Found problems: 186
Let $n$ be a positive integer. For any $k$, denote by $a_k$ the number of permutations of $\{1,2,\dots,n\}$ with exactly $k$ disjoint cycles. (For example, if $n=3$ then $a_2=3$ since $(1)(23)$, $(2)(31)$, $(3)(12)$ are the only such permutations.) Evaluate
\[ a_n n^n + a_{n-1} n^{n-1} + \dots + a_1 n. \][i]Proposed by Sammy Luo[/i]
For any positive integer n, evaluate $$\sum_{i=0}^{\lfloor \frac{n+1}{2} \rfloor}
{n-i+1 \choose i}$$
, where $\lfloor n \rfloor$ is the greatest integer less than or equal to $n$ .
Consider the paths of length $16$ that go from the lower left corner to the upper right corner of an $8\times 8$ grid. Find the number of such paths that change direction exactly $4$ times.
Find the number of all $ 6$-digit natural numbers such that the sum of their digits is $ 10$ and each of the digits $ 0,1,2,3$ occurs at least once in them.
[14 points out of 100 for the 6 problems]
[i]25 problems for 30 minutes[/i]
[b]p1.[/b] Compute $2^2 + 0 \cdot 0 + 2^2 + 3^3$.
[b]p2.[/b] How many total letters (not necessarily distinct) are there in the names Jerry, Justin, Jackie, Jason, and Jeffrey?
[b]p3.[/b] What is the remainder when $20232023$ is divided by $50$?
[b]p4.[/b] Let $ABCD$ be a square. The fraction of the area of $ABCD$ that is the area of the intersection of triangles $ABD$ and $ABC$ can be expressed as $\frac{a}{b}$ , where $a$ and $b$ relatively prime positive integers. Find $a + b$.
[b]p5.[/b] Raymond is playing basketball. He makes a total of $15$ shots, all of which are either worth $2$ or $3$ points. Given he scored a total of $40$ points, how many $2$-point shots did he make?
[b]p6.[/b] If a fair coin is flipped $4$ times, the probability that it lands on heads more often than tails is $\frac{a}{b}$ , where $a$ and $b$ relatively prime positive integers. Find $a + b$.
[b]p7.[/b] What is the sum of the perfect square divisors of $640$?
[b]p8.[/b] A regular hexagon and an equilateral triangle have the same perimeter. The ratio of the area between the hexagon and equilateral triangle can be expressed in the form $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$.
[b]p9.[/b] If a cylinder has volume $1024\pi$, radius of $r$ and height $h$, how many ordered pairs of integers $(r, h)$ are possible?
[b]p10.[/b] Pump $A$ can fill up a balloon in $3$ hours, while pump $B$ can fill up a balloon in $5$ hours. Pump $A$ starts filling up a balloon at $12:00$ PM, and pump $B$ is added alongside pump $A$ at a later time. If the balloon is completely filled at $2:00$ PM, how many minutes after $12:00$ PM was Pump $B$ added?
[b]p11.[/b] For some positive integer $k$, the product $81 \cdot k$ has $20$ factors. Find the smallest possible value of $k$.
[b]p12.[/b] Two people wish to sit in a row of fifty chairs. How many ways can they sit in the chairs if they do not want to sit directly next to each other and they do not want to sit with exactly one empty chair between them?
[b]p13.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $2$ and $M$ be the midpoint of $BC$. Let $P$ be a point in the same plane such that $2PM = BC$. The minimum value of $AP$ can be expressed as $\sqrt{a}-b$, where $a$ and $b$ are positive integers such that $a$ is not divisible by any perfect square aside from $1$. Find $a + b$.
[b]p14.[/b] What are the $2022$nd to $2024$th digits after the decimal point in the decimal expansion of $\frac{1}{27}$ , expressed as a $3$ digit number in that order (i.e the $2022$nd digit is the hundreds digit, $2023$rd digit is the tens digit, and $2024$th digit is the ones digit)?
[b]p15.[/b] After combining like terms, how many terms are in the expansion of $(xyz+xy+yz+xz+x+y+z)^{20}$?
