Found problems: 34
MBMT Geometry Rounds, 2019
[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide]
[b]D1.[/b] Triangle $ABC$ has $AB = 3$, $BC = 4$, and $\angle B = 90^o$. Find the area of triangle $ABC$.
[b]D2 / L1.[/b] Let $ABCDEF$ be a regular hexagon. Given that $AD = 5$, find $AB$.
[b]D3.[/b] Caroline glues two pentagonal pyramids to the top and bottom of a pentagonal prism so that the pentagonal faces coincide. How many edges does Caroline’s figure have?
[b]D4 / L3.[/b] The hour hand of a clock is $6$ inches long, and the minute hand is $10$ inches long. Find the area of the region swept out by the hands from $8:45$ AM to $9:15$ AM of a single day, in square inches.
[b]D5 / L2.[/b] Circles $A$, $B$, and $C$ are all externally tangent, with radii $1$, $10$, and $100$, respectively. What is the radius of the smallest circle entirely containing all three circles?
[b]D6.[/b] Four parallel lines are drawn such that they are equally spaced and pass through the four vertices of a unit square. Find the distance between any two consecutive lines.
[b]D7 / L4.[/b] In rectangle $ABCD$, $AB = 2$ and $AD > AB$. Two quarter circles are drawn inside of $ABCD$ with centers at $A$ and $C$ that pass through $B$ and $D$, respectively. If these two quarter circles are tangent, find the area inside of $ABCD$ that is outside both of the quarter circles.
[b]D8 / L6.[/b] Triangle $ABC$ is equilateral. A circle passes through $A$ and is tangent to side $BC$. It intersects sides AB and $AC$ again at $E$ and $F$, respectively. If $AE = 10$ and $AF = 11$, find $AB$.
[b]L5.[/b] Find the area of a triangle with side lengths $\sqrt{2}$, $\sqrt{58}$, and $2\sqrt{17}$.
[b]L7.[/b] Triangle $ABC$ has area $80$. Point $D$ is in the interior of $\vartriangle ABC$ such that $AD =6$, $BD = 4$, $CD = 16$, and the area of $\vartriangle ADC = 48$. Determine the area of $\vartriangle ADB$.
[b]L8. [/b]Given two points $A$ and $B$ in the plane with $AB = 1$, define $f(C)$ to be the circumcenter of triangle $ABC$, if it exists. Find the number of points $X$ so that $f^{2019}(X) = X$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Geometry Rounds, 2016
[hide=E stands for Euclid, L stands for Lobachevsky]they had two problem sets under those two names[/hide]
[b]E1.[/b] What is the perimeter of a rectangle if its area is $24$ and one side length is $6$?
[b]E2.[/b] John moves 3 miles south, then $2$ miles west, then $7$ miles north, and then $5$ miles east. What is the length of the shortest path, in miles, from John's current position to his original position?
[b]E3.[/b] An equilateral triangle $ABC$ is drawn with side length $2$. The midpoints of sides $AB$, $BC$, and $CA$ are constructed, and are connected to form a triangle. What is the perimeter of the newly formed triangle?
[b]E4.[/b] Let triangle $ABC$ have sides $AB = 74$ and $AC = 5$. What is the sum of all possible integral side lengths of BC?
[b]E5.[/b] What is the area of quadrilateral $ABCD$ on the coordinate plane with $A(1, 0)$, $B(0, 1)$, $C(1, 3)$, and $D(5, 2)$?
[b]E6 / L1.[/b] Let $ABCD$ be a square with side length $30$. A circle centered at the center of $ABCD$ with diameter $34$ is drawn. Let $E$ and $F$ be the points at which the circle intersects side $AB$. What is $EF$?
[b]E7 / L2.[/b] What is the area of the quadrilateral bounded by $|2x| + |3y| = 6$?
[b]E8.[/b] A circle $O$ with radius $2$ has a regular hexagon inscribed in it. Upon the sides of the hexagon, equilateral triangles of side length $2$ are erected outwards. Find the area of the union of these triangles and circle $O$.
[b]L3.[/b] Right triangle $ABC$ has hypotenuse $AB$. Altitude $CD$ divides $AB$ into segments $AD$ and $DB$, with $AD = 20$ and $DB = 16$. What is the area of triangle $ABC$?
[b]L4.[/b] Circle $O$ has chord $AB$. Extend $AB$ past $B$ to a point $C$. A ray from $C$ is drawn, and this ray intersects circle $O$. Let point $D$ be the point of intersection of the ray and the circle that is closest to point $C$. Given $AB = 20$, $BC = 16$, and $OA = \frac{201}{6}$ , find the longest possible length of $CD$.
[b]L5.[/b] Consider a circular cone with vertex $A$. The cone's height is $4$ and the radius of its base is $3$. Inscribe a sphere inside the cone. Find the ratio of the volume of the cone to the volume of the sphere.
[b]L6.[/b] A disk of radius $\frac12$ is randomly placed on the coordinate plane. What is the probability that it contains a lattice point (point with integer coordinates)?
[b]L7.[/b] Let $ABC$ be an equilateral triangle of side length $2$. Let $D$ be the midpoint of $BC$, and let $P$ be a variable point on $AC$. By moving $P$ along $AC$, what is the minimum perimeter of triangle $BDP$?
[b]L8.[/b] Let $ABCD$ be a rectangle with $AB = 8$ and $BC = 9$. Let $DEFG$ be a rhombus, where $G$ is on line $BC$ and $A$ is on line $EF$. If $m\angle EFG = 30^o, what is $DE$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Guts Rounds, 2023
[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide]
[u]Set 1[/u]
[b]B1 / G1[/b] Find $20^3 + 2^2 + 3^1$.
[b]B2[/b] A piece of string of length $10$ is cut $4$ times into strings of equal length. What is the length of each small piece of string?
[b]B3 / G2[/b] What is the smallest perfect square that is also a perfect cube?
[b]B4[/b] What is the probability a $5$-sided die with sides labeled from $1$ through $5$ rolls an odd number?
[b]B5 / G3[/b] Hanfei spent $14$ dollars on chicken nuggets at McDonalds. $4$ nuggets cost $3$ dollars, $6$ nuggets cost $4$ dollars, and $12$ nuggets cost $9$ dollars. How many chicken nuggets did Hanfei buy?
[u]Set 2[/u]
[b]B6[/b] What is the probability a randomly chosen positive integer less than or equal to $15$ is prime?
[b]B7[/b] Andrew flips a fair coin with sides labeled 0 and 1 and also rolls a fair die with sides labeled $1$ through $6$. What is the probability that the sum is greater than $5$?
[b]B8 / G4[/b] What is the radius of a circle with area $4$?
[b]B9[/b] What is the maximum number of equilateral triangles on a piece of paper that can share the same corner?
[b]B10 / G5[/b] Bob likes to make pizzas. Bab also likes to make pizzas. Bob can make a pizza in $20$ minutes. Bab can make a pizza in $30$ minutes. If Bob and Bab want to make $50$ pizzas in total, how many hours would that take them?
[u]Set 3[/u]
[b]B11 / G6[/b] Find the area of an equilateral rectangle with perimeter $20$.
[b]B12 / G7[/b] What is the minimum possible number of divisors that the sum of two prime numbers greater than $2$ can have?
[b]B13 / G8[/b] Kwu and Kz play rock-paper-scissors-dynamite, a variant of the classic rock-paperscissors in which dynamite beats rock and paper but loses to scissors. The standard rock-paper-scissors rules apply, where rock beats scissors, paper beats rock, and scissors beats paper. If they throw out the same option, they keep playing until one of them wins. If Kz randomly throws out one of the four options with equal probability, while Kwu only throws out dynamite, what is the probability Kwu wins?
[b]B14 / G9[/b] Aven has $4$ distinct baguettes in a bag. He picks three of the bagged baguettes at random and lays them on a table in random order. How many possible orderings of three baguettes are there on the table?
[b]B15 / G10[/b] Find the largest $7$-digit palindrome that is divisible by $11$.
PS. You should use hide for answers. Rest problems have been posted [url=https://artofproblemsolving.com/community/c3h3132170p28376644]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Guts Rounds, 2022
[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide]
[u]Set 1[/u]
[b]D1 / Z1.[/b] What is $1 + 2 \cdot 3$?
[b]D2.[/b] What is the average of the first $9$ positive integers?
[b]D3 / Z2.[/b] A square of side length $2$ is cut into $4$ congruent squares. What is the perimeter of one of the $4$ squares?
[b]D4.[/b] Find the ratio of a circle’s circumference squared to the area of the circle.
[b]D5 / Z3.[/b] $6$ people split a bag of cookies such that they each get $21$ cookies. Kyle comes and demands his share of cookies. If the $7$ people then re-split the cookies equally, how many cookies does Kyle get?
[u]Set 2[/u]
[b]D6.[/b] How many prime numbers are perfect squares?
[b]D7.[/b] Josh has an unfair $4$-sided die numbered $1$ through $4$. The probability it lands on an even number is twice the probability it lands on an odd number. What is the probability it lands on either $1$ or $3$?
[b]D8.[/b] If Alice consumes $1000$ calories every day and burns $500$ every night, how many days will it take for her to first reach a net gain of $5000$ calories?
[b]D9 / Z4.[/b] Blobby flips $4$ coins. What is the probability he sees at least one heads and one tails?
