] Let $A,B,C$, and $D$ be consecutive vertices of a regular polygon. If $\angle ACD = 120^o$, how many sides does the polygon have?
[b]p7.[/b] Fred and George have a fair $8$-sided die with the numbers $0, 1, 2, 9, 2, 0, 1, 1$ written on the sides. If Fred and George each roll the die once, what is the probability that Fred rolls a larger number than George?
[b]p8.[/b] Find the smallest positive integer $t$ such that $(23t)^3 - (20t)^3 - (3t)^3$ is a perfect square.
[b]p9.[/b] In triangle $ABC$, $AC = 8$ and $AC < AB$. Point $D$ lies on side BC with $\angle BAD = \angle CAD$. Let $M$ be the midpoint of $BC$. The line passing through $M$ parallel to $AD$ intersects lines $AB$ and $AC$ at $F$ and $E$, respectively. If $EF =\sqrt2$ and $AF = 1$, what is the length of segment $BC$? (See the following diagram.)
[img]https://cdn.artofproblemsolving.com/attachments/2/3/4b5dd0ae28b09f5289fb0e6c72c7cbf421d025.png[/img]
[b]p10.[/b] There are $2011$ evenly spaced points marked on a circular table. Three segments are randomly drawn between pairs of these points such that no two segments share an endpoint on the circle. What is the probability that each of these segments intersects the other two?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Accuracy Rounds, 2023
[b]p1.[/b] Minseo writes all of the divisors of $1,000,000$ on the whiteboard. She then erases all of the numbers which have the digit $0$ in their decimal representation. How many numbers are left?
[b]p2.[/b] $n < 100$ is an odd integer and can be expressed as $3k - 2$ and $5m + 1$ for positive integers $k$ and $m$. Find the sum of all possible values of $n$.
[b]p3.[/b] Mr. Pascal is a math teacher who has the license plate $SQUARE$. However, at night, a naughty student scrambles Mr. Pascal’s license plate to $UQRSEA$. The math teacher luckily has an unscrambler that is able to move license plate letters. The unscrambler swaps the positions of any two adjacent letters. What is the minimum number of times Mr. Pascal must use the unscrambler to restore his original license plate?
[b]p4.[/b] Find the number of distinct real numbers $x$ which satisfy $x^2 + 4 \lfloor x \rfloor + 4 = 0$.
[b]p5.[/b] All four faces of tetrahedron $ABCD$ are acute. The distances from point $D$ to $\overline{BC}$, $\overline{CA}$ and $\overline{AB}$ are all $7$, and the distance from point $D$ to face $ABC$ is $5$. Given that the volume of tetrahedron $ABCD$ is $60$, find the surface area of tetrahedron $ABCD$.
[b]p6.[/b] Forrest has a rectangular piece of paper with a width of $5$ inches and a height of $3$ inches. He wants to cut the paper into five rectangular pieces, each of which has a width of $1$ inch and a distinct integer height between $1$ and $5$ inches, inclusive. How many ways can he do so? (One possible way is shown below.)
[img]https://cdn.artofproblemsolving.com/attachments/7/3/205afe28276f9df1c6bcb45fff6313c6c7250f.png[/img]
[b]p7.[/b] In convex quadrilateral $ABCD$, $AB = CD = 5$, $BC = 4$ and $AD = 8$. If diagonal $\overline{AC}$ bisects $\angle DAB$, find the area of quadrilateral $ABCD$.
[b]p8.[/b] Let $x$ and $y$ be real numbers such that $$x + y = x^3 + y^3 + \frac34 = \frac{1}{8xy}.$$ Find the value of $x + y$.
[b]p9.[/b] Four blue squares and four red parallelograms are joined edge-to-edge alternately to form a ring of quadrilateral as shown. The areas of three of the red parallelograms are shown. Find the area of the fourth red parallelogram.
[img]https://cdn.artofproblemsolving.com/attachments/9/c/911a8d53604f639e2f9bd72b59c7f50e43e258.png[/img]
[b]p10.[/b] Define $f(x, n) =\sum_{d|n}\frac{x^n-1}{x^d-1}$ . For how many integers $n$ between $1$ and $2023$ inclusive is $f(3, n)$ an odd integer?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Team Rounds, 2016
[b]p1.[/b] Lisa is playing the piano at a tempo of $80$ beats per minute. If four beats make one measure of her rhythm, how many seconds are in one measure?
[b]p2.[/b] Compute the smallest integer $n > 1$ whose base-$2$ and base-$3$ representations both do not contain the digit $0$.
[b]p3.[/b] In a room of $24$ people, $5/6$ of the people are old, and $5/8$ of the people are male. At least how many people are both old and male?
[b]p4.[/b] Juan chooses a random even integer from $1$ to $15$ inclusive, and Gina chooses a random odd integer from $1$ to $15$ inclusive. What is the probability that Juan’s number is larger than Gina’s number? (They choose all possible integers with equal probability.)
[b]p5.[/b] Set $S$ consists of all positive integers less than or equal to $ 2016$. Let $A$ be the subset of $S$ consisting of all multiples of $6$. Let $B$ be the subset of $S$ consisting of all multiples of $7$. Compute the ratio of the number of positive integers in $A$ but not $B$ to the number of integers in $B$ but not $A$.
[b]p6.[/b] Three peas form a unit equilateral triangle on a flat table. Sebastian moves one of the peas a distance $d$ along the table to form a right triangle. Determine the minimum possible value of $d$.
[b]p7.[/b] Oumar is four times as old as Marta. In $m$ years, Oumar will be three times as old as Marta will be. In another $n$ years after that, Oumar will be twice as old as Marta will be. Compute the ratio $m/n$.
[b]p8.[/b] Compute the area of the smallest square in which one can inscribe two non-overlapping equilateral triangles with side length $ 1$.
[b]p9.[/b] Teemu, Marcus, and Sander are signing documents. If they all work together, they would finish in $6$ hours. If only Teemu and Sander work together, the work would be finished in 8 hours. If only Marcus and Sander work together, the work would be finished in $10$ hours. How many hours would Sander take to finish signing if he worked alone?
[b]p10.[/b]Triangle $ABC$ has a right angle at $B$. A circle centered at $B$ with radius $BA$ intersects side $AC$ at a point $D$ different from $A$. Given that $AD = 20$ and $DC = 16$, find the length of $BA$.
[b]p11.[/b] A regular hexagon $H$ with side length $20$ is divided completely into equilateral triangles with side length $ 1$. How many regular hexagons with sides parallel to the sides of $H$ are formed by lines in the grid?
[b]p12[/b]. In convex pentagon $PEARL$, quadrilateral $PERL$ is a trapezoid with side $PL$ parallel to side $ER$. The areas of triangle $ERA$, triangle $LAP$, and trapezoid $PERL$ are all equal. Compute the ratio $\frac{PL}{ER}$.
[b]p13.[/b] Let $m$ and $n$ be positive integers with $m < n$. The first two digits after the decimal point in the decimal representation of the fraction $m/n$ are $74$. What is the smallest possible value of $n$?
[b]p14.[/b] Define functions $f(x, y) = \frac{x + y}{2} - \sqrt{xy}$ and $g(x, y) = \frac{x + y}{2} + \sqrt{xy}$. Compute $g (g (f (1, 3), f (5, 7)), g (f (3, 5), f (7, 9)))$.
[b]p15.[/b] Natalia plants two gardens in a $5 \times 5$ grid of points. Each garden is the interior of a rectangle with vertices on grid points and sides parallel to the sides of the grid. How many unordered pairs of two non-overlapping rectangles can Nataliia choose as gardens? (The two rectangles may share an edge or part of an edge but should not share an interior point.)
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Speed Rounds, 2019
[i]20 problems for 25 minutes.[/i]
[b]p1.[/b] Given that $a + 19b = 3$ and $a + 1019b = 5$, what is $a + 2019b$?
[b]p2.[/b] How many multiples of $3$ are there between $2019$ and $2119$, inclusive?
[b]p3.[/b] What is the maximum number of quadrilaterals a $12$-sided regular polygon can be quadrangulated into? Here quadrangulate means to cut the polygon along lines from vertex to vertex, none of which intersect inside the polygon, to form pieces which all have exactly $4$ sides.
[b]p4.[/b] What is the value of $|2\pi - 7| + |2\pi - 6|$, rounded to the nearest hundredth?