[b]p16.[/b] Let $ABCD$ be a trapezoid with $AB \parallel CD$ where $AB > CD$, $\angle B = 90^o$, and $BC = 12$. A line $k$ is dropped from $A$, perpendicular to line $CD$, and another line $\ell$ is dropped from $C$, perpendicular to line $AD$. $k$ and $\ell$ intersect at $X$. If $\vartriangle AXC$ is an equilateral triangle, the area of $ABCD$ can be written as $m\sqrt{n}$, where $m$ and $n$ are positive integers such that $n$ is not divisible by any perfect square aside from $1$. Find $m + n$.
[b]p17.[/b] If real numbers $x$ and $y$ satisfy $2x^2 + y^2 = 8x$, maximize the expression $x^2 + y^2 + 4x$.
[b]p18.[/b] Let $f(x)$ be a monic quadratic polynomial with nonzero real coefficients. Given that the minimum value of $f(x)$ is one of the roots of $f(x)$, and that $f(2022) = 1$, there are two possible values of $f(2023)$. Find the larger of these two values.
[b]p19.[/b] I am thinking of a positive integer. After realizing that it is four more than a multiple of $3$, four less than a multiple of $4$, four more than a multiple of 5, and four less than a multiple of $7$, I forgot my number. What is the smallest possible value of my number?
[b]p20.[/b] How many ways can Aston, Bryan, Cindy, Daniel, and Evan occupy a row of $14$ chairs such that none of them are sitting next to each other?
[b]p21.[/b] Let $x$ be a positive real number. The minimum value of $\frac{1}{x^2} +\sqrt{x}$ can be expressed in the form \frac{a}{b^{(c/d)}} , where $a$, $b$, $c$, $d$ are all positive integers, $a$ and $b$ are relatively prime, $c$ and $d$ are relatively prime, and $b$ is not divisible by any perfect square. Find $a + b + c + d$.
[b]p22.[/b] For all $x > 0$, the function $f(x)$ is defined as $\lfloor x \rfloor \cdot (x + \{x\})$. There are $24$ possible $x$ such that $f(x)$ is an integer between $2000$ and $2023$, inclusive. If the sum of these $24$ numbers equals $N$, then find $\lfloor N \rfloor$.
Note: Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$, called the floor function. Also, $\{x\}$ is defined as $x - \lfloor x \rfloor$, called the fractional part function.
[b]p23.[/b] Let $ABCD$ be a rectangle with $AD = 1$. Let $P$ be a point on diagonal $\overline{AC}$, and let $\omega$ and $\xi$ be the circumcircles of $\vartriangle APB$ and $\vartriangle CPD$, respectively. Line $\overleftrightarrow{AD}$ is extended, intersecting $\omega$ at $X$, and $\xi$ at $Y$ . If $AX = 5$ and $DY = 2$, find $[ABCD]^2$.
Note: $[ABCD]$ denotes the area of the polygon $ABCD$.
[b]p24.[/b] Alice writes all of the three-digit numbers on a blackboard (it’s a pretty big blackboard). Let $X_a$ be the set of three-digit numbers containing a somewhere in its representation, where a is a string of digits. (For example, $X_{12}$ would include $12$, $121$, $312$, etc.) If Bob then picks a value of $a$ at random so $0 \le a \le 999$, the expected number of elements in $X_a$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find$ m + n$.
[b]p25.[/b] Let $f(x) = x^5 + 2x^4 - 2x^3 + 4x^2 + 5x + 6$ and $g(x) = x^4 - x^3 + x^2 - x + 1$. If $a$, $b$, $c$, $d$ are the roots of $g(x)$, then find $f(a) + f(b) + f(c) + f(d)$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Find the number of ordered triples of positive integers $(a, b, c)$ such that
\[6a + 10b + 15c = 3000.\]
Let $S$ be the set of all 3-tuples $(a,b,c)$ that satisfy $a+b+c=3000$ and $a,b,c>0$. If one of these 3-tuples is chosen at random, what's the probability that $a,b$ or $c$ is greater than or equal to 2,500?
Find the number of degree $8$ polynomials $f (x)$ with nonnegative integer coefficients satisfying both $f (1) = 16$ and $f (-1) = 8$.
Let $S$ be the set of all polynomials of the form $z^3+az^2+bz+c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $\left\lvert z \right\rvert = 20$ or $\left\lvert z \right\rvert = 13$.