[b]D10.[/b] Lillian has $n$ jars and $48$ marbles. If George steals one jar from Lillian, she can fill each jar with $8$ marbles. If George steals $3$ jars, Lillian can fill each jar to maximum capacity. How many marbles can each jar fill?
[u]Set 3[/u]
[b]D11 / Z6.[/b] How many perfect squares less than $100$ are odd?
[b]D12.[/b] Jash and Nash wash cars for cash. Jash gets $\$6$ for each car, while Nash gets $\$11$ per car. If Nash has earned $\$1$ more than Jash, what is the least amount of money that Nash could have earned?
[b]D13 / Z5.[/b] The product of $10$ consecutive positive integers ends in $3$ zeros. What is the minimum possible value of the smallest of the $10$ integers?
[b]D14 / Z7.[/b] Guuce continually rolls a fair $6$-sided dice until he rolls a $1$ or a $6$. He wins if he rolls a $6$, and loses if he rolls a $1$. What is the probability that Guuce wins?
[b]D15 / Z8.[/b] The perimeter and area of a square with integer side lengths are both three digit integers. How many possible values are there for the side length of the square?
PS. You should use hide for answers. D.16-30/Z.9-14, 17, 26-30 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916250p26045695]here [/url]and Z.15-25 [url=https://artofproblemsolving.com/community/c3h2916258p26045774]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Team Rounds, 2020.43
Let $\sigma_k(n)$ be the sum of the $k^{th}$ powers of the divisors of $n$. For all $k \ge 2$ and all $n \ge 3$, we have that $$\frac{\sigma_k(n)}{n^{k+2}} (2020n + 2019)^2 > m.$$ Find the largest possible value of $m$.
MBMT Team Rounds, 2020.42
$\vartriangle ABC$ has side lengths $AB = 4$ and $AC = 9$. Angle bisector $AD$ bisects angle $A$ and intersects $BC$ at $D$. Let $k$ be the ratio $\frac{BD}{AB}$ . Given that the length $AD$ is an integer, find the sum of all possible $k^2$
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MBMT Team Rounds, 2019
[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide]
[b]D1.[/b] What is the solution to the equation $3 \cdot x \cdot 5 = 4 \cdot 5 \cdot 6$?
[b]D2.[/b] Mr. Rose is making Platonic solids! If there are five different types of Platonic solids, and each Platonic solid can be one of three colors, how many different colored Platonic solids can Mr. Rose make?
[b]D3.[/b] What fraction of the multiples of $5$ between $1$ and $100$ inclusive are also multiples of $20$?
[b]D4.[/b] What is the maximum number of times a circle can intersect a triangle?
[b]D5 / L1.[/b] At an interesting supermarket, the nth apple you purchase costs $n$ dollars, while pears are $3$ dollars each. Given that Layla has exactly enough money to purchase either $k$ apples or $2k$ pears for $k > 0$, how much money does Layla have?
[b]D6 / L3.[/b] For how many positive integers $1 \le n \le 10$ does there exist a prime $p$ such that the sum of the digits of $p$ is $n$?
[b]D7 / L2.[/b] Real numbers $a, b, c$ are selected uniformly and independently at random between $0$ and $1$. What is the probability that $a \ge b \le c$?
[b]D8.[/b] How many ordered pairs of positive integers $(x, y)$ satisfy $lcm(x, y) = 500$?
[b]D9 / L4.[/b] There are $50$ dogs in the local animal shelter. Each dog is enemies with at least $2$ other dogs. Steven wants to adopt as many dogs as possible, but he doesn’t want to adopt any pair of enemies, since they will cause a ruckus. Considering all possible enemy networks among the dogs, find the maximum number of dogs that Steven can possibly adopt.
[b]D10 / L7.[/b] Unit circles $a, b, c$ satisfy $d(a, b) = 1$, $d(b, c) = 2$, and $d(c, a) = 3,$ where $d(x, y)$ is defined to be the minimum distance between any two points on circles $x$ and $y$. Find the radius of the smallest circle entirely containing $a$, $b$, and $c$.
[b]D11 / L8.[/b] The numbers $1$ through $5$ are written on a chalkboard. Every second, Sara erases two numbers $a$ and $b$ such that $a \ge b$ and writes $\sqrt{a^2 - b^2}$ on the board. Let M and m be the maximum and minimum possible values on the board when there is only one number left, respectively. Find the ordered pair $(M, m)$.
[b]D12 / L9.[/b] $N$ people stand in a line. Bella says, “There exists an assignment of nonnegative numbers to the $N$ people so that the sum of all the numbers is $1$ and the sum of any three consecutive people’s numbers does not exceed $1/2019$.” If Bella is right, find the minimum value of $N$ possible.
[b]D13 / L10.[/b] In triangle $\vartriangle ABC$, $D$ is on $AC$ such that $BD$ is an altitude, and $E$ is on $AB$ such that $CE$ is an altitude. Let F be the intersection of $BD$ and $CE$. If $EF = 2FC$, $BF = 8DF$, and $DC = 3$, then find the area of $\vartriangle CDF$.
[b]D14 / L11.[/b] Consider nonnegative real numbers $a_1, ..., a_6$ such that $a_1 +... + a_6 = 20$. Find the minimum possible value of $$\sqrt{a^2_1 + 1^2} +\sqrt{a^2_2 + 2^2} +\sqrt{a^2_3 + 3^2} +\sqrt{a^2_4 + 4^2} +\sqrt{a^2_5 + 5^2} +\sqrt{a^2_6 + 6^2}.$$
[b]D15 / L13.[/b] Find an $a < 1000000$ so that both $a$ and $101a$ are triangular numbers. (A triangular number is a number that can be written as $1 + 2 +... + n$ for some $n \ge 1$.)
Note: There are multiple possible answers to this problem. You only need to find one.
[b]L6.[/b] How many ordered pairs of positive integers $(x, y)$, where $x$ is a perfect square and $y$ is a perfect cube, satisfy $lcm(x, y) = 81000000$?
[b]L12.[/b] Given two points $A$ and $B$ in the plane with $AB = 1$, define $f(C)$ to be the incenter of triangle $ABC$, if it exists. Find the area of the region of points $f(f(X))$ where $X$ is arbitrary.
[b]L14.[/b] Leptina and Zandar play a game. At the four corners of a square, the numbers $1, 2, 3$, and $4$ are written in clockwise order. On Leptina’s turn, she must swap a pair of adjacent numbers. On Zandar’s turn, he must choose two adjacent numbers $a$ and $b$ with $a \ge b$ and replace $a$ with $ a - b$. Zandar wants to reduce the sum of the numbers at the four corners of the square to $2$ in as few turns as possible, and Leptina wants to delay this as long as possible. If Leptina goes first and both players play optimally, find the minimum number of turns Zandar can take after which Zandar is guaranteed to have reduced the sum of the numbers to $2$.
[b]L15.[/b] There exist polynomials $P, Q$ and real numbers $c_0, c_1, c_2, ... , c_{10}$ so that the three polynomials $P, Q$, and $$c_0P^{10} + c_1P^9Q + c_2P^8Q^2 + ... + c_{10}Q^{10}$$ are all polynomials of degree 2019. Suppose that $c_0 = 1$, $c_1 = -7$, $c_2 = 22$. Find all possible values of $c_{10}$.
Note: The answer(s) are rational numbers. It suffices to give the prime factorization(s) of the numerator(s) and denominator(s).
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Geometry Rounds, 2023
[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide]
[b]B1.[/b] If the values of two angles in a triangle are $60$ and $75$ degrees respectively, what is the measure of the third angle?
[b]B2.[/b] Square $ABCD$ has side length $1$. What is the area of triangle $ABC$?
[b]B3 / G1.[/b] An equilateral triangle and a square have the same perimeter. If the side length of the equilateral triangle is $8$, what is the square’s side length?
[b]B4 / G2.[/b] What is the maximum possible number of sides and diagonals of equal length in a quadrilateral?
[b]B5.[/b] A square of side length $4$ is put within a circle such that all $4$ corners lie on the circle. What is the diameter of the circle?
[b]B6 / G3.[/b] Patrick is rafting directly across a river $20$ meters across at a speed of $5$ m/s. The river flows in a direction perpendicular to Patrick’s direction at a rate of $12$ m/s. When Patrick reaches the shore on the other end of the river, what is the total distance he has traveled?
[b]B7 / G4.[/b] Quadrilateral $ABCD$ has side lengths $AB = 7$, $BC = 15$, $CD = 20$, and $DA = 24$. It has a diagonal length of $BD = 25$. Find the measure, in degrees, of the sum of angles $ABC$ and $ADC$.
[b]B8 / G5.[/b] What is the largest $P$ such that any rectangle inscribed in an equilateral triangle of side length $1$ has a perimeter of at least $P$?
[b]G6.[/b] A circle is inscribed in an equilateral triangle with side length $s$. Points $A$,$B$,$C$,$D$,$E$,$F$ lie on the triangle such that line segments $AB$, $CD$, and $EF$ are parallel to a side of the triangle, and tangent to the circle. If the area of hexagon $ABCDEF = \frac{9\sqrt3}{2}$ , find $s$.