[b]p5.[/b] In the town of EMCCxeter, there is a $30\%$ chance that it will snow on Saturday, and independently, a $40\%$ chance that it will snow on Sunday. What is the probability that it snows exactly once that weekend, as a percentage?
[b]p6.[/b] Define $n!$ to be the product of all integers between $1$ and $n$ inclusive. Compute $\frac{2019!}{2017!} \times \frac{2016!}{2018!}$ .
[b]p7.[/b] There are $2019$ people standing in a row, and they are given positions $1$, $2$, $3$, $...$, $2019$ from left to right. Next, everyone in an odd position simultaneously leaves the row, and the remaining people are assigned new positions from $1$ to $1009$, again from left to right. This process is then repeated until one person remains. What was this person's original position?
[b]p8.[/b] The product $1234\times 4321$ contains exactly one digit not in the set $\{1, 2, 3, 4\}$. What is this digit?
[b]p9.[/b] A quadrilateral with positive area has four integer side lengths, with shortest side $1$ and longest side $9$. How many possible perimeters can this quadrilateral have?
[b]p10.[/b] Define $s(n)$ to be the sum of the digits of $n$ when expressed in base $10$, and let $\gamma (n)$ be the sum of $s(d)$ over all natural number divisors $d$ of $n$. For instance, $n = 11$ has two divisors, $1$ and $11$, so $\gamma (11) = s(1) + s(11) = 1 + (1 + 1) = 3$. Find the value of $\gamma (2019)$.
[b]p11.[/b] How many five-digit positive integers are divisible by $9$ and have $3$ as the tens digit?
[b]p12.[/b] Adam owns a large rectangular block of cheese, that has a square base of side length $15$ inches, and a height of $4$ inches. He wants to remove a cylindrical cheese chunk of height $4$, by making a circular hole that goes through the top and bottom faces, but he wants the surface area of the leftover cheese block to be the same as before. What should the diameter of his hole be, in inches?
[i]Αddendum on 1/26/19: the hole must have non-zero diameter.
[/i]
[b]p13.[/b] Find the smallest prime that does not divide $20! + 19! + 2019!$.
[b]p14.[/b] Convex pentagon $ABCDE$ has angles $\angle ABC = \angle BCD = \angle DEA = \angle EAB$ and angle $\angle CDE = 60^o$. Given that $BC = 3$, $CD = 4$, and $DE = 5$, find $EA$.
[i]Addendum on 1/26/19: ABCDE is specified to be convex.
[/i]
[b]p15.[/b] Sophia has $3$ pairs of red socks, $4$ pairs of blue socks, and $5$ pairs of green socks. She picks out two individual socks at random: what is the probability she gets a pair with matching color?
[b]p16.[/b] How many real roots does the function $f(x) = 2019^x - 2019x - 2019$ have?
[b]p17.[/b] A $30-60-90$ triangle is placed on a coordinate plane with its short leg of length $6$ along the $x$-axis, and its long leg along the $y$-axis. It is then rotated $90$ degrees counterclockwise, so that the short leg now lies along the $y$-axis and long leg along the $x$-axis. What is the total area swept out by the triangle during this rotation?
[b]p18.[/b] Find the number of ways to color the unit cells of a $2\times 4$ grid in four colors such that all four colors are used and every cell shares an edge with another cell of the same color.
[b]p19.[/b] Triangle $\vartriangle ABC$ has centroid $G$, and $X, Y,Z$ are the centroids of triangles $\vartriangle BCG$, $\vartriangle ACG$, and $\vartriangle ABG$, respectively. Furthermore, for some points $D,E, F$, vertices $A,B,C$ are themselves the centroids of triangles $\vartriangle DBC$, $\vartriangle ECA$, and $\vartriangle FAB$, respectively. If the area of $\vartriangle XY Z = 7$, what is the area of $\vartriangle DEF$?
[b]p20.[/b] Fhomas orders three $2$-gallon jugs of milk from EMCCBay for his breakfast omelette. However, every jug is actually shipped with a random amount of milk (not necessarily an integer), uniformly distributed between $0$ and $2$ gallons. If Fhomas needs $2$ gallons of milk for his breakfast omelette, what is the probability he will receive enough milk?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Speed Rounds, 2012
[i]20 problems for 20 minutes.[/i]
[b]p1.[/b] Evaluate $=\frac{1}{2 \cdot 3 \cdot 4}+\frac{1}{3 \cdot 4 \cdot 5}$.
[b]p2.[/b] A regular hexagon and a regular $n$-sided polygon have the same perimeter. If the ratio of the side length of the hexagon to the side length of the $n$-sided polygon is $2 : 1$, what is $n$?
[b]p3.[/b] How many nonzero digits are there in the decimal representation of $2 \cdot 10\cdot 500 \cdot 2500$?
[b]p4.[/b] When the numerator of a certain fraction is increased by $2012$, the value of the fraction increases by $2$. What is the denominator of the fraction?
[b]p5.[/b] Sam did the computation $1 - 10 \cdot a + 22$, where $a$ is some real number, except he messed up his order of operations and computed the multiplication last; that is, he found the value of $(1 - 10) \cdot (a + 22)$ instead. Luckily, he still ended up with the right answer. What is $a$?
[b]p6.[/b] Let $n! = n \cdot(n-1) \cdot\cdot\cdot 2 \cdot 1$. For how many integers $n$ between $1$ and $100$ inclusive is $n!$ divisible by $36$?
[b]p7.[/b] Simplify the expression $\sqrt{\frac{3 \cdot 27^3}{27 \cdot 3^3}}$
[b]p8.[/b] Four points $A,B,C,D$ lie on a line in that order such that $\frac{AB}{CB}=\frac{AD}{CD}$ . Let $M$ be the midpoint of segment $AC$. If $AB = 6$, $BC = 2$, compute $MB \cdot MD$.
[b]p9.[/b] Allan has a deck with $8$ cards, numbered $1$, $1$, $2$, $2$, $3$, $3$, $4$, $4$. He pulls out cards without replacement, until he pulls out an even numbered card, and then he stops. What is the probability that he pulls out exactly $2$ cards?
[b]p10.[/b] Starting from the sequence $(3, 4, 5, 6, 7, 8, ... )$, one applies the following operation repeatedly. In each operation, we change the sequence $$(a_1, a_2, a_3, ... , a_{a_1-1}, a_{a_1} , a_{a_1+1},...)$$ to the sequence $$(a_2, a_3, ... , a_{a_1} , a_1, a_{a_1+1}, ...) .$$ (In other words, for a sequence starting with$ x$, we shift each of the next $x-1$ term to the left by one, and put x immediately to the right of these numbers, and keep the rest of the terms unchanged. For example, after one operation, the sequence is $(4, 5, 3, 6, 7, 8, ... )$, and after two operations, the sequence becomes $(5, 3, 6, 4, 7, 8,... )$. How many operations will it take to obtain a sequence of the form $(7, ... )$ (that is, a sequence starting with $7$)?
[b]p11.[/b] How many ways are there to place $4$ balls into a $4\times 6$ grid such that no column or row has more than one ball in it? (Rotations and reflections are considered distinct.)
[b]p12.[/b] Point $P$ lies inside triangle $ABC$ such that $\angle PBC = 30^o$ and $\angle PAC = 20^o$. If $\angle APB$ is a right angle, find the measure of $\angle BCA$ in degrees.
[b]p13.[/b] What is the largest prime factor of $9^3 - 4^3$?
[b]p14.[/b] Joey writes down the numbers $1$ through $10$ and crosses one number out. He then adds the remaining numbers. What is the probability that the sum is less than or equal to $47$?
[b]p15.[/b] In the coordinate plane, a lattice point is a point whose coordinates are integers. There is a pile of grass at every lattice point in the coordinate plane. A certain cow can only eat piles of grass that are at most $3$ units away from the origin. How many piles of grass can she eat?
[b]p16.[/b] A book has 1000 pages numbered $1$, $2$, $...$ , $1000$. The pages are numbered so that pages $1$ and $2$ are back to back on a single sheet, pages $3$ and $4$ are back to back on the next sheet, and so on, with pages $999$ and $1000$ being back to back on the last sheet. How many pairs of pages that are back to back (on a single sheet) share no digits in the same position? (For example, pages $9$ and $10$, and pages $89$ and $90$.)
[b]p17.[/b] Find a pair of integers $(a, b)$ for which $\frac{10^a}{a!}=\frac{10^b}{b!}$ and $a < b$.