Find the number of second-degree polynomials $ f(x)$ with integer coefficients and integer zeros for which $ f(0)\equal{}2010$.
Three fair six-sided dice are thrown. Determine the probability that the sum of the numbers on the three top faces is $6$.
[u]Round 1[/u]
[b]p1.[/b] Solve the maze
[img]https://cdn.artofproblemsolving.com/attachments/8/c/6439816a52b5f32c3cb415e2058556edb77c80.png[/img]
[b]p2.[/b] Billiam can write a problem in $30$ minutes, Jerry can write a problem in $10$ minutes, and Evin can write a problem in $20$ minutes. Billiam begins writing problems alone at $3:00$ PM until Jerry joins himat $4:00$ PM, and Evin joins both of them at $4:30$ PM. Given that they write problems until the end of math team at $5:00$ PM, how many full problems have they written in total?
[b]p3.[/b] How many pairs of positive integers $(n,k)$ are there such that ${n \choose k}= 6$?
[u]Round 2 [/u]
[b]p4.[/b] Find the sumof all integers $b > 1$ such that the expression of $143$ in base $b$ has an even number of digits and all digits are the same.
[b]p5.[/b] Ιni thinks that $a \# b = a^2 - b$ and $a \& b = b^2 - a$, while Mimi thinks that $a \# b = b^2 - a$ and $a \& b = a^2 - b$. Both Ini and Mimi try to evaluate $6 \& (3 \# 4)$, each using what they think the operations $\&$ and $\#$ mean. What is the positive difference between their answers?
[b]p6.[/b] A unit square sheet of paper lies on an infinite grid of unit squares. What is the maximum number of grid squares that the sheet of paper can partially cover at once? A grid square is partially covered if the area of the grid square under the sheet of paper is nonzero - i.e., lying on the edge only does not count.
[u]Round 3[/u]
[b]p7.[/b] Maya wants to buy lots of burgers. A burger without toppings costs $\$4$, and every added topping increases the price by 50 cents. There are 5 different toppings for Maya to choose from, and she can put any combination of toppings on each burger. How much would it cost forMaya to buy $1$ burger for each distinct set of toppings? Assume that the order in which the toppings are stacked onto the burger does not matter.
[b]p8.[/b] Consider square $ABCD$ and right triangle $PQR$ in the plane. Given that both shapes have area $1$, $PQ =QR$, $PA = RB$, and $P$, $A$, $B$ and $R$ are collinear, find the area of the region inside both square $ABCD$ and $\vartriangle PQR$, given that it is not $0$.
[b]p9.[/b] Find the sum of all $n$ such that $n$ is a $3$-digit perfect square that has the same tens digit as $\sqrt{n}$, but that has a different ones digit than $\sqrt{n}$.
[u]Round 4[/u]
[b]p10.[/b] Jeremy writes the string: $$LMTLMTLMTLMTLMTLMT$$ on a whiteboard (“$LMT$” written $6$ times). Find the number of ways to underline $3$ letters such that from left to right the underlined letters spell LMT.
[b]p11.[/b] Compute the remainder when $12^{2022}$ is divided by $1331$.
[b]p12.[/b] What is the greatest integer that cannot be expressed as the sum of $5$s, $23$s, and $29$s?
[u]Round 5 [/u]
[b]p13.[/b] Square $ABCD$ has point $E$ on side $BC$, and point $F$ on side $CD$, such that $\angle EAF = 45^o$. Let $BE = 3$, and $DF = 4$. Find the length of $FE$.
[b]p14.[/b] Find the sum of all positive integers $k$ such that
$\bullet$ $k$ is the power of some prime.
$\bullet$ $k$ can be written as $5654_b$ for some $b > 6$.
[b]p15.[/b] If $\sqrt[3]{x} + \sqrt[3]{y} = 2$ and $x + y = 20$, compute $\max \,(x, y)$.
PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167372p28825861]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
A candy store has $100$ pieces of candy to give away. When you get to the store, there are five people in front of you, numbered from $1$ to $5$. The $i$th person in line considers the set of positive integers congruent to $i$ modulo $5$ which are at most the number of pieces of candy remaining. If this set is empty, then they take no candy. Otherwise they pick an element of this set and take that many pieces of candy. For example, the first person in line will pick an integer from the set $\{1,6,\dots,96\}$ and take that many pieces of candy. How many ways can the first five people take their share of candy so that after they are done there are at least $35$ pieces of candy remaining?