[b]G7.[/b] Let $\vartriangle ABC$ be such that $\angle A = 105^o$, $\angle B = 45^o$, $\angle C = 30^o$. Let $M$ be the midpoint of $AC$. What is $\angle MBC$?
[b]G8.[/b] Points $A$, $B$, and $C$ lie on a circle centered at $O$ with radius $10$. Let the circumcenter of $\vartriangle AOC$ be $P$. If $AB = 16$, find the minimum value of $PB$.
[i]The circumcenter of a triangle is the intersection point of the three perpendicular bisectors of the sides.
[/i]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Team Rounds, 2022
[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide]
[b]D1.[/b] The product of two positive integers is $5$. What is their sum?
[b]D2.[/b] Gavin is $4$ feet tall. He walks $5$ feet before falling forward onto a cushion. How many feet is the top of Gavin’s head from his starting point?
[b]D3.[/b] How many times must Nathan roll a fair $6$-sided die until he can guarantee that the sum of his rolls is greater than $6$?
[b]D4 / Z1.[/b] What percent of the first $20$ positive integers are divisible by $3$?
[b]D5.[/b] Let $a$ be a positive integer such that $a^2 + 2a + 1 = 36$. Find $a$.
[b]D6 / Z2.[/b] It is said that a sheet of printer paper can only be folded in half $7$ times. A sheet of paper is $8.5$ inches by $11$ inches. What is the ratio of the paper’s area after it has been folded in half $7$ times to its original area?
[b]D7 / Z3.[/b] Boba has an integer. They multiply the number by $8$, which results in a two digit integer. Bubbles multiplies the same original number by 9 and gets a three digit integer. What was the original number?
[b]D8.[/b] The average number of letters in the first names of students in your class of $24$ is $7$. If your teacher, whose first name is Blair, is also included, what is the new class average?
[b]D9 / Z4.[/b] For how many integers $x$ is $9x^2$ greater than $x^4$?
[b]D10 / Z5.[/b] How many two digit numbers are the product of two distinct prime numbers ending in the same digit?
[b]D11 / Z6.[/b] A triangle’s area is twice its perimeter. Each side length of the triangle is doubled,and the new triangle has area $60$. What is the perimeter of the new triangle?
[b]D12 / Z7.[/b] Let $F$ be a point inside regular pentagon $ABCDE$ such that $\vartriangle FDC$ is equilateral. Find $\angle BEF$.
[b]D13 / Z8.[/b] Carl, Max, Zach, and Amelia sit in a row with $5$ seats. If Amelia insists on sitting next to the empty seat, how many ways can they be seated?
[b]D14 / Z9.[/b] The numbers $1, 2, ..., 29, 30$ are written on a whiteboard. Gumbo circles a bunch of numbers such that for any two numbers he circles, the greatest common divisor of the two numbers is the same as the greatest common divisor of all the numbers he circled. Gabi then does the same. After this, what is the least possible number of uncircled numbers?
[b]D15 / Z10.[/b] Via has a bag of veggie straws, which come in three colors: yellow, orange, and green. The bag contains $8$ veggie straws of each color. If she eats $22$ veggie straws without considering their color, what is the probability she eats all of the yellow veggie straws?
[b]Z11.[/b] We call a string of letters [i]purple[/i] if it is in the form $CVCCCV$ , where $C$s are placeholders for (not necessarily distinct) consonants and $V$s are placeholders for (not necessarily distinct) vowels. If $n$ is the number of purple strings, what is the remainder when $n$ is divided by $35$? The letter $y$ is counted as a vowel.
[b]Z12.[/b] Let $a, b, c$, and d be integers such that $a+b+c+d = 0$ and $(a+b)(c+d)(ab+cd) = 28$. Find $abcd$.
[b]Z13.[/b] Griffith is playing cards. A $13$-card hand with Aces of all $4$ suits is known as a godhand. If Griffith and $3$ other players are dealt $13$-card hands from a standard $52$-card deck, then the probability that Griffith is dealt a godhand can be expressed in simplest form as $\frac{a}{b}$. Find $a$.
[b]Z14.[/b] For some positive integer $m$, the quadratic $x^2 + 202200x + 2022m$ has two (not necessarily distinct) integer roots. How many possible values of $m$ are there?
[b]Z15.[/b] Triangle $ABC$ with altitudes of length $5$, $6$, and $7$ is similar to triangle $DEF$. If $\vartriangle DEF$ has integer side lengths, find the least possible value of its perimeter.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Guts Rounds, 2019
[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide]
[u]Set 1[/u]
[b]D.1 / L.1[/b] Find the units digit of $3^{1^{3^{3^7}}}$.
[b]D.2[/b] Find the positive solution to the equation $x^3 - x^2 = x - 1$.
[b]D.3[/b] Points $A$ and $B$ lie on a unit circle centered at O and are distance $1$ apart. What is the degree measure of $\angle AOB$?
[b]D.4[/b] A number is a perfect square if it is equal to an integer multiplied by itself. How many perfect squares are there between $1$ and $2019$, inclusive?
[b]D.5[/b] Ted has four children of ages $10$, $12$, $15$, and $17$. In fifteen years, the sum of the ages of his children will be twice Ted’s age in fifteen years. How old is Ted now?
[u]Set 2[/u]
[b]D.6[/b] Mr. Schwartz is on the show Wipeout, and is standing on the first of $5$ balls, all in a row. To reach the finish, he has to jump onto each of the balls and collect the prize on the final ball. The probability that he makes a jump from a ball to the next is $1/2$, and if he doesn’t make the jump he will wipe out and no longer be able to finish. Find the probability that he will finish.
[b]D.7 / L. 5[/b] Kevin has written $5$ MBMT questions. The shortest question is $5$ words long, and every other question has exactly twice as many words as a different question. Given that no two questions have the same number of words, how many words long is the longest question?
[b]D.8 / L. 3[/b] Square $ABCD$ with side length $1$ is rolled into a cylinder by attaching side $AD$ to side $BC$. What is the volume of that cylinder?
[b]D.9 / L.4[/b] Haydn is selling pies to Grace. He has $4$ pumpkin pies, $3$ apple pies, and $1$ blueberry pie. If Grace wants $3$ pies, how many different pie orders can she have?
[b]D.10[/b] Daniel has enough dough to make $8$ $12$-inch pizzas and $12$ $8$-inch pizzas. However, he only wants to make $10$-inch pizzas. At most how many $10$-inch pizzas can he make?
[u]Set 3[/u]
[b]D.11 / L.2[/b] A standard deck of cards contains $13$ cards of each suit (clubs, diamonds, hearts, and spades). After drawing $51$ cards from a randomly ordered deck, what is the probability that you have drawn an odd number of clubs?
[b]D.12 / L. 7[/b] Let $s(n)$ be the sum of the digits of $n$. Let $g(n)$ be the number of times s must be applied to n until it has only $1$ digit. Find the smallest n greater than $2019$ such that $g(n) \ne g(n + 1)$.
[b]D.13 / L. 8[/b] In the Montgomery Blair Meterology Tournament, individuals are ranked (without ties) in ten categories. Their overall score is their average rank, and the person with the lowest overall score wins. Alice, one of the $2019$ contestants, is secretly told that her score is $S$. Based on this information, she deduces that she has won first place, without tying with anyone. What is the maximum possible value of $S$?
[b]D.14 / L. 9[/b] Let $A$ and $B$ be opposite vertices on a cube with side length $1$, and let $X$ be a point on that cube. Given that the distance along the surface of the cube from $A$ to $X$ is $1$, find the maximum possible distance along the surface of the cube from $B$ to $X$.
[b]D.15[/b] A function $f$ with $f(2) > 0$ satisfies the identity $f(ab) = f(a) + f(b)$ for all $a, b > 0$. Compute $\frac{f(2^{2019})}{f(23)}$.
PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here [/url] and L10,16-30 [url=https://artofproblemsolving.com/community/c3h2790825p24541816]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Guts Rounds, 2017
[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide]
[u]Set 4[/u]
[b]R4.16 / P1.4[/b] Adam and Becky are building a house. Becky works twice as fast as Adam does, and they both work at constant speeds for the same amount of time each day. They plan to finish building in $6$ days. However, after $2$ days, their friend Charlie also helps with building the house. Because of this, they finish building in just $5$ days. What fraction of the house did Adam build?
[b]R4.17[/b] A bag with $10$ items contains both pencils and pens. Kanye randomly chooses two items from the bag, with replacement. Suppose the probability that he chooses $1$ pen and $1$ pencil is $\frac{21}{50}$ . What are all possible values for the number of pens in the bag?
[b]R4.18 / P2.8[/b] In cyclic quadrilateral $ABCD$, $\angle ABD = 40^o$, and $\angle DAC = 40^o$. Compute the measure of $\angle ADC$ in degrees. (In cyclic quadrilaterals, opposite angles sum up to $180^o$.)
[b]R4.19 / P2.6[/b] There is a strange random number generator which always returns a positive integer between $1$ and $7500$, inclusive. Half of the time, it returns a uniformly random positive integer multiple of $25$, and the other half of the time, it returns a uniformly random positive integer that isn’t a multiple of $25$. What is the probability that a number returned from the generator is a multiple of $30$?
[b]R4.20 / P2.7[/b] Julia is shopping for clothes. She finds $T$ different tops and $S$ different skirts that she likes, where $T \ge S > 0$. Julia can either get one top and one skirt, just one top, or just one skirt. If there are $50$ ways in which she can make her choice, what is $T - S$?