[b]p18.[/b] Find all ordered pairs $(x, y)$ of real numbers satisfying
$$\begin{cases}
-x^2 + 3y^2 - 5x + 7y + 4 = 0 \\
2x^2 - 2y^2 - x + y + 21 = 0 \end{cases}$$
[b]p19.[/b] There are six blank fish drawn in a line on a piece of paper. Lucy wants to color them either red or blue, but will not color two adjacent fish red. In how many ways can Lucy color the fish?
[b]p20.[/b] There are sixteen $100$-gram balls and sixteen $99$-gram balls on a table (the balls are visibly indistinguishable). You are given a balance scale with two sides that reports which side is heavier or that the two sides have equal weights. A weighing is defined as reading the result of the balance scale: For example, if you place three balls on each side, look at the result, then add two more balls to each side, and look at the result again, then two weighings have been performed. You wish to pick out two different sets of balls (from the $32$ balls) with equal numbers of balls in them but different total weights. What is the minimal number of weighings needed to ensure this?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Team Rounds, 2015
[b]p1.[/b] Nicky is studying biology and has a tank of $17$ lizards. In one day, he can either remove $5$ lizards or add $2$ lizards to his tank. What is the minimum number of days necessary for Nicky to get rid of all of the lizards from his tank?
[b]p2.[/b] What is the maximum number of spheres with radius $1$ that can fit into a sphere with radius $2$?
[b]p3.[/b] A positive integer $x$ is sunny if $3x$ has more digits than $x$. If all sunny numbers are written in increasing order, what is the $50$th number written?
[b]p4.[/b] Quadrilateral $ABCD$ satisfies $AB = 4$, $BC = 5$, $DA = 4$, $\angle DAB = 60^o$, and $\angle ABC = 150^o$. Find the area of $ABCD$.
[b]p5. [/b]Totoro wants to cut a $3$ meter long bar of mixed metals into two parts with equal monetary value. The left meter is bronze, worth $10$ zoty per meter, the middle meter is silver, worth $25$ zoty per meter, and the right meter is gold, worth $40$ zoty per meter. How far, in meters, from the left should Totoro make the cut?
[b]p6.[/b] If the numbers $x_1, x_2, x_3, x_4$, and $x5$ are a permutation of the numbers $1, 2, 3, 4$, and $5$, compute the maximum possible value of $$|x_1 - x_2| + |x_2 - x_3| + |x_3 - x_4| + |x_4 - x_5|.$$
[b]p7.[/b] In a $3 \times 4$ grid of $12$ squares, find the number of paths from the top left corner to the bottom right corner that satisfy the following two properties:
$\bullet$ The path passes through each square exactly once.
$\bullet$ Consecutive squares share a side.
Two paths are considered distinct if and only if the order in which the twelve squares are visited is different. For instance, in the diagram below, the two paths drawn are considered the same.
[img]https://cdn.artofproblemsolving.com/attachments/7/a/bb3471bbde1a8f58a61d9dd69c8527cfece05a.png[/img]
[b]p8.[/b] Scott, Demi, and Alex are writing a computer program that is $25$ ines long. Since they are working together on one computer, only one person may type at a time. To encourage collaboration, no person can type two lines in a row, and everyone must type something. If Scott takes $10$ seconds to type one line, Demi takes $15$ seconds, and Alex takes $20$ seconds, at least how long, in seconds, will it take them to finish the program?
[b]p9.[/b] A hand of four cards of the form $(c, c, c + 1, c + 1)$ is called a tractor. Vinjai has a deck consisting of four of each of the numbers $7$, $8$, $9$ and $10$. If Vinjai shuffles and draws four cards from his deck, compute the probability that they form a tractor.
[b]p10. [/b]The parabola $y = 2x^2$ is the wall of a fortress. Totoro is located at $(0, 4)$ and fires a cannonball in a straight line at the closest point on the wall. Compute the y-coordinate of the point on the wall that the cannonball hits.
[b]p11. [/b]How many ways are there to color the squares of a $10$ by $10$ grid with black and white such that in each row and each column there are exactly two black squares and between the two black squares in a given row or column there are exactly [b]4[/b] white squares? Two configurations that are the same under rotations or reflections are considered different.
[b]p12.[/b] In rectangle $ABCD$, points $E$ and $F$ are on sides $AB$ and $CD$, respectively, such that $AE = CF > AD$ and $\angle CED = 90^o$. Lines $AF, BF, CE$ and $DE$ enclose a rectangle whose area is $24\%$ of the area of $ABCD$. Compute $\frac{BF}{CE}$ .
[b]p13.[/b] Link cuts trees in order to complete a quest. He must cut $3$ Fenwick trees, $3$ Splay trees and $3$ KD trees. If he must also cut 3 trees of the same type in a row at some point during his quest, in how many ways can he cut the trees and complete the quest? (Trees of the same type are indistinguishable.)
[b]p14.[/b] Find all ordered pairs (a, b) of positive integers such that $\sqrt{64a + b^2} + 8 = 8\sqrt{a} + b$.
[b]p15.[/b] Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle BCD = 108^o$, $\angle CDE = 168^o$ and $AB =BC = CD = DE$. Find the measure of $\angle AEB$
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Speed Rounds, 2018
[i]20 problems for 25 minutes.[/i]
[b]p1.[/b] What is $2018 - 3018 + 4018$?
[b]p2.[/b] What is the smallest integer greater than $100$ that is a multiple of both $6$ and $8$?
[b]p3.[/b] What positive real number can be expressed as both $\frac{b}{a}$ and $a:b$ in base $10$ for nonzero digits $a$ and $b$? Express your answer as a decimal.
[b]p4.[/b] A non-degenerate triangle has sides of lengths $1$, $2$, and $\sqrt{n}$, where $n$ is a positive integer. How many possible values of $n$ are there?
[b]p5.[/b] When three integers are added in pairs, and the results are $20$, $18$, and $x$. If all three integers sum to $31$, what is $x$?
[b]p6.[/b] A cube's volume in cubic inches is numerically equal to the sum of the lengths of all its edges, in inches. Find the surface area of the cube, in square inches.
[b]p7.[/b] A $12$ hour digital clock currently displays$ 9 : 30$. Ignoring the colon, how many times in the next hour will the clock display a palindrome (a number that reads the same forwards and backwards)?
[b]p8.[/b] SeaBay, an online grocery store, offers two different types of egg cartons. Small egg cartons contain $12$ eggs and cost $3$ dollars, and large egg cartons contain $18$ eggs and cost $4$ dollars. What is the maximum number of eggs that Farmer James can buy with $10$ dollars?
[b]p9.[/b] What is the sum of the $3$ leftmost digits of $\underbrace{999...9}_{2018\,\,\ 9' \,\,s}\times 12$?
[b]p10.[/b] Farmer James trisects the edges of a regular tetrahedron. Then, for each of the four vertices, he slices through the plane containing the three trisection points nearest to the vertex. Thus, Farmer James cuts off four smaller tetrahedra, which he throws away. How many edges does the remaining shape have?
[b]p11.[/b] Farmer James is ordering takeout from Kristy's Krispy Chicken. The base cost for the dinner is $\$14.40$, the sales tax is $6.25\%$, and delivery costs $\$3.00$ (applied after tax). How much did Farmer James pay, in dollars?
[b]p12.[/b] Quadrilateral $ABCD$ has $ \angle ABC = \angle BCD = \angle BDA = 90^o$. Given that $BC = 12$ and $CD = 9$, what is the area of $ABCD$?
[b]p13.[/b] Farmer James has $6$ cards with the numbers $1-6$ written on them. He discards a card and makes a $5$ digit number from the rest. In how many ways can he do this so that the resulting number is divisible by $6$?
[b]p14.[/b] Farmer James has a $5 \times 5$ grid of points. What is the smallest number of triangles that he may draw such that each of these $25$ points lies on the boundary of at least one triangle?
[b]p15.[/b] How many ways are there to label these $15$ squares from $1$ to $15$ such that squares $1$ and $2$ are adjacent, squares $2$ and $3$ are adjacent, and so on?
[img]https://cdn.artofproblemsolving.com/attachments/e/a/06dee288223a16fbc915f8b95c9e4f2e4e1c1f.png[/img]
[b]p16.[/b] On Farmer James's farm, there are three henhouses located at $(4, 8)$, $(-8,-4)$, $(8,-8)$. Farmer James wants to place a feeding station within the triangle formed by these three henhouses. However, if the feeding station is too close to any one henhouse, the hens in the other henhouses will complain, so Farmer James decides the feeding station cannot be within 6 units of any of the henhouses. What is the area of the region where he could possibly place the feeding station?