In how many ways we can choose 3 non empty and non intersecting subsets from $ (1,2,\ldots,2008)$.
A candy store has $100$ pieces of candy to give away. When you get to the store, there are five people in front of you, numbered from $1$ to $5$. The $i$th person in line considers the set of positive integers congruent to $i$ modulo $5$ which are at most the number of pieces of candy remaining. If this set is empty, then they take no candy. Otherwise they pick an element of this set and take that many pieces of candy. For example, the first person in line will pick an integer from the set $\{1,6,\dots,96\}$ and take that many pieces of candy. How many ways can the first five people take their share of candy so that after they are done there are at least $35$ pieces of candy remaining?
[u]Round 1[/u]
[b]1.1[/b] If the sum of two non-zero integers is $28$, then find the largest possible ratio of these integers.
[b]1.2[/b] If Tom rolls a eight-sided die where the numbers $1$ − $8$ are all on a side, let $\frac{m}{n}$ be the probability that the number is a factor of $16$ where $m, n$ are relatively prime positive integers. Find $m + n$.
[b]1.3[/b] The average score of $35$ second graders on an IQ test was $180$ while the average score of $70$ adults was $90$. What was the total average IQ score of the adults and kids combined?
[u]Round 2[/u]
[b]2.1[/b] So far this year, Bob has gotten a $95$ and a 98 in Term $1$ and Term $2$. How many different pairs of Term $3$ and Term $4$ grades can Bob get such that he finishes with an average of $97$ for the whole year? Bob can only get integer grades between $0$ and $100$, inclusive.
[b]2.2[/b] If a complement of an angle $M$ is one-third the measure of its supplement, then what would be the measure (in degrees) of the third angle of an isosceles triangle in which two of its angles were equal to the measure of angle $M$?
[b]2.3[/b] The distinct symbols $\heartsuit, \diamondsuit, \clubsuit$ and $\spadesuit$ each correlate to one of $+, -, \times , \div$, not necessarily in that given order. Given that $$((((72 \,\, \,\, \diamondsuit \,\, \,\,36) \,\, \,\,\spadesuit \,\, \,\,0 ) \,\, \,\, \diamondsuit \,\, \,\, 32) \,\, \,\, \clubsuit \,\, \,\, 3)\,\, \,\, \heartsuit \,\, \,\, 2 = \,\, \,\, 6,$$ what is the value of $$(((((64 \,\, \,\, \spadesuit \,\, \,\, 8) \heartsuit \,\, \,\, 6) \,\, \,\, \spadesuit \,\, \,\, 5) \,\, \,\, \heartsuit \,\, \,\, 1) \,\, \,\, \clubsuit \,\, \,\, 7) \,\, \,\, \diamondsuit \,\, \,\, 1?$$
[u]Round 3[/u]
[b]3.1[/b] How many ways can $5$ bunnies be chosen from $7$ male bunnies and $9$ female bunnies if a majority of female bunnies is required? All bunnies are distinct from each other.
[b]3.2[/b] If the product of the LCM and GCD of two positive integers is $2021$, what is the product of the two positive integers?
[b]3.3[/b] The month of April in ABMC-land is $50$ days long. In this month, on $44\%$ of the days it rained, and on $28\%$ of the days it was sunny. On half of the days it was sunny, it rained as well. The rest of the days were cloudy. How many days were cloudy in April in ABMC-land?
[u]Round 4[/u]
[b]4.1[/b] In how many ways can $4$ distinct dice be rolled such that a sum of $10$ is produced?
[b]4.2[/b] If $p, q, r$ are positive integers such that $p^3\sqrt{q}r^2 = 50$, find the sum of all possible values of $pqr$.
[b]4.3[/b] Given that numbers $a, b, c$ satisfy $a + b + c = 0$, $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}= 10$, and $ab + bc + ac \ne 0$, compute the value of $\frac{-a^2 - b^2 - a^2}{ab + bc + ac}$.
PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2826137p24988781]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
[u]Part 1[/u]
[b]p1.[/b] Two kids $A$ and $B$ play a game as follows: From a box containing $n$ marbles ($n > 1$), they alternately take some marbles for themselves, such that:
1. $A$ goes first.