[u]Set 5[/u]
[b]R5.21[/b] A $5 \times 5 \times 5$ cube’s surface is completely painted blue. The cube is then completely split into $ 1 \times 1 \times 1$ cubes. What is the average number of blue faces on each $ 1 \times 1 \times 1$ cube?
[b]R5.22 / P2.10[/b] Find the number of values of $n$ such that a regular $n$-gon has interior angles with integer degree measures.
[b]R5.23[/b] $4$ positive integers form an geometric sequence. The sum of the $4$ numbers is $255$, and the average of the second and the fourth number is $102$. What is the smallest number in the sequence?
[b]R5.24[/b] Let $S$ be the set of all positive integers which have three digits when written in base $2016$ and two digits when written in base $2017$. Find the size of $S$.
[b]R5.25 / P3.12[/b] In square $ABCD$ with side length $13$, point $E$ lies on segment $CD$. Segment $AE$ divides $ABCD$ into triangle $ADE$ and quadrilateral $ABCE$. If the ratio of the area of $ADE$ to the area of $ABCE$ is $4 : 11$, what is the ratio of the perimeter of $ADE$ to the perimeter of $ABCE$?
[u]Set 6[/u]
[b]R6.26 / P6.25[/b] Submit a decimal n to the nearest thousandth between $0$ and $200$. Your score will be $\min (12, S)$, where $S$ is the non-negative difference between $n$ and the largest number less than or equal to $n$ chosen by another team (if you choose the smallest number, $S = n$). For example, 1.414 is an acceptable answer, while $\sqrt2$ and $1.4142$ are not.
[b]R6.27 / P6.27[/b] Guang is going hard on his YNA project. From $1:00$ AM Saturday to $1:00$ AM Sunday, the probability that he is not finished with his project $x$ hours after $1:00$ AM on Saturday is $\frac{1}{x+1}$ . If Guang does not finish by 1:00 AM on Sunday, he will stop procrastinating and finish the project immediately. Find the expected number of minutes $A$ it will take for him to finish his project.
An estimate of $E$ will earn $12 \cdot 2^{-|E-A|/60}$ points.
[b]R6.28 / P6.28[/b] All the diagonals of a regular $100$-gon (a regular polygon with $100$ sides) are drawn. Let $A$ be the number of distinct intersection points between all the diagonals. Find $A$.
An estimate of $E$ will earn $12 \cdot \left(16 \log_{10}\left(\max \left(\frac{E}{A},\frac{A}{E}\right)\right)+ 1\right)^{-\frac12}$ or $0$ points if this expression is undefined.
[b]R6.29 / P6.29 [/b]Find the smallest positive integer $A$ such that the following is true: if every integer $1, 2, ..., A$ is colored either red or blue, then no matter how they are colored, there are always 6 integers among them forming an increasing arithmetic progression that are all colored the same color.
An estimate of $E$ will earn $12 min \left(\frac{E}{A},\frac{A}{E}\right)$ points or $0$ points if this expression is undefined.
[b]R6.30 / P6.30[/b] For all integers $n \ge 2$, let $f(n)$ denote the smallest prime factor of $n$. Find $A =\sum^{10^6}_{n=2}f(n)$.
In other words, take the smallest prime factor of every integer from $2$ to $10^6$ and sum them all up to get $A$.
You may find the following values helpful: there are $78498$ primes below $10^6$, $9592$ primes below $10^5$, $1229$ primes below $10^4$, and $168$ primes below $10^3$.
An estimate of $E$ will earn $\max \left(0, 12-4 \log_{10}(max \left(\frac{E}{A},\frac{A}{E}\right)\right)$ or $0$ points if this expression is undefined.
PS. You should use hide for answers. R1-15 /P1-5 have been posted [url=https://artofproblemsolving.com/community/c3h2786721p24495629]here[/url], and P11-25 [url=https://artofproblemsolving.com/community/c3h2786880p24497350]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Team Rounds, 2023
[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide]
[b]B1[/b] What is the sum of the first $5$ positive integers?
[b]B2[/b] Bread picks a number $n$. He finds out that if he multiplies $n$ by $23$ and then subtracts $20$, he gets $46279$. What is $n$?
[b]B3[/b] A [i]Harshad [/i] Number is a number that is divisible by the sum of its digits. For example, $27$ is divisible by $2 + 7 = 9$. Only one two-digit multiple of $9$ is not a [i]Harshad [/i] Number. What is this number?
[b]B4 / G1[/b] There are $5$ red balls and 3 blue balls in a bag. Alice randomly picks a ball out of the bag and then puts it back in the bag. Bob then randomly picks a ball out of the bag. What is the probability that Alice gets a red ball and Bob gets a blue ball, assuming each ball is equally likely to be chosen?
[b]B5[/b] Let $a$ be a $1$-digit positive integer and $b$ be a $3$-digit positive integer. If the product of $a$ and $b$ is a$ 4$-digit integer, what is the minimum possible value of the sum of $a$ and $b$?
[b]B6 / G2[/b] A circle has radius $6$. A smaller circle with the same center has radius $5$. What is the probability that a dart randomly placed inside the outer circle is outside the inner circle?
[b]B7[/b] Call a two-digit integer “sus” if its digits sum to $10$. How many two-digit primes are sus?
[b]B8 / G3[/b] Alex and Jeff are playing against Max and Alan in a game of tractor with $2$ standard decks of $52$ cards. They take turns taking (and keeping) cards from the combined decks. At the end of the game, the $5$s are worth $5$ points, the $10$s are worth $10$ points, and the kings are worth 10 points. Given that a team needs $50$ percent more points than the other to win, what is the minimal score Alan and Max need to win?
[b]B9 / G4[/b] Bob has a sandwich in the shape of a rectangular prism. It has side lengths $10$, $5$, and $5$. He cuts the sandwich along the two diagonals of a face, resulting in four pieces. What is the volume of the largest piece?
[b]B10 / G5[/b] Aven makes a rectangular fence of area $96$ with side lengths $x$ and $y$. John makesva larger rectangular fence of area 186 with side lengths $x + 3$ and $y + 3$. What is the value of $x + y$?
[b]B11 / G6[/b] A number is prime if it is only divisible by itself and $1$. What is the largest prime number $n$ smaller than $1000$ such that $n + 2$ and $n - 2$ are also prime?
Note: $1$ is not prime.
[b]B12 / G7[/b] Sally has $3$ red socks, $1$ green sock, $2$ blue socks, and $4$ purple socks. What is the probability she will choose a pair of matching socks when only choosing $2$ socks without replacement?
[b]B13 / G8[/b] A triangle with vertices at $(0, 0)$,$ (3, 0)$, $(0, 6)$ is filled with as many $1 \times 1$ lattice squares as possible. How much of the triangle’s area is not filled in by the squares?
[b]B14 / G10[/b] A series of concentric circles $w_1, w_2, w_3, ...$ satisfy that the radius of $w_1 = 1$ and the radius of $w_n =\frac34$ times the radius of $w_{n-1}$. The regions enclosed in $w_{2n-1}$ but not in $w_{2n}$ are shaded for all integers $n > 0$. What is the total area of the shaded regions?
[b]B15 / G12[/b] $10$ cards labeled 1 through $10$ lie on a table. Kevin randomly takes $3$ cards and Patrick randomly takes 2 of the remaining $7$ cards. What is the probability that Kevin’s largest card is smaller than Patrick’s largest card, and that Kevin’s second-largest card is smaller than Patrick’s smallest card?
[b]G9[/b] Let $A$ and $B$ be digits. If $125A^2 + B161^2 = 11566946$. What is $A + B$?
[b]G11[/b] How many ordered pairs of integers $(x, y)$ satisfy $y^2 - xy + x = 0$?
[b]G13[/b] $N$ consecutive integers add to $27$. How many possible values are there for $N$?
[b]G14[/b] A circle with center O and radius $7$ is tangent to a pair of parallel lines $\ell_1$ and $\ell_2$. Let a third line tangent to circle $O$ intersect $\ell_1$ and $\ell_2$ at points $A$ and $B$. If $AB = 18$, find $OA + OB$.
[b]G15[/b] Let $$ M =\prod ^{42}_{i=0}(i^2 - 5).$$ Given that $43$ doesn’t divide $M$, what is the remainder when M is divided by $43$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Geometry Rounds, 2022
[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide]
[b]D1.[/b] A Giant Hopper is $200$ meters away from you. It can hop $50$ meters. How many hops would it take for it to reach you?
[b]D2.[/b] A rope of length $6$ is used to form the edges of an equilateral triangle (a triangle with equal side lengths). What is the length of one of these edges?
[b]D3 / Z1.[/b] Point $E$ is on side $AB$ of rectangle $ABCD$. Find the area of triangle $ECD$ divided by the area of rectangle $ABCD$.
[b]D4 / Z2.[/b] Garb and Grunt have two rectangular pastures of area $30$. Garb notices that his has a side length of $3$, while Grunt’s has a side length of $5$. What’s the positive difference between the perimeters of their pastures?
[b]D5.[/b] Let points $A$ and $B$ be on a circle with radius $6$ and center $O$. If $\angle AOB = 90^o$, find the area of triangle $AOB$.