[b]p17.[/b] At Eggs-Eater Academy, every student attends at least one of $3$ clubs. $8$ students attend frying club, $12$ students attend scrambling club, and $20$ students attend poaching club. Additionally, $10$ students attend at least two clubs, and $3$ students attend all three clubs. How many students are there in total at Eggs-Eater Academy?
[b]p18.[/b] Let $x, y, z$ be real numbers such that $8^x = 9$, $27^y = 25$, and $125^z = 128$. What is the value of $xyz$?
[b]p19.[/b] Let $p$ be a prime number and $x, y$ be positive integers. Given that $9xy = p(p + 3x + 6y)$, find the maximum possible value of $p^2 + x^2 + y^2$.
[b]p20.[/b] Farmer James's hens like to drop eggs. Hen Hao drops $6$ eggs uniformly at random in a unit square. Farmer James then draws the smallest possible rectangle (by area), with sides parallel to the sides of the square, that contain all $6$ eggs. What is the probability that at least one of the $6$ eggs is a vertex of this rectangle?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Guts Rounds, 2024
[u]Round 1[/u]
[b]p1.[/b] When Shiqiao sells a bale of kale, he makes $x$ dollars, where $$x =\frac{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8}{3 + 4 + 5 + 6}.$$ Find $x$.
[b]p2.[/b] The fraction of Shiqiao’s kale that has gone rotten is equal to $$\sqrt{ \frac{100^2}{99^2} -\frac{100}{99}}.$$
Find the fraction of Shiqiao’s kale that has gone rotten.
[b]p3.[/b] Shiqiao is growing kale. Each day the number of kale plants doubles, but $4$ of his kale plants die afterwards. He starts with $6$ kale plants. Find the number of kale plants Shiqiao has after five days.
[u]Round 2[/u]
[b]p4.[/b] Today the high is $68$ degrees Fahrenheit. If $C$ is the temperature in Celsius, the temperature in Fahrenheit is equal to $1.8C + 32$. Find the high today in Celsius.
[b]p5.[/b] The internal angles in Evan’s triangle are all at most $68$ degrees. Find the minimum number of degrees an angle of Evan’s triangle could measure.
[b]p6.[/b] Evan’s room is at $68$ degrees Fahrenheit. His thermostat has two buttons, one to increase the temperature by one degree, and one to decrease the temperature by one degree. Find the number of combinations of $10$ button presses Evan can make so that the temperature of his room never drops below $67$ degrees or rises above $69$ degrees.
[u]Round 3[/u]
[b]p7.[/b] In a digital version of the SAT, there are four spaces provided for either a digit $(0-9)$, a fraction sign $(\/)$, or a decimal point $(.)$. The answer must be in simplest form and at most one space can be a non-digit character. Determine the largest fraction which, when expressed in its simplest form, fits within this space, but whose exact decimal representation does not.
[b]p8.[/b] Rounding Rox picks a real number $x$. When she rounds x to the nearest hundred, its value increases by $2.71828$. If she had instead rounded $x$ to the nearest hundredth, its value would have decreased by $y$. Find $y$.
[b]p9.[/b] Let $a$ and $b$ be real numbers satisfying the system of equations $$\begin{cases}
a + \lfloor b \rfloor = 2.14 \\
\lfloor a \rfloor + b = 2.72 \end{cases}$$ Determine $a + b$.
[u]Round 4[/u]
[b]p10.[/b] Carol and Lily are playing a game with two unfair coins, both of which have a $1/4$ chance of landing on heads. They flip both coins. If they both land on heads, Lily loses the game, and if they both land on tails, Carol loses the game. If they land on different sides, Carol and Lily flip the coins again. They repeat this until someone loses the game. Find the probability that Lily loses the game.
[b]p11.[/b] Dongchen is carving a circular coin design. He carves a regular pentagon of side length $1$ such that all five vertices of the pentagon are on the rim of the coin. He then carves a circle inside the pentagon so that the circle is tangent to all five sides of the pentagon. Find the area of the region between the smaller circle and the rim of the coin.
[b]p12.[/b] Anthony flips a fair coin six times. Find the probability that at some point he flips $2$ heads in a row.
PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3248731p29808147]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Speed Rounds, 2021
[i]20 problems for 25 minutes.[/i]
[b]p1.[/b] Evaluate $20 \times 21 + 2021$.
[b]p2.[/b] Let points $A$, $B$, $C$, and $D$ lie on a line in that order. Given that $AB = 5CD$ and $BD = 2BC$, compute $\frac{AC}{BD}$.
[b]p3.[/b] There are $18$ students in Vincent the Bug's math class. Given that $11$ of the students take U.S. History, $15$ of the students take English, and $2$ of the students take neither, how many students take both U.S. History and English?
[b]p4.[/b] Among all pairs of positive integers $(x, y)$ such that $xy = 12$, what is the least possible value of $x + y$?
[b]p5.[/b] What is the smallest positive integer $n$ such that $n! + 1$ is composite?
[b]p6.[/b] How many ordered triples of positive integers $(a, b,c)$ are there such that $a + b + c = 6$?
[b]p7.[/b] Thomas orders some pizzas and splits each into $8$ slices. Hungry Yunseo eats one slice and then finds that she is able to distribute all the remaining slices equally among the $29$ other math club students. What is the fewest number of pizzas that Thomas could have ordered?
[b]p8.[/b] Stephanie has two distinct prime numbers $a$ and $b$ such that $a^2-9b^2$ is also a prime. Compute $a + b$.
[b]p9.[/b] Let $ABCD$ be a unit square and $E$ be a point on diagonal $AC$ such that $AE = 1$. Compute $\angle BED$, in degrees.
[b]p10.[/b] Sheldon wants to trace each edge of a cube exactly once with a pen. What is the fewest number of continuous strokes that he needs to make? A continuous stroke is one that goes along the edges and does not leave the surface of the cube.
[b]p11.[/b] In base $b$, $130_b$ is equal to $3n$ in base ten, and $1300_b$ is equal to $n^2$ in base ten. What is the value of $n$, expressed in base ten?
[b]p12.[/b] Lin is writing a book with $n$ pages, numbered $1,2,..., n$. Given that $n > 20$, what is the least value of $n$ such that the average number of digits of the page numbers is an integer?
[b]p13.[/b] Max is playing bingo on a $5\times 5$ board. He needs to fill in four of the twelve rows, columns, and main diagonals of his bingo board to win. What is the minimum number of boxes he needs to fill in to win?
[b]p14.[/b] Given that $x$ and $y$ are distinct real numbers such that $x^2 + y = y^2 + x = 211$, compute the value of $|x - y|$.
[b]p15.[/b] How many ways are there to place 8 indistinguishable pieces on a $4\times 4$ checkerboard such that there are two pieces in each row and two pieces in each column?
[b]p16.[/b] The Manhattan distance between two points $(a, b)$ and $(c, d)$ in the plane is defined to be $|a - c| + |b - d|$. Suppose Neil, Neel, and Nail are at the points $(5, 3)$, $(-2,-2)$ and $(6, 0)$, respectively, and wish to meet at a point $(x, y)$ such that their Manhattan distances to$ (x, y)$ are equal. Find $10x + y$.
[b]p17.[/b] How many positive integers that have a composite number of divisors are there between $1$ and $100$, inclusive?
[b]p18.[/b] Find the number of distinct roots of the polynomial $$(x - 1)(x - 2) ... (x - 90)(x^2 - 1)(x^2 - 2) ... (x^2 - 90)(x^4 - 1)(x^4 - 2)...(x^4 - 90)$$.
[b]p19.[/b] In triangle $ABC$, let $D$ be the foot of the altitude from $ A$ to $BC$. Let $P,Q$ be points on $AB$, $AC$, respectively, such that $PQ$ is parallel to $BC$ and $\angle PDQ = 90^o$. Given that $AD = 25$, $BD = 9$, and $CD = 16$, compute $111 \times PQ$.
[b]p20.[/b] The simplified fraction with numerator less than $1000$ that is closest but not equal to $\frac{47}{18}$ is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Accuracy Rounds, 2018
[b]p1.[/b] On SeaBay, green herring costs $\$2.50$ per pound, blue herring costs $\$4.00$ per pound, and red herring costs $\$5,85$ per pound. What must Farmer James pay for $12$ pounds of green herring and $7$ pounds of blue herring, in dollars?