2. The number of marbles taken by $A$ in his first turn, denoted by $k$, must be between $1$ and $n$, inclusive.
3. The number of marbles taken in a turn by any player must be between $1$ and $k$, inclusive.
The winner is the one who takes the last marble. What is the sum of all $n$ for which $B$ has a winning strategy?
[b]p2.[/b] How many ways can your rearrange the letters of "Alejandro" such that it contains exactly one pair of adjacent vowels?
[b]p3.[/b] Assuming real values for $p, q, r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $q + 6i$, and the product of the other two roots is $3 - 4i$. Find the smallest value of $q$.
[b]p4.[/b] Lisa has a $3$D box that is $48$ units long, $140$ units high, and $126$ units wide. She shines a laser beam into the box through one of the corners, at a $45^o$ angle with respect to all of the sides of the box. Whenever the laser beam hits a side of the box, it is reflected perfectly, again at a $45^o$ angle. Compute the distance the laser beam travels until it hits one of the eight corners of the box.
[u]Part 2[/u]
[b]p5.[/b] How many ways can you divide a heptagon into five non-overlapping triangles such that the vertices of the triangles are vertices of the heptagon?
[b]p6.[/b] Let $a$ be the greatest root of $y = x^3 + 7x^2 - 14x - 48$. Let $b$ be the number of ways to pick a group of $a$ people out of a collection of $a^2$ people. Find $\frac{b}{2}$ .
[b]p7.[/b] Consider the equation
$$1 -\frac{1}{d}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c},$$
with $a, b, c$, and $d$ being positive integers. What is the largest value for $d$?
[b]p8.[/b] The number of non-negative integers $x_1, x_2,..., x_{12}$ such that $$x_1 + x_2 + ... + x_{12} \le 17$$
can be expressed in the form ${a \choose b}$ , where $2b \le a$. Find $a + b$.
[u]Part 3[/u]
[b]p9.[/b] In the diagram below, $AB$ is tangent to circle $O$. Given that $AC = 15$, $AB = 27/2$, and $BD = 243/34$, compute the area of $\vartriangle ABC$.
[img]https://cdn.artofproblemsolving.com/attachments/b/f/b403e5e188916ac4fb1b0ba74adb7f1e50e86a.png[/img]
[b]p10.[/b] If
$$\left[2^{\log x}\right]^{[x^{\log 2}]^{[2^{\log x}]...}}= 2, $$
where $\log x$ is the base-$10$ logarithm of $x$, then it follows that $x =\sqrt{n}$. Compute $n^2$.
[b]p11.[/b]
[b]p12.[/b] Find $n$ in the equation $$133^5 + 110^5 + 84^5 + 27^5 = n^5, $$ where $n$ is an integer less than $170$.
[u]Part 4[/u]
[b]p13.[/b] Let $x$ be the answer to number $14$, and $z$ be the answer to number $16$. Define $f(n)$ as the number of distinct two-digit integers that can be formed from digits in $n$. For example, $f(15) = 4$ because the integers $11$, $15$, $51$, $55$ can be formed from digits of $15$. Let $w$ be such that $f(3xz - w) = w$. Find $w$.
[b]p14.[/b] Let $w$ be the answer to number $13$ and $z$ be the answer to number $16$. Let $x$ be such that the coefficient of $a^xb^x$ in $(a + b)^{2x}$ is $5z^2 + 2w - 1$. Find $x$.
[b]p15.[/b] Let $w$ be the answer to number $13$, $x$ be the answer to number $14$, and $z$ be the answer to number $16$. Let $A$, $B$, $C$, $D$ be points on a circle, in that order, such that $\overline{AD}$ is a diameter of the circle. Let $E$ be the intersection of $\overleftrightarrow{AB}$ and $\overleftrightarrow{DC}$, let $F$ be the intersection of $\overleftrightarrow{AC}$ and $\overleftrightarrow{BD}$, and let $G$ be the intersection of $\overleftrightarrow{EF}$ and $\overleftrightarrow{AD}$. Now, let $AE = 3x$, $ED = w^2 - w + 1$, and $AD = 2z$. If $FG = y$, find $y$.
[b]p16.[/b] Let $