[b]D6 / Z3.[/b] A scalene triangle (the $3$ side lengths are all different) has integer angle measures (in degrees). What is the largest possible difference between two angles in the triangle?
[b]D7.[/b] Square $ABCD$ has side length $6$. If triangle $ABE$ has area $9$, find the sum of all possible values of the distance from $E$ to line $CD$.
[b]D8 / Z4.[/b] Let point $E$ be on side $\overline{AB}$ of square $ABCD$ with side length $2$. Given $DE = BC+BE$, find $BE$.
[b]Z5.[/b] The two diagonals of rectangle $ABCD$ meet at point $E$. If $\angle AEB = 2\angle BEC$, and $BC = 1$, find the area of rectangle $ABCD$.
[b]Z6.[/b] In $\vartriangle ABC$, let $D$ be the foot of the altitude from $A$ to $BC$. Additionally, let $X$ be the intersection of the angle bisector of $\angle ACB$ and $AD$. If $BD = AC = 2AX = 6$, find the area of $ABC$.
[b]Z7.[/b] Let $\vartriangle ABC$ have $\angle ABC = 40^o$. Let $D$ and $E$ be on $\overline{AB}$ and $\overline{AC}$ respectively such that DE is parallel to $\overline{BC}$, and the circle passing through points $D$, $E$, and $C$ is tangent to $\overline{AB}$. If the center of the circle is $O$, find $\angle DOE$.
[b]Z8.[/b] Consider $\vartriangle ABC$ with $AB = 3$, $BC = 4$, and $AC = 5$. Let $D$ be a point of $AC$ other than $A$ for which $BD = 3$, and $E$ be a point on $BC$ such that $\angle BDE = 90^o$. Find $EC$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Team Rounds, 2020.44
Let $a_n =\sum_{d|n} \frac{1}{2^{d+ \frac{n}{d}}}$. In other words, $a_n$ is the sum of $\frac{1}{2^{d+ \frac{n}{d}}}$ over all divisors $d$ of $n$.
Find
$$\frac{\sum_{k=1} ^{\infty}ka_k}{\sum_{k=1}^{\infty} a_k} =\frac{a_1 + 2a_2 + 3a_3 + ....}{a_1 + a_2 + a_3 +....}$$
MBMT Team Rounds, 2018
[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide]
[b]C1.[/b] Mr. Pham flips $2018$ coins. What is the difference between the maximum and minimum number of heads that can appear?
[b]C2 / G1.[/b] Brandon wants to maximize $\frac{\Box}{\Box} +\Box$ by placing the numbers $1$, $2$, and $3$ in the boxes. If each number may only be used once, what is the maximum value attainable?
[b]C3.[/b] Guang has $10$ cents consisting of pennies, nickels, and dimes. What are all the possible numbers of pennies he could have?
[b]C4.[/b] The ninth edition of Campbell Biology has $1464$ pages. If Chris reads from the beginning of page $426$ to the end of page$449$, what fraction of the book has he read?
[b]C5 / G2.[/b] The planet Vriky is a sphere with radius $50$ meters. Kyerk starts at the North Pole, walks straight along the surface of the sphere towards the equator, runs one full circle around the equator, and returns to the North Pole. How many meters did Kyerk travel in total throughout his journey?
[b]C6 / G3.[/b] Mr. Pham is lazy and decides Stan’s quarter grade by randomly choosing an integer from $0$ to $100$ inclusive. However, according to school policy, if the quarter grade is less than or equal to $50$, then it is bumped up to $50$. What is the probability that Stan’s final quarter grade is $50$?
[b]C7 / G5.[/b] What is the maximum (finite) number of points of intersection between the boundaries of a equilateral triangle of side length $1$ and a square of side length $20$?
[b]C8.[/b] You enter the MBMT lottery, where contestants select three different integers from $1$ to $5$ (inclusive). The lottery randomly selects two winning numbers, and tickets that contain both of the winning numbers win. What is the probability that your ticket will win?
[b]C9 / G7.[/b] Find a possible solution $(B, E, T)$ to the equation $THE + MBMT = 2018$, where $T, H, E, M, B$ represent distinct digits from $0$ to $9$.
[b]C10.[/b] $ABCD$ is a unit square. Let $E$ be the midpoint of $AB$ and $F$ be the midpoint of $AD$. $DE$ and $CF$ meet at $G$. Find the area of $\vartriangle EFG$.
[b]C11.[/b] The eight numbers $2015$, $2016$, $2017$, $2018$, $2019$, $2020$, $2021$, and $2022$ are split into four groups of two such that the two numbers in each pair differ by a power of $2$. In how many different ways can this be done?
[b]C12 / G4.[/b] We define a function f such that for all integers $n, k, x$, we have that $$f(n, kx) = k^n f(n, x) and f(n + 1, x) = xf(n, x).$$ If $f(1, k) = 2k$ for all integers $k$, then what is $f(3, 7)$?
[b]C13 / G8.[/b] A sequence of positive integers is constructed such that each term is greater than the previous term, no term is a multiple of another term, and no digit is repeated in the entire sequence. An example of such a sequence would be $4$, $79$, $1035$. How long is the longest possible sequence that satisfies these rules?
[b]C14 / G11.[/b] $ABC$ is an equilateral triangle of side length $8$. $P$ is a point on side AB. If $AC +CP = 5 \cdot AP$, find $AP$.
[b]C15.[/b] What is the value of $(1) + (1 + 2) + (1 + 2 + 3) + ... + (1 + 2 + ... + 49 + 50)$?
[b]G6.[/b] An ant is on a coordinate plane. It starts at $(0, 0)$ and takes one step each second in the North, South, East, or West direction. After $5$ steps, what is the probability that the ant is at the point $(2, 1)$?
[b]G10.[/b] Find the set of real numbers $S$ so that $$\prod_{c\in S}(x^2 + cxy + y^2) = (x^2 - y^2)(x^{12} - y^{12}).$$
[b]G12.[/b] Given a function $f(x)$ such that $f(a + b) = f(a) + f(b) + 2ab$ and $f(3) = 0$, find $f\left( \frac12 \right)$.
[b]G13.[/b] Badville is a city on the infinite Cartesian plane. It has $24$ roads emanating from the origin, with an angle of $15$ degrees between each road. It also has beltways, which are circles centered at the origin with any integer radius. There are no other roads in Badville. Steven wants to get from $(10, 0)$ to $(3, 3)$. What is the minimum distance he can take, only going on roads?
[b]G14.[/b] Team $A$ and Team $B$ are playing basketball. Team A starts with the ball, and the ball alternates between the two teams. When a team has the ball, they have a $50\%$ chance of scoring $1$ point. Regardless of whether or not they score, the ball is given to the other team after they attempt to score. What is the probability that Team $A$ will score $5$ points before Team $B$ scores any?
[b]G15.[/b] The twelve-digit integer $$\overline{A58B3602C91D},$$ where $A, B, C, D$ are digits with $A > 0$, is divisible by $10101$. Find $\overline{ABCD}$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Guts Rounds, 2022
[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide]
[b]Z15.[/b] Let $AOB$ be a quarter circle with center $O$ and radius $4$. Let $\omega_1$ and $\omega_2$ be semicircles inside $AOB$ with diameters $OA$ and $OB$, respectively. Find the area of the region within $AOB$ but outside of $\omega_1$ and $\omega_2$.
[u]Set 4[/u]
[b]Z16.[/b] Integers $a, b, c$ form a geometric sequence with an integer common ratio. If $c = a + 56$, find $b$.
[b]Z17 / D24.[/b] In parallelogram $ABCD$, $\angle A \cdot \angle C - \angle B \cdot \angle D = 720^o$ where all angles are in degrees. Find the value of $\angle C$.
[b]Z18.[/b] Steven likes arranging his rocks. A mountain formation is where the sequence of rocks to the left of the tallest rock increase in height while the sequence of rocks to the right of the tallest rock decrease in height. If his rocks are $1, 2, . . . , 10$ inches in height, how many mountain formations are possible?
For example: the sequences $(1-3-5-6-10-9-8-7-4-2)$ and $(1-2-3-4-5-6-7-8-9-10)$ are considered mountain formations.
[b]Z19.[/b] Find the smallest $5$-digit multiple of $11$ whose sum of digits is $15$.
[b]Z20.[/b] Two circles, $\omega_1$ and $\omega_2$, have radii of $2$ and $8$, respectively, and are externally tangent at point $P$. Line $\ell$ is tangent to the two circles, intersecting $\omega_1$ at $A$ and $\omega_2$ at $B$. Line $m$ passes through $P$ and is tangent to both circles. If line $m$ intersects line $\ell$ at point $Q$, calculate the length of $P Q$.
[u]Set 5[/u]
[b]Z21.[/b] Sen picks a random $1$ million digit integer. Each digit of the integer is placed into a list. The probability that the last digit of the integer is strictly greater than twice the median of the digit list is closest to $\frac{1}{a}$, for some integer $a$. What is $a$?
[b]Z22.[/b] Let $6$ points be evenly spaced on a circle with center $O$, and let $S$ be a set of $7$ points: the $6$ points on the circle and $O$. How many equilateral polygons (not self-intersecting and not necessarily convex) can be formed using some subset of $S$ as vertices?