[b]p2.[/b] A triangle has side lengths $3$, $4$, and $6$. A second triangle, similar to the first one, has one side of length $12$. Find the sum of all possible lengths of the second triangle's longest side.
[b]p3.[/b] Hen Hao runs two laps around a track. Her overall average speed for the two laps was $20\%$ slower than her average speed for just the first lap. What is the ratio of Hen Hao's average speed in the first lap to her average speed in the second lap?
[b]p4.[/b] Square $ABCD$ has side length $2$. Circle $\omega$ is centered at $A$ with radius $2$, and intersects line $AD$ at distinct points $D$ and $E$. Let $X$ be the intersection of segments $EC$ and $AB$, and let $Y$ be the intersection of the minor arc $DB$ with segment $EC$. Compute the length of $XY$ .
[b]p5.[/b] Hen Hao rolls $4$ tetrahedral dice with faces labeled $1$, $2$, $3$, and $4$, and adds up the numbers on the faces facing down. Find the probability that she ends up with a sum that is a perfect square.
[b]p6.[/b] Let $N \ge 11$ be a positive integer. In the Eggs-Eater Lottery, Farmer James needs to choose an (unordered) group of six different integers from $1$ to $N$, inclusive. Later, during the live drawing, another group of six numbers from $1$ to $N$ will be randomly chosen as winning numbers. Farmer James notices that the probability he will choose exactly zero winning numbers is the same as the probability that he will choose exactly one winning number. What must be the value of $N$?
[b]p7.[/b] An egg plant is a hollow cylinder of negligible thickness with radius $2$ and height $h$. Inside the egg plant, there is enough space for four solid spherical eggs of radius $1$. What is the minimum possible value for $h$?
[b]p8.[/b] Let $a_1, a_2, a_3, ...$ be a geometric sequence of positive reals such that $a_1 < 1$ and $(a_{20})^{20} = (a_{18})^{18}$. What is the smallest positive integer n such that the product $a_1a_2a_3...a_n$ is greater than $1$?
[b]p9.[/b] In parallelogram $ABCD$, the angle bisector of $\angle DAB$ meets segment $BC$ at $E$, and $AE$ and $BD$ intersect at $P$. Given that $AB = 9$, $AE = 16$, and $EP = EC$, find $BC$.
[b]p10.[/b] Farmer James places the numbers $1, 2,..., 9$ in a $3\times 3$ grid such that each number appears exactly once in the grid. Let $x_i$ be the product of the numbers in row $i$, and $y_i$ be the product of the numbers in column $i$. Given that the unordered sets $\{x_1, x_2, x_3\}$ and $\{y_1, y_2, y_3\}$ are the same, how many possible arrangements could Farmer James have made?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Guts Rounds, 2022
[u]Round 5[/u]
[b]p13.[/b] Find the number of six-digit positive integers that satisfy all of the following conditions:
(i) Each digit does not exceed $3$.
(ii) The number $1$ cannot appear in two consecutive digits.
(iii) The number $2$ cannot appear in two consecutive digits.
[b]p14.[/b] Find the sum of all distinct prime factors of $103040301$.
[b]p15.[/b] Let $ABCA'B'C'$ be a triangular prism with height $3$ where bases $ABC$ and $A'B'C'$ are equilateral triangles with side length $\sqrt6$. Points $P$ and $Q$ lie inside the prism so that $ABCP$ and $A'B'C'Q$ are regular tetrahedra. The volume of the intersection of these two tetrahedra can be expressed in the form $\frac{\sqrt{m}}{n}$ , where $m$ and $n$ are positive integers and $m$ is not divisible by the square of any prime. Find $m + n$.
[u]Round 6[/u]
[b]p16.[/b] Let $a_0, a_1, ...$ be an infinite sequence such that $a^2_n -a_{n-1}a_{n+1} = a_n -a_{n-1}$ for all positive integers $n$. Given that $a_0 = 1$ and $a_1 = 4$, compute the smallest positive integer $k$ such that $a_k$ is an integer multiple of $220$.
[b]p17.[/b] Vincent the Bug is on an infinitely long number line. Every minute, he jumps either $2$ units to the right with probability $\frac23$ or $3$ units to the right with probability $\frac13$ . The probability that Vincent never lands exactly $15$ units from where he started can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$?
[b]p18.[/b] Battler and Beatrice are playing the “Octopus Game.” There are $2022$ boxes lined up in a row, and inside one of the boxes is an octopus. Beatrice knows the location of the octopus, but Battler does not. Each turn, Battler guesses one of the boxes, and Beatrice reveals whether or not the octopus is contained in that box at that time. Between turns, the octopus teleports to an adjacent box and secretly communicates to Beatrice where it teleported to. Find the least positive integer $B$ such that Battler has a strategy to guarantee that he chooses the box containing the octopus in at most $B$ guesses.
[u]Round 7[/u]
[b]p19.[/b] Given that $f(x) = x^2-2$ the number $f(f(f(f(f(f(f(2.5)))))))$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Find the greatest positive integer $n$ such that $2^n$ divides $ab+a+b-1$.
[b]p20.[/b] In triangle $ABC$, the shortest distance between a point on the $A$-excircle $\omega$ and a point on the $B$-excircle $\Omega$ is $2$. Given that $AB = 5$, the sum of the circumferences of $\omega$ and $\Omega$ can be written in the form $\frac{m}{n}\pi$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? (Note: The $A$-excircle is defined to be the circle outside triangle $ABC$ that is tangent to the rays $\overrightarrow{AB}$ and $\overrightarrow{AC}$ and to the side $ BC$. The $B$-excircle is defined similarly for vertex $B$.)
[b]p21.[/b] Let $a_0, a_1, ...$ be an infinite sequence such that $a_0 = 1$, $a_1 = 1$, and there exists two fixed integer constants $x$ and $y$ for which $a_{n+2}$ is the remainder when $xa_{n+1}+ya_n$ is divided by $15$ for all nonnegative integers $n$. Let $t$ be the least positive integer such that $a_t = 1$ and $a_{t+1} = 1$ if such an integer exists, and let $t = 0$ if such an integer does not exist. Find the maximal value of t over all possible ordered pairs $(x, y)$.
[u]Round 8[/u]
[b]p22.[/b] A mystic square is a $3$ by $3$ grid of distinct positive integers such that the least common multiples of the numbers in each row and column are the same. Let M be the least possible maximal element in a mystic square and let $N$ be the number of mystic squares with $M$ as their maximal element. Find $M + N$.
[b]p23.[/b] In triangle $ABC$, $AB = 27$, $BC = 23$, and $CA = 34$. Let $X$ and $Y$ be points on sides $ AB$ and $AC$, respectively, such that $BX = 16$ and $CY = 7$. Given that $O$ is the circumcenter of $BXY$ , the value of $CO^2$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
[b]p24.[/b] Alan rolls ten standard fair six-sided dice, and multiplies together the ten numbers he obtains. Given that the probability that Alan’s result is a perfect square is $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers, compute $a$.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2949416p26408251]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Accuracy Rounds, 2014
[b]p1.[/b] Chad lives on the third floor of an apartment building with ten floors. He leaves his room and goes up two floors, goes down four floors, goes back up five floors, and finally goes down one floor, where he finds Jordan's room. On which floor does Jordan live?
[b]p2.[/b] A real number $x$ satisfies the equation $2014x + 1337 = 1337x + 2014$. What is $x$?
[b]p3.[/b] Given two points on the plane, how many distinct regular hexagons include both of these points as vertices?
[b]p4.[/b] Jordan has six different files on her computer and needs to email them to Chad. The sizes of these files are $768$, $1024$, $2304$, $2560$, $4096$, and $7680$ kilobytes. Unfortunately, the email server holds a limit of $S$ kilobytes on the total size of the attachments per email, where $S$ is a positive integer. It is additionally given that all of the files are indivisible. What is the maximum value of S for which it will take Jordan at least three emails to transmit all six files to Chad?
[b]p5.[/b] If real numbers $x$ and $y$ satisfy $(x + 2y)^2 + 4(x + 2y + 2 - xy) = 0$, what is $x + 2y$?