[b]Z23.[/b] For a positive integer $n$, define $r_n$ recursively as follows: $r_n = r^2_{n-1} + r^2_{n-2} + ... + r^2_0$,where $r_0 = 1$. Find the greatest integer less than $$\frac{r_2}{r^2_1}+\frac{r_3}{r^2_2}+ ...+\frac{r_{2023}}{r^2_{2022}}.$$
[b]Z24.[/b] Arnav starts at $21$ on the number line. Every minute, if he was at $n$, he randomly teleports to $2n^2$, $n^2$, or $\frac{n^2}{4}$ with equal chance. What is the probability that Arnav only ever steps on integers?
[b]Z25.[/b] Let $ABCD$ be a rectangle inscribed in circle $\omega$ with $AB = 10$. If $P$ is the intersection of the tangents to $\omega$ at $C$ and $D$, what is the minimum distance from $P$ to $AB$?
PS. You should use hide for answers. D.1-15 / Z.1-8 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916240p26045561]here [/url]and D.16-30/Z.9-14, 17, 26-30 [url=https://artofproblemsolving.com/community/c3h2916250p26045695]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Geometry Rounds, 2018
[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide]
[b]C1.[/b] A circle has circumference $6\pi$. Find the area of this circle.
[b]C2 / G2.[/b] Points $A$, $B$, and $C$ are on a line such that $AB = 6$ and $BC = 11$. Find all possible values of $AC$.
[b]C3.[/b] A trapezoid has area $84$ and one base of length $5$. If the height is $12$, what is the length of the other base?
[b]C4 / G1.[/b] $27$ cubes of side length 1 are arranged to form a $3 \times 3 \times 3$ cube. If the corner $1 \times 1 \times 1$ cubes are removed, what fraction of the volume of the big cube is left?
[b]C5.[/b] There is a $50$-foot tall wall and a $300$-foot tall guard tower $50$ feet from the wall. What is the minimum $a$ such that a flat “$X$” drawn on the ground $a$ feet from the side of the wall opposite the guard tower is visible from the top of the guard tower?
[b]C6.[/b] Steven’s pizzeria makes pizzas in the shape of equilateral triangles. If a pizza with side length 8 inches will feed 2 people, how many people will a pizza of side length of 16 inches feed?
[b]C7 / G3.[/b] Consider rectangle $ABCD$, with $1 = AB < BC$. The angle bisector of $\angle DAB$ intersects $\overline{BC}$ at $E$ and $\overline{DC}$ at $F$. If $FE = FD$, find $BC$.
[b]C8 / G6.[/b] $\vartriangle ABC$. is a right triangle with $\angle A = 90^o$. Square $ADEF$ is drawn, with $D$ on $\overline{AB}$, $F$ on $\overline{AC}$, and $E$ inside $\vartriangle ABC$. Point $G$ is chosen on $\overline{BC}$ such that $EG$ is perpendicular to $BC$. Additionally, $DE = EG$. Given that $\angle C = 20^o$, find the measure of $\angle BEG$.
[b]G4.[/b] Consider a lamp in the shape of a hollow cylinder with the circular faces removed with height $48$ cm and radius $7$ cm. A point source of light is situated at the center of the lamp. The lamp is held so that the bottom of the lamp is at a height $48$ cm above an infinite flat sheet of paper. What is the area of the illuminated region on the flat sheet of paper, in $cm^2$?
[img]https://cdn.artofproblemsolving.com/attachments/c/6/6e5497a67ae5ff5a7bff7007834a4271ce3ca7.png[/img]
[b]G5.[/b] There exist two triangles $ABC$ such that $AB = 13$, $BC = 12\sqrt2$, and $\angle C = 45^o$. Find the positive difference between their areas.
[b]G7.[/b] Let $ABC$ be an equilateral triangle with side length $2$. Let the circle with diameter $AB$ be $\Gamma$. Consider the two tangents from $C$ to $\Gamma$, and let the tangency point closer to $A$ be $D$. Find the area of $\angle CAD$.
[b]G8.[/b] Let $ABC$ be a triangle with $\angle A = 60^o$, $AB = 37$, $AC = 41$. Let $H$ and $O$ be the orthocenter and circumcenter of $ABC$, respectively. Find $OH$.
[i]The orthocenter of a triangle is the intersection point of the three altitudes. The circumcenter of a triangle is the intersection point of the three perpendicular bisectors of the sides.[/i]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Team Rounds, 2020.41
What are the last two digits of $$2^{3^{4^{...^{2019}}}} ?$$
MBMT Guts Rounds, 2022
[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide]
[u]Set 4[/u]
[b]D16.[/b] The cooking club at Blair creates $14$ croissants and $21$ danishes. Daniel chooses pastries randomly, stopping when he gets at least one croissant and at least two danishes. How many pastries must he choose to guarantee that he has one croissant and two danishes?
[b]D17.[/b] Each digit in a $3$ digit integer is either $1, 2$, or $4$ with equal probability. What is the probability that the hundreds digit is greater than the sum of the tens digit and the ones digit?
[b]D18 / Z11.[/b] How many two digit numbers are there such that the product of their digits is prime?
[b]D19 / Z9.[/b] In the coordinate plane, a point is selected in the rectangle defined by $-6 \le x \le 4$ and $-2 \le y \le 8$. What is the largest possible distance between the point and the origin, $(0, 0)$?
[b]D20 / Z10.[/b] The sum of two numbers is $6$ and the sum of their squares is $32$. Find the product of the two numbers.
[u]Set 5[/u]
[b]D21 / Z12.[/b] Triangle $ABC$ has area $4$ and $\overline{AB} = 4$. What is the maximum possible value of $\angle ACB$?
[b]D22 / Z13.[/b] Let $ABCD$ be an iscoceles trapezoid with $AB = CD$ and M be the midpoint of $AD$. If $\vartriangle ABM$ and $\vartriangle MCD$ are equilateral, and $BC = 4$, find the area of trapezoid $ABCD$.
[b]D23 / Z14.[/b] Let $x$ and $y$ be positive real numbers that satisfy $(x^2 + y^2)^2 = y^2$. Find the maximum possible value of $x$.
[b]D24 / Z17.[/b] In parallelogram $ABCD$, $\angle A \cdot \angle C - \angle B \cdot \angle D = 720^o$ where all angles are in degrees. Find the value of $\angle C$.
[b]D25.[/b] The number $12ab9876543$ is divisible by $101$, where $a, b$ represent digits between $0$ and $9$. What is $10a + b$?
[u]Set 6[/u]
[b]D26 / Z26.[/b] For every person who wrote a problem that appeared on the final MBMT tests, take the number of problems they wrote, and then take that number’s factorial, and finally multiply all these together to get $n$. Estimate the greatest integer $a$ such that $2^a$ evenly divides $n$.
[b]D27 / Z27.[/b] Circles of radius $5$ are centered at each corner of a square with side length $6$. If a random point $P$ is chosen randomly inside the square, what is the probability that $P$ lies within all four circles?
[b]D28 / Z28.[/b] Mr. Rose’s evil cousin, Mr. Caulem, has teaches a class of three hundred bees. Every week, he tries to disrupt Mr. Rose’s $4$th period by sending three of his bee students to fly around and make human students panic. Unfortunately, no pair of bees can fly together twice, as then Mr. Rose will become suspicious and trace them back to Mr. Caulem. What’s the largest number of weeks Mr. Caulem can disrupt Mr. Rose’s class?
[b]D29 / Z29. [/b]Two blind brothers Beard and Bored are driving their tractors in the middle of a field facing north, and both are $10$ meters west from a roast turkey. Beard, can turn exactly $0.7^o$ and Bored can turn exactly $0.2^o$ degrees. Driving at a consistent $2$ meters per second, they drive straight until they notice the smell of the turkey getting farther away, and then turn right and repeat until they get to the turkey.
Suppose Beard gets to the Turkey in about $818.5$ seconds. Estimate the amount of time it will take Bored.
[b]D30 / Z30.[/b] Let a be the probability that $4$ randomly chosen positive integers have no common divisor except for $1$. Estimate $300a$. Note that the integers $1, 2, 3, 4$ have no common divisor except for $1$.
Remark. This problem is asking you to find $300 \lim_{n\to \infty} a_n$, if $a_n$ is defined to be the probability that $4$ randomly chosen integers from $\{1, 2, ..., n\}$ have greatest common divisor $1$.
PS. You should use hide for answers. D.1-15 / Z.1-8 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916240p26045561]here [/url]and Z.15-25 [url=https://artofproblemsolving.com/community/c3h2916258p26045774]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Geometry Rounds, 2015
[hide=F stands for Fermat, E stands for Euler]they had two problem sets under those two names[/hide]
[b]F1.[/b] A circle has area $\pi$. Find the circumference of the circle.
[b]F2.[/b] In triangle $ABC$, $AB = 5$, $BC = 12$, and $\angle B = 90^o$. Compute $AC$.
[b]F3 / E1.[/b] A square has area $2015$. Find the length of the square's diagonal.
[b]F4.[/b] I have two cylindrical candles. The first candle has diameter $1$ and height $1$. The second candle has diameter $2$ and height $2$. Both candles are lit at $1:00$ PM and both burn at the same constant rate (volume per time period). The first candle burns out at $1:50$ PM. When does the second candle burn out? Specify AM or PM.
[b]F5 / E2.[/b] In triangle $ABC$, $BC$ has length $12$, the altitude from $A$ to $BC$ has length $6$, and the altitude from $C$ to $AB$ has length $8$. Compute the length of $AB$.
[b]F6 / E3.[/b] Let $ABC$ be an isosceles triangle with base $AC$. Suppose that there exists a point $D$ on side $AB$ such that $AC = CD = BD$. Find the degree measure of $\angle ABC$.
[b]F7 / E6.[/b] In concave quadrilateral $ABCD$, $\angle ABC = 60^o$ and $\angle ADC = 240^o$. If $AD = CD = 4$, compute $BD$.
[b]F8 / E7.[/b] A circle of radius $5$ is inscribed in an isosceles trapezoid with legs of length $14$. Compute the area of the trapezoid.
[b]E4.[/b] The Egyptian goddess Isil has a staff consisting of a pole with a circle on top. The length of the pole is $32$ inches, and the tangent segment from the bottom of the pole to the circle is $40$ inches. Find the radius of the circle, in inches.
[img]https://cdn.artofproblemsolving.com/attachments/5/b/01ea1819aa58c4bde105b9b885f658b3823494.png[/img]
[b]E5.[/b] The two concentric circles shown below have radii $1$ and $2$. A chord of the larger circle that is tangent to the smaller circle is drawn. Find the area of the shaded region, bounded by the chord and the larger circle.
[img]https://cdn.artofproblemsolving.com/attachments/e/5/425735d2717548552fda8363c201dc8043da13.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Guts Rounds, 2017
[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide]
[u] Set 1[/u]
[b]R1.1 / P1.1[/b] Find $291 + 503 - 91 + 492 - 103 - 392$.
[b]R1.2[/b] Let the operation $a$ & $b$ be defined to be $\frac{a-b}{a+b}$. What is $3$ & $-2$?
[b]R1.3[/b]. Joe can trade $5$ apples for $3$ oranges, and trade $6$ oranges for $5$ bananas. If he has $20$ apples, what is the largest number of bananas he can trade for?
[b]R1.4[/b] A cone has a base with radius $3$ and a height of $5$. What is its volume? Express your answer in terms of $\pi$.
[b]R1.5[/b] Guang brought dumplings to school for lunch, but by the time his lunch period comes around, he only has two dumplings left! He tries to remember what happened to the dumplings. He first traded $\frac34$ of his dumplings for Arman’s samosas, then he gave $3$ dumplings to Anish, and lastly he gave David $\frac12$ of the dumplings he had left. How many dumplings did Guang bring to school?
[u]Set 2[/u]
[b]R2.6 / P1.3[/b] In the recording studio, Kanye has $10$ different beats, $9$ different manuscripts, and 8 different samples. If he must choose $1$ beat, $1$ manuscript, and $1$ sample for his new song, how many selections can he make?
[b]R2.7[/b] How many lines of symmetry does a regular dodecagon (a polygon with $12$ sides) have?
[b]R2.8[/b] Let there be numbers $a, b, c$ such that $ab = 3$ and $abc = 9$. What is the value of $c$?
[b]R2.9[/b] How many odd composite numbers are there between $1$ and $20$?
[b]R2.10[/b] Consider the line given by the equation $3x - 5y = 2$. David is looking at another line of the form ax - 15y = 5, where a is a real number. What is the value of a such that the two lines do not intersect at any point?
[u]Set 3[/u]
[b]R3.11[/b] Let $ABCD$ be a rectangle such that $AB = 4$ and $BC = 3$. What is the length of BD?
[b]R3.12[/b] Daniel is walking at a constant rate on a $100$-meter long moving walkway. The walkway moves at $3$ m/s. If it takes Daniel $20$ seconds to traverse the walkway, find his walking speed (excluding the speed of the walkway) in m/s.
[b]R3.13 / P1.3[/b] Pratik has a $6$ sided die with the numbers $1, 2, 3, 4, 6$, and $12$ on the faces. He rolls the die twice and records the two numbers that turn up on top. What is the probability that the product of the two numbers is less than or equal to $12$?
[b]R3.14 / P1.5[/b] Find the two-digit number such that the sum of its digits is twice the product of its digits.
[b]R3.15[/b] If $a^2 + 2a = 120$, what is the value of $2a^2 + 4a + 1$?
PS. You should use hide for answers. R16-30 /P6-10/ P26-30 have been posted [url=https://artofproblemsolving.com/community/c3h2786837p24497019]here[/url], and P11-25 [url=https://artofproblemsolving.com/community/c3h2786880p24497350]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Guts Rounds, 2018
[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide]
[u]Set 4[/u]
[b]G.16[/b] A number $k$ is the product of exactly three distinct primes (in other words, it is of the form $pqr$, where $p, q, r$ are distinct primes). If the average of its factors is $66$, find $k$.
[b]G.17[/b] Find the number of lattice points contained on or within the graph of $\frac{x^2}{3} +\frac{y^2}{2}= 12$. Lattice points are coordinate points $(x, y)$ where $x$ and $y$ are integers.
[b]G.18 / C.23[/b] How many triangles can be made from the vertices and center of a regular hexagon? Two congruent triangles with different orientations are considered distinct.
[b]G.19[/b] Cindy has a cone with height $15$ inches and diameter $16$ inches. She paints one-inch thick bands of paint in circles around the cone, alternating between red and blue bands, until the whole cone is covered with paint. If she starts from the bottom of the cone with a blue strip, what is the ratio of the area of the cone covered by red paint to the area of the cone covered by blue paint?
[b]G.20 / C.25[/b] An even positive integer $n$ has an odd factorization if the largest odd divisor of $n$ is also the smallest odd divisor of n greater than 1. Compute the number of even integers $n$ less than $50$ with an odd factorization.
[u] Set 5[/u]
[b]G.21[/b] In the magical tree of numbers, $n$ is directly connected to $2n$ and $2n + 1$ for all nonnegative integers n. A frog on the magical tree of numbers can move from a number $n$ to a number connected to it in $1$ hop. What is the least number of hops that the frog can take to move from $1000$ to $2018$?
[b]G.22[/b] Stan makes a deal with Jeff. Stan is given 1 dollar, and every day for $10$ days he must either double his money or burn a perfect square amount of money. At first Stan thinks he has made an easy $1024$ dollars, but then he learns the catch - after $10$ days, the amount of money he has must be a multiple of $11$ or he loses all his money. What is the largest amount of money Stan can have after the $10$ days are up?
[b]G.23[/b] Let $\Gamma_1$ be a circle with diameter $2$ and center $O_1$ and let $\Gamma_2$ be a congruent circle centered at a point $O_2 \in \Gamma_1$. Suppose $\Gamma_1$ and $\Gamma_2$ intersect at $A$ and $B$. Let $\Omega$ be a circle centered at $A$ passing through $B$. Let $P$ be the intersection of $\Omega$ and $\Gamma_1$ other than $B$ and let $Q$ be the intersection of $\Omega$ and ray $\overrightarrow{AO_1}$. Define $R$ to be the intersection of $PQ$ with $\Gamma_1$. Compute the length of $O_2R$.
[b]G.24[/b] $8$ people are at a party. Each person gives one present to one other person such that everybody gets a present and no two people exchange presents with each other. How many ways is this possible?
[b]G.25[/b] Let $S$ be the set of points $(x, y)$ such that $y = x^3 - 5x$ and $x = y^3 - 5y$. There exist four points in $S$ that are the vertices of a rectangle. Find the area of this rectangle.
PS. You should use hide for answers. C1-15/ G1-10 have been posted [url=https://artofproblemsolving.com/community/c3h2790674p24540132]here [/url] and C16-30/G10-15, G25-30 [url=https://artofproblemsolving.com/community/c3h2790676p24540145]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]
MBMT Guts Rounds, 2018
[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide]
[u] Set 4[/u]
[b]C.16 / G.6[/b] Let $a, b$, and $c$ be real numbers. If $a^3 + b^3 + c^3 = 64$ and $a + b = 0$, what is the value of $c$?
[b]C.17 / G.8[/b] Bender always turns $60$ degrees clockwise. He walks $3$ meters, turns, walks $2$ meters, turns, walks $1$ meter, turns, walks $4$ meters, turns, walks $1$ meter, and turns. How many meters does Bender have to walk to get back to his original position?
[b]C.18 / G.13[/b] Guang has $4$ identical packs of gummies, and each pack has a red, a blue, and a green gummy. He eats all the gummies so that he finishes one pack before going on to the next pack, but he never eats two gummies of the same color in a row. How many different ways can Guang eat the gummies?
[b]C.19[/b] Find the sum of all digits $q$ such that there exists a perfect square that ends in $q$.
[b]C.20 / G.14[/b] The numbers $5$ and $7$ are written on a whiteboard. Every minute Stev replaces the two numbers on the board with their sum and difference. After $2017$ minutes the product of the numbers on the board is $m$. Find the number of factors of $m$.
[u]Set 5[/u]
[b]C.21 / G.10[/b] On the planet Alletas, $\frac{32}{33}$ of the people with silver hair have purple eyes and $\frac{8}{11}$ of the people with purple eyes have silver hair. On Alletas, what is the ratio of the number of people with purple eyes to the number of people with silver hair?