[b]p6.[/b] While playing table tennis against Jordan, Chad came up with a new way of scoring. After the first point, the score is regarded as a ratio. Whenever possible, the ratio is reduced to its simplest form. For example, if Chad scores the first two points of the game, the score is reduced from $2:0$ to $1:0$. If later in the game Chad has $5$ points and Jordan has $9$, and Chad scores a point, the score is automatically reduced from $6:9$ to $2:3$. Chad's next point would tie the game at $1:1$. Like normal table tennis, a player wins if he or she is the first to obtain $21$ points. However, he or she does not win if after his or her receipt of the $21^{st}$ point, the score is immediately reduced. Chad and Jordan start at $0:0$ and finish the game using this rule, after which Jordan notes a curiosity: the score was never reduced. How many possible games could they have played? Two games are considered the same if and only if they include the exact same sequence of scoring.
[b]p7.[/b] For a positive integer $m$, we define $m$ as a factorial number if and only if there exists a positive integer $k$ for which $m = k \cdot (k - 1) \cdot ... \cdot 2 \cdot 1$. We define a positive integer $n$ as a Thai number if and only if $n$ can be written as both the sum of two factorial numbers and the product of two factorial numbers. What is the sum of the five smallest Thai numbers?
[b]p8.[/b] Chad and Jordan are in the Exeter Space Station, which is a triangular prism with equilateral bases. Its height has length one decameter and its base has side lengths of three decameters. To protect their station against micrometeorites, they install a force field that contains all points that are within one decameter of any point of the surface of the station. What is the volume of the set of points within the force field and outside the station, in cubic decameters?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Accuracy Rounds, 2017
[b]p1.[/b] Chris goes to Matt's Hamburger Shop to buy a hamburger. Each hamburger must contain exactly one bread, one lettuce, one cheese, one protein, and at least one condiment. There are two kinds of bread, two kinds of lettuce, three kinds of cheese, three kinds of protein, and six different condiments: ketchup, mayo, mustard, dill pickles, jalape~nos, and Matt's Magical Sunshine Sauce. How many different hamburgers can Chris make?
[b]p2.[/b] The degree measures of the interior angles in convex pentagon $NICKY$ are all integers and form an increasing arithmetic sequence in some order. What is the smallest possible degree measure of the pentagon's smallest angle?
[b]p3.[/b] Daniel thinks of a two-digit positive integer $x$. He swaps its two digits and gets a number $y$ that is less than $x$. If $5$ divides $x-y$ and $7$ divides $x+y$, find all possible two-digit numbers Daniel could have in mind.
[b]p4.[/b] At the Lio Orympics, a target in archery consists of ten concentric circles. The radii of the circles are $1$, $2$, $3$, $...$, $9$, and $10$ respectively. Hitting the innermost circle scores the archer $10$ points, the next ring is worth $9$ points, the next ring is worth 8 points, all the way to the outermost ring, which is worth $1$ point. If a beginner archer has an equal probability of hitting any point on the target and never misses the target, what is the probability that his total score after making two shots is even?
[b]p5.[/b] Let $F(x) = x^2 + 2x - 35$ and $G(x) = x^2 + 10x + 14$. Find all distinct real roots of $F(G(x)) = 0$.
[b]p6.[/b] One day while driving, Ivan noticed a curious property on his car's digital clock. The sum of the digits of the current hour equaled the sum of the digits of the current minute. (Ivan's car clock shows $24$-hour time; that is, the hour ranges from $0$ to $23$, and the minute ranges from $0$ to $59$.) For how many possible times of the day could Ivan have observed this property?
[b]p7.[/b] Qi Qi has a set $Q$ of all lattice points in the coordinate plane whose $x$- and $y$-coordinates are between $1$ and $7$ inclusive. She wishes to color $7$ points of the set blue and the rest white so that each row or column contains exactly $1$ blue point and no blue point lies on or below the line $x + y = 5$. In how many ways can she color the points?
[b]p8.[/b] A piece of paper is in the shape of an equilateral triangle $ABC$ with side length $12$. Points $A_B$ and $B_A$ lie on segment $AB$, such that $AA_B = 3$, and $BB_A = 3$. Define points $B_C$ and $C_B$ on segment $BC$ and points $C_A$ and $A_C$ on segment $CA$ similarly. Point $A_1$ is the intersection of $A_CB_C$ and $A_BC_B$. Define $B_1$ and $C_1$ similarly. The three rhombi - $AA_BA_1A_C$,$BB_CB_1B_A$, $CC_AC_1C_B$ - are cut from triangle $ABC$, and the paper is folded along segments $A_1B_1$, $B_1C_1$, $C_1A_1$, to form a tray without a top. What is the volume of this tray?
[b]p9.[/b] Define $\{x\}$ as the fractional part of $x$. Let $S$ be the set of points $(x, y)$ in the Cartesian coordinate plane such that $x + \{x\} \le y$, $x \ge 0$, and $y \le 100$. Find the area of $S$.
[b]p10.[/b] Nicky likes dolls. He has $10$ toy chairs in a row, and he wants to put some indistinguishable dolls on some of these chairs. (A chair can hold only one doll.) He doesn't want his dolls to get lonely, so he wants each doll sitting on a chair to be adjacent to at least one other doll. How many ways are there for him to put any number (possibly none) of dolls on the chairs? Two ways are considered distinct if and only if there is a chair that has a doll in one way but does not have one in the other.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Accuracy Rounds, 2012
[b]p1.[/b] An $18$oz glass of apple juice is $6\%$ sugar and a $6$oz glass of orange juice is $12\%$ sugar. The two glasses are poured together to create a cocktail. What percent of the cocktail is sugar?
[b]p2.[/b] Find the number of positive numbers that can be expressed as the difference of two integers between $-2$ and $2012$ inclusive.
[b]p3.[/b] An annulus is defined as the region between two concentric circles. Suppose that the inner circle of an annulus has radius $2$ and the outer circle has radius $5$. Find the probability that a randomly chosen point in the annulus is at most $3$ units from the center.
[b]p4.[/b] Ben and Jerry are walking together inside a train tunnel when they hear a train approaching. They decide to run in opposite directions, with Ben heading towards the train and Jerry heading away from the train. As soon as Ben finishes his $1200$ meter dash to the outside, the front of the train enters the tunnel. Coincidentally, Jerry also barely survives, with the front of the train exiting the tunnel as soon as he does. Given that Ben and Jerry both run at $1/9$ of the train’s speed, how long is the tunnel in meters?
[b]p5.[/b] Let $ABC$ be an isosceles triangle with $AB = AC = 9$ and $\angle B = \angle C = 75^o$. Let $DEF$ be another triangle congruent to $ABC$. The two triangles are placed together (without overlapping) to form a quadrilateral, which is cut along one of its diagonals into two triangles. Given that the two resulting triangles are incongruent, find the area of the larger one.
[b]p6.[/b] There is an infinitely long row of boxes, with a Ditto in one of them. Every minute, each existing Ditto clones itself, and the clone moves to the box to the right of the original box, while the original Ditto does not move. Eventually, one of the boxes contains over $100$ Dittos. How many Dittos are in that box when this first happens?
[b]p7.[/b] Evaluate $$26 + 36 + 998 + 26 \cdot 36 + 26 \cdot 998 + 36 \cdot 998 + 26 \cdot 36 \cdot 998.$$
[b]p8. [/b]There are $15$ students in a school. Every two students are either friends or not friends. Among every group of three students, either all three are friends with each other, or exactly one pair of them are friends. Determine the minimum possible number of friendships at the school.
[b]p9.[/b] Let $f(x) = \sqrt{2x + 1 + 2\sqrt{x^2 + x}}$. Determine the value of $$\frac{1}{f(1)}+\frac{1}{f(1)}+\frac{1}{f(3)}+...+\frac{1}{f(24)}.$$
[b]p10.[/b] In square $ABCD$, points $E$ and $F$ lie on segments $AD$ and $CD$, respectively. Given that $\angle EBF = 45^o$, $DE = 12$, and $DF = 35$, compute $AB$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Team Rounds, 2013
[b]p1.[/b] Determine the number of ways to place $4$ rooks on a $4 \times 4$ chessboard such that:
(a) no two rooks attack one another, and
(b) the main diagonal (the set of squares marked $X$ below) does not contain any rooks.
[img]https://cdn.artofproblemsolving.com/attachments/e/e/e3aa96de6c8ed468c6ef3837e66a0bce360d36.png[/img]
The rooks are indistinguishable and the board cannot be rotated. (Two rooks attack each other if they are in the same row or column.)
[b]p2.[/b] Seven students, numbered $1$ to $7$ in counter-clockwise order, are seated in a circle. Fresh Mann has 100 erasers, and he wants to distribute them to the students, albeit unfairly. Starting with person $ 1$ and proceeding counter-clockwise, Fresh Mann gives $i$ erasers to student $i$; for example, he gives $ 1$ eraser to student $ 1$, then $2$ erasers to student $2$, et cetera. He continues around the circle until he does not have enough erasers to give to the next person. At this point, determine the number of erasers that Fresh Mann has.
[b]p3.[/b] Let $ABC$ be a triangle with $AB = AC = 17$ and $BC = 24$. Approximate $\angle ABC$ to the nearest multiple of $10$ degrees.
[b]p4.[/b] Define a sequence of rational numbers $\{x_n\}$ by $x_1 =\frac35$ and for $n \ge 1$, $x_{n+1} = 2 - \frac{1}{x_n}$ . Compute the product $x_1x_2x_3... x_{2013}$.
[b]p5.[/b] In equilateral triangle $ABC$, points $P$ and $R$ lie on segment $AB$, points $I$ and $M$ lie on segment $BC$, and points $E$ and $S$ lie on segment $CA$ such that $PRIMES$ is a equiangular hexagon. Given that $AB = 11$, $PR = 2$, $IM = 3$, and $ES = 5$, compute the area of hexagon $PRIMES$.
[b]p6.[/b] Let $f(a, b) = \frac{a^2}{a+b}$ . Let $A$ denote the sum of $f(i, j)$ over all pairs of integers $(i, j)$ with $1 \le i < j \le 10$; that is,
$$A = (f(1, 2) + f(1, 3) + ...+ f(1, 10)) + (f(2, 3) + f(2, 4) +... + f(2, 10)) +... + f(9, 10).$$
Similarly, let $B$ denote the sum of $f(i, j)$ over all pairs of integers $(i, j)$ with $1 \le j < i \le 10$, that is, $$B = (f(2, 1) + f(3, 1) + ... + f(10, 1)) + (f(3, 2) + f(4, 2) +... + f(10, 2)) +... + f(10, 9).$$ Compute $B - A$.
[b]p7.[/b] Fresh Mann has a pile of seven rocks with weights $1, 1, 2, 4, 8, 16$, and $32$ pounds and some integer X between $1$ and $64$, inclusive. He would like to choose a set of the rocks whose total weight is exactly $X$ pounds. Given that he can do so in more than one way, determine the sum of all possible values of $X$. (The two $1$-pound rocks are indistinguishable.)
[b]p8.[/b] Let $ABCD$ be a convex quadrilateral with $AB = BC = CA$. Suppose that point $P$ lies inside the quadrilateral with $AP = PD = DA$ and $\angle PCD = 30^o$. Given that $CP = 2$ and $CD = 3$, compute $CA$.
[b]p9.[/b] Define a sequence of rational numbers $\{x_n\}$ by $x_1 = 2$, $x_2 = \frac{13}{2}$ , and for $n \ge 1$, $x_{n+2} = 3 -\frac{3}{x_{n+1}}+\frac{1}{x_nx_{n+1}}$. Compute $x_{100}$.
[b]p10.[/b] Ten prisoners are standing in a line. A prison guard wants to place a hat on each prisoner. He has two colors of hats, red and blue, and he has $10$ hats of each color. Determine the number of ways in which the prison guard can place hats such that among any set of consecutive prisoners, the number of prisoners with red hats and the number of prisoners with blue hats differ by at most $2$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Speed Rounds, 2022
[i]20 problems for 25 minutes.[/i]
[b]p1.[/b] Compute $(2 + 0)(2 + 2)(2 + 0)(2 + 2)$.
[b]p2.[/b] Given that $25\%$ of $x$ is $120\%$ of $30\%$ of $200$, find $x$.
[b]p3.[/b] Jacob had taken a nap. Given that he fell asleep at $4:30$ PM and woke up at $6:23$ PM later that same day, for how many minutes was he asleep?
[b]p4.[/b] Kevin is painting a cardboard cube with side length $12$ meters. Given that he needs exactly one can of paint to cover the surface of a rectangular prism that is $2$ meters long, $3$ meters wide, and $6$ meters tall, how many cans of paint does he need to paint the surface of his cube?
[b]p5.[/b] How many nonzero digits does $200 \times 25 \times 8 \times 125 \times 3$ have?
[b]p6.[/b] Given two real numbers $x$ and $y$, define $x \# y = xy + 7x - y$. Compute the absolute value of $0 \# (1 \# (2 \# (3 \# 4)))$.
[b]p7.[/b] A $3$-by-$5$ rectangle is partitioned into several squares of integer side length. What is the fewest number of such squares? Squares in this partition must not overlap and must be contained within the rectangle.
[b]p8.[/b] Points $A$ and $B$ lie in the plane so that $AB = 24$. Given that $C$ is the midpoint of $AB$, $D$ is the midpoint of $BC$, $E$ is the midpoint of $AD$, and $F$ is the midpoint of $BD$, find the length of segment $EF$.
[b]p9.[/b] Vincent the Bug and Achyuta the Anteater are climbing an infinitely tall vertical bamboo stalk. Achyuta begins at the bottom of the stalk and climbs up at a rate of $5$ inches per second, while Vincent begins somewhere along the length of the stalk and climbs up at a rate of $3$ inches per second. After climbing for $t$ seconds, Achyuta is half as high above the ground as Vincent. Given that Achyuta catches up to Vincent after another $160$ seconds, compute $t$.
[b]p10.[/b] What is the minimum possible value of $|x - 2022| + |x - 20|$ over all real numbers $x$?
[b]p11.[/b] Let $ABCD$ be a rectangle. Lines $\ell_1$ and $\ell_2$ divide $ABCD$ into four regions such that $\ell_1$ is parallel to $AB$ and line $\ell_2$ is parallel to $AD$. Given that three of the regions have area $6$, $8$, and $12$, compute the sum of all possible areas of the fourth region.
[b]p12.[/b] A diverse number is a positive integer that has two or more distinct prime factors. How many diverse numbers are less than $50$?
[b]p13.[/b] Let $x$, $y$, and $z$ be real numbers so that $(x+y)(y +z) = 36$ and $(x+z)(x+y) = 4$. Compute $y^2 -x^2$.
[b]p14.[/b] What is the remainder when $ 1^{10} + 3^{10} + 7^{10}$ is divided by $58$?
[b]p15.[/b] Let $A = (0, 1)$, $B = (3, 5)$, $C = (1, 4)$, and $D = (3, 4)$ be four points in the plane. Find the minimum possible value of $AP + BP + CP + DP$ over all points $P$ in the plane.
[b]p16.[/b] In trapezoid $ABCD$, points $E$ and $F$ lie on sides $BC$ and $AD$, respectively, such that $AB \parallel CD \parallel EF$. Given that $AB = 3$, $EF = 5$, and $CD = 6$, the ratio $\frac{[ABEF]}{[CDFE]}$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. (Note: $[F]$ denotes the area of $F$.)
[b]p17.[/b] For sets $X$ and $Y$ , let $|X \cap Y |$ denote the number of elements in both $X$ and $Y$ and $|X \cup Y|$ denote the number of elements in at least one of $X$ or $Y$ . How many ordered pairs of subsets $(A,B)$ of $\{1, 2, 3,..., 8\}$ are there such that $|A \cap B| = 2$ and $|A \cup B| = 5$?
[b]p18.[/b] A tetromino is a polygon composed of four unit squares connected orthogonally (that is, sharing a edge). A tri-tetromino is a polygon formed by three orthogonally connected tetrominoes. What is the maximum possible perimeter of a tri-tetromino?
[b]p19.[/b] The numbers from $1$ through $2022$, inclusive, are written on a whiteboard. Every day, Hermione erases two numbers $a$ and $b$ and replaces them with $ab+a+b$. After some number of days, there is only one number $N$ remaining on the whiteboard. If $N$ has $k$ trailing nines in its decimal representation, what is the maximum possible value of $k$?
[b]p20.[/b] Evaluate $5(2^2 + 3^2) + 7(3^2 + 4^2) + 9(4^2 + 5^2) + ... + 199(99^2 + 100^2)$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Speed Rounds, 2020
[i]20 problems for 25 minutes.[/i]
[b]p1.[/b] What is $20 \div 2 - 0 \times 1 + 2 \times 5$?