[b]C.22 / G.15[/b] Let $P$ be a point on $y = -1$. Let the clockwise rotation of $P$ by $60^o$ about $(0, 0)$ be $P'$. Find the minimum possible distance between $P'$ and $(0, -1)$.
[b]C.23 / G.18[/b] How many triangles can be made from the vertices and center of a regular hexagon? Two congruent triangles with different orientations are considered distinct.
[b]C.24[/b] Jeremy and Kevin are arguing about how cool a sweater is on a scale of $1-5$. Jeremy says “$3$”, and Kevin says “$4$”. Jeremy angrily responds “$3.5$”, to which Kevin replies “$3.75$”. The two keep going at it, responding with the average of the previous two ratings. What rating will they converge to (and settle on as the coolness of the sweater)?
[b]C.25 / G.20[/b] An even positive integer $n$ has an [i]odd factorization[/i] if the largest odd divisor of $n$ is also the smallest odd divisor of $n$ greater than $1$. Compute the number of even integers $n$ less than $50$ with an odd factorization.
[u]Set 6[/u]
[b]C.26 / G.26[/b] When $2018! = 2018 \times 2017 \times ... \times 1$ is multiplied out and written as an integer, find the number of $4$’s.
If the correct answer is $A$ and your answer is $E$, you will receive $12 \min\, \, (A/E, E/A)^3$points.
[b]C.27 / G.27[/b] A circle of radius $10$ is cut into three pieces of equal area with two parallel cuts. Find the width of the center piece.
[img]https://cdn.artofproblemsolving.com/attachments/e/2/e0ab4a2d51052ee364dd14336677b053a40352.png[/img]
If the correct answer is $A$ and your answer is $E$, you will receive $\max \, \,(0, 12 - 6|A - E|)$points.
[b]C.28 / G.28[/b] An equilateral triangle of side length $1$ is randomly thrown onto an infinite set of lines, spaced $1$ apart. On average, how many times will the boundary of the triangle intersect one of the lines?
[img]https://cdn.artofproblemsolving.com/attachments/0/1/773c3d3e0dfc1df54945824e822feaa9c07eb7.png[/img]
For example, in the above diagram, the boundary of the triangle intersects the lines in $2$ places.
If the correct answer is $A$ and your answer is $E$, you will receive $\max\, \,(0, 12-120|A-E|/A)$ points.
[b]C.29 / G.29[/b] Call an ordered triple of integers $(a, b, c)$ nice if there exists an integer $x$ such that $ax^2 + bx + c = 0$. How many nice triples are there such that $-100 \le a, b, c \le 100$?
If the correct answer is $A$ and your answer is $E$, you will receive $12 \min\, \,(A/E, E/A)$ points.
[b]C.30 / G.30[/b] Let $f(i)$ denote the number of MBMT volunteers to be born in the $i$th state to join the United States. Find the value of $1f(1) + 2f(2) + 3f(3) + ... + 50f(50)$.
Note 1: Maryland was the $7$th state to join the US.
Note 2: Last year’s MBMT competition had $42$ volunteers.
If the correct answer is $A$ and your answer is $E$, you will receive $\max\, \,(0, 12 - 500(|A -E|/A)^2)$ points.
PS. You should use hide for answers. C1-15/ G1-10 have been posted [url=https://artofproblemsolving.com/community/c3h2790674p24540132]here [/url] and G16-25 [url=https://artofproblemsolving.com/community/c3h2790679p24540159]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Team Rounds, 2017
[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide]
[b]R1.[/b] What is $11^2 - 9^2$?
[b]R2.[/b] Write $\frac{9}{15}$ as a decimal.
[b]R3.[/b] A $90^o$ sector of a circle is shaded, as shown below. What percent of the circle is shaded?
[b]R4.[/b] A fair coin is flipped twice. What is the probability that the results of the two flips are different?
[b]R5.[/b] Wayne Dodson has $55$ pounds of tungsten. If each ounce of tungsten is worth $75$ cents, and there are $16$ ounces in a pound, how much money, in dollars, is Wayne Dodson’s tungsten worth?
[b]R6.[/b] Tenley Towne has a collection of $28$ sticks. With these $28$ sticks he can build a tower that has $1$ stick in the top row, $2$ in the next row, and so on. Let $n$ be the largest number of rows that Tenley Towne’s tower can have. What is n?
[b]R7.[/b] What is the sum of the four smallest primes?
[b]R8 / P1.[/b] Let $ABC$ be an isosceles triangle such that $\angle B = 42^o$. What is the sum of all possible degree measures of angle $A$?
[b]R9.[/b] Consider a line passing through $(0, 0)$ and $(4, 8)$. This line passes through the point $(2, a)$. What is the value of $a$?
[b]R10 / P2.[/b] Brian and Stan are playing a game. In this game, Brian rolls a fair six-sided die, while Stan rolls a fair four-sided die. Neither person shows the other what number they rolled. Brian tells Stan, “The number I rolled is guaranteed to be higher than the number you rolled.” Stan now has to guess Brian’s number. If Stan plays optimally, what is the probability that Stan correctly guesses the number that Brian rolled?
[b]R11.[/b] Guang chooses $4$ distinct integers between $0$ and $9$, inclusive. How many ways can he choose the integers such that every pair of chosen integers sums up to an even number?
[b]R12 / P4.[/b] David is trying to write a problem for MBMT. He assigns degree measures to every interior angle in a convex $n$-gon, and it so happens that every angle he assigned is less than $144$ degrees. He tells Pratik the value of $n$ and the degree measures in the $n$-gon, and to David’s dismay, Pratik claims that such an $n$-gon does not exist. What is the smallest value of $n \ge 3$ such that Pratik’s claim is necessarily true?
[b]R13 / P3.[/b] Consider a triangle $ABC$ with side lengths of $5$, $5$, and $2\sqrt5$. There exists a triangle with side lengths of $5, 5$, and $x$ ($x \ne 2\sqrt5$) which has the same area as $ABC$. What is the value of $x$?
[b]R14 / P5.[/b] A mother has $11$ identical apples and $9$ identical bananas to distribute among her $3$ kids. In how many ways can the fruits be allocated so that each child gets at least one apple and one banana?
[b]R15 / P7.[/b] Find the sum of the five smallest positive integers that cannot be represented as the sum of two not necessarily distinct primes.
[b]P6.[/b] Srinivasa Ramanujan has the polynomial $P(x) = x^5 - 3x^4 - 5x^3 + 15x^2 + 4x - 12$. His friend Hardy tells him that $3$ is one of the roots of $P(x)$. What is the sum of the other roots of $P(x)$?
[b]P8.[/b] $ABC$ is an equilateral triangle with side length $10$. Let $P$ be a point which lies on ray $\overrightarrow{BC}$ such that $PB = 20$. Compute the ratio $\frac{PA}{PC}$.
[b]P9.[/b] Let $ABC$ be a triangle such that $AB = 10$, $BC = 14$, and $AC = 6$. The median $CD$ and angle bisector $CE$ are both drawn to side $AB$. What is the ratio of the area of triangle $CDE$ to the area of triangle $ABC$?
[b]P10.[/b] Find all integer values of $x$ between $0$ and $2017$ inclusive, which satisfy $$2016x^{2017} + 990x^{2016} + 2x + 17 \equiv 0 \,\,\, (mod \,\,\, 2017).$$
[b]P11.[/b] Let $x^2 + ax + b$ be a quadratic polynomial with positive integer roots such that $a^2 - 2b = 97$. Compute $a + b$.
[b]P12.[/b] Let $S$ be the set $\{2, 3, ... , 14\}$. We assign a distinct number from $S$ to each side of a six-sided die. We say a numbering is predictable if prime numbers are always opposite prime numbers and composite numbers are always opposite composite numbers. How many predictable numberings are there? (Rotations of a die are not distinct)
[b]P13.[/b] In triangle $ABC$, $AB = 10$, $BC = 21$, and $AC = 17$. $D$ is the foot of the altitude from $A$ to $BC$, $E$ is the foot of the altitude from $D$ to $AB$, and $F$ is the foot of the altitude from $D$ to $AC$. Find the area of the smallest circle that contains the quadrilateral $AEDF$.
[b]P14.[/b] What is the greatest distance between any two points on the graph of $3x^2 + 4y^2 + z^2 - 12x + 8y + 6z = -11$?
[b]P15.[/b] For a positive integer $n$, $\tau (n)$ is defined to be the number of positive divisors of $n$. Given this information, find the largest positive integer $n$ less than $1000$ such that $$\sum_{d|n} \tau (d) = 108.$$ In other words, we take the sum of $\tau (d)$ for every positive divisor $d$ of $n$, which has to be $108$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
MBMT Team Rounds, 2020.45
In the Flatland Congress there are senators who are on committees. Each senator is on at least one committee, and each committee has at least one senator. The rules for forming committees are as follows:
$\bullet$ For any pair of senators, there is exactly one committee which contains both senators.
$\bullet$ For any two committees, there is exactly one senator who is on both committees.
$\bullet$ There exist a set of four senators, no three of whom are all on the same committee.
$\bullet$ There exists a committee with exactly $6$ senators.
If there are at least $25$ senators in this Congress, compute the minimum possible number of senators $s$ and minimum number of committees $c$ in this Congress. Express your answer in the form $(s, c)$.