[b]p2.[/b] Today is Saturday, January $25$, $2020$. Exactly four hundred years from today, January $25$, $2420$, is again a Saturday. How many weekend days (Saturdays and Sundays) are in February, $2420$? (January has $31$ days and in year $2040$, February has $29$ days.)
[b]p3.[/b] Given that there are four people sitting around a circular table, and two of them stand up, what is the probability that the two of them were originally sitting next to each other?
[b]p4.[/b] What is the area of a triangle with side lengths $5$, $5$, and $6$?
[b]p5.[/b] Six people go to OBA Noodles on Main Street. Each person has $1/2$ probability to order Duck Noodle Soup, $1/3$ probability to order OBA Ramen, and $1/6$ probability to order Kimchi Udon Soup. What is the probability that three people get Duck Noodle Soup, two people get OBA Ramen, and one person gets Kimchi Udon Soup?
[b]p6.[/b] Among all positive integers $a$ and $b$ that satisfy $a^b = 64$, what is the minimum possible value of $a+b$?
[b]p7.[/b] A positive integer $n$ is called trivial if its tens digit divides $n$. How many two-digit trivial numbers are there?
[b]p8.[/b] Triangle $ABC$ has $AB = 5$, $BC = 13$, and $AC = 12$. Square $BCDE$ is constructed outside of the triangle. The perpendicular line from $A$ to side $DE$ cuts the square into two parts. What is the positive difference in their areas?
[b]p9.[/b] In an increasing arithmetic sequence, the first, third, and ninth terms form an increasing geometric sequence (in that order). Given that the first term is $5$, find the sum of the first nine terms of the arithmetic sequence.
[b]p10.[/b] Square $ABCD$ has side length $1$. Let points $C'$ and $D'$ be the reflections of points $C$ and $D$ over lines $AB$ and $BC$, respectively. Let P be the center of square $ABCD$. What is the area of the concave quadrilateral $PD'BC'$?
[b]p11.[/b] How many four-digit palindromes are multiples of $7$? (A palindrome is a number which reads the same forwards and backwards.)
[b]p12.[/b] Let $A$ and $B$ be positive integers such that the absolute value of the difference between the sum of the digits of $A$ and the sum of the digits of $(A + B)$ is $14$. What is the minimum possible value for $B$?
[b]p13.[/b] Clark writes the following set of congruences: $x \equiv a$ (mod $6$), $x \equiv b$ (mod $10$), $x \equiv c$ (mod $15$), and he picks $a$, $b$, and $c$ to be three randomly chosen integers. What is the probability that a solution for $x$ exists?
[b]p14.[/b] Vincent the bug is crawling on the real number line starting from $2020$. Each second, he may crawl from $x$ to $x - 1$, or teleport from $x$ to $\frac{x}{3}$ . What is the least number of seconds needed for Vincent to get to $0$?
[b]p15.[/b] How many positive divisors of $2020$ do not also divide $1010$?
[b]p16.[/b] A bishop is a piece in the game of chess that can move in any direction along a diagonal on which it stands. Two bishops attack each other if the two bishops lie on the same diagonal of a chessboard. Find the maximum number of bishops that can be placed on an $8\times 8$ chessboard such that no two bishops attack each other.
[b]p17.[/b] Let $ABC$ be a right triangle with hypotenuse $20$ and perimeter $41$. What is the area of $ABC$?
[b]p18.[/b] What is the remainder when $x^{19} + 2x^{18} + 3x^{17} +...+ 20$ is divided by $x^2 + 1$?
[b]p19.[/b] Ben splits the integers from $1$ to $1000$ into $50$ groups of $20$ consecutive integers each, starting with $\{1, 2,...,20\}$. How many of these groups contain at least one perfect square?
[b]p20.[/b] Trapezoid $ABCD$ with $AB$ parallel to $CD$ has $AB = 10$, $BC = 20$, $CD = 35$, and $AD = 15$. Let $AD$ and $BC$ intersect at $P$ and let $AC$ and $BD$ intersect at $Q$. Line $PQ$ intersects $AB$ at $R$. What is the length of $AR$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
EMCC Guts Rounds, 2011
[u]Round 1[/u]
[b]p1.[/b] In order to make good salad dressing, Bob needs a $0.9\%$ salt solution. If soy sauce is $15\%$ salt, how much water, in mL, does Bob need to add to $3$ mL of pure soy sauce in order to have a good salad dressing?
[b]p2.[/b] Alex the Geologist is buying a canteen before he ventures into the desert. The original cost of a canteen is $\$20$, but Alex has two coupons. One coupon is $\$3$ off and the other is $10\%$ off the entire remaining cost. Alex can use the coupons in any order. What is the least amount of money he could pay for the canteen?
[b]p3.[/b] Steve and Yooni have six distinct teddy bears to split between them, including exactly $1$ blue teddy bear and $1$ green teddy bear. How many ways are there for the two to divide the teddy bears, if Steve gets the blue teddy bear and Yooni gets the green teddy bear? (The two do not necessarily have to get the same number of teddy bears, but each teddy bear must go to a person.)
[u]Round 2[/u]
[b]p4.[/b] In the currency of Mathamania, $5$ wampas are equal to $3$ kabobs and $10$ kabobs are equal to $2$ jambas. How many jambas are equal to twenty-five wampas?
[b]p5.[/b] A sphere has a volume of $81\pi$. A new sphere with the same center is constructed with a radius that is $\frac13$ the radius of the original sphere. Find the volume, in terms of $\pi$, of the region between the two spheres.
[b]p6.[/b] A frog is located at the origin. It makes four hops, each of which moves it either $1$ unit to the right or $1$ unit to the left. If it also ends at the origin, how many $4$-hop paths can it take?
[u]Round 3[/u]
[b]p7.[/b] Nick multiplies two consecutive positive integers to get $4^5 - 2^5$ . What is the smaller of the two numbers?
[b]p8.[/b] In rectangle $ABCD$, $E$ is a point on segment $CD$ such that $\angle EBC = 30^o$ and $\angle AEB = 80^o$. Find $\angle EAB$, in degrees.
[b]p9.[/b] Mary’s secret garden contains clones of Homer Simpson and WALL-E. A WALL-E clone has $4$ legs. Meanwhile, Homer Simpson clones are human and therefore have $2$ legs each. A Homer Simpson clone always has $5$ donuts, while a WALL-E clone has $2$. In Mary’s secret garden, there are $184$ donuts and $128$ legs. How many WALL-E clones are there?
[u]Round 4[/u]
[b]p10.[/b] Including Richie, there are $6$ students in a math club. Each day, Richie hangs out with a different group of club mates, each of whom gives him a dollar when he hangs out with them. How many dollars will Richie have by the time he has hung out with every possible group of club mates?
[b]p11.[/b] There are seven boxes in a line: three empty, three holding $\$10$ each, and one holding the jackpot of $\$1, 000, 000$. From the left to the right, the boxes are numbered $1, 2, 3, 4, 5, 6$ and $7$, in that order.
You are told the following:
$\bullet$ No two adjacent boxes hold the same contents.
$\bullet$ Box $4$ is empty.
$\bullet$ There is one more $\$10$ prize to the right of the jackpot than there is to the left.
Which box holds the jackpot?
[b]p12.[/b] Let $a$ and $b$ be real numbers such that $a + b = 8$. Let $c$ be the minimum possible value of $x^2 + ax + b$ over all real numbers $x$. Find the maximum possible value of $c$ over all such $a$ and $b$.
[u]Round 5[/u]
[b]p13.[/b] Let $ABCD$ be a rectangle with $AB = 10$ and $BC = 12$. Let M be the midpoint of $CD$, and $P$ be a point on $BM$ such that $BP = BC$. Find the area of $ABPD$.
[b]p14.[/b] The number $19$ has the following properties:
$\bullet$ It is a $2$-digit positive integer.
$\bullet$ It is the two leading digits of a $4$-digit perfect square, because $1936 = 44^2$.
How many numbers, including $19$, satisfy these two conditions?
[b]p15.[/b] In a $3 \times 3$ grid, each unit square is colored either black or white. A coloring is considered “nice” if there is at most one white square in each row or column. What is the total number of nice colorings? Rotations and reflections of a coloring are considered distinct. (For example, in the three squares shown below, only the rightmost one has a nice coloring.
[img]https://cdn.artofproblemsolving.com/attachments/e/4/e6932c822bec77aa0b07c98d1789e58416b912.png[/img]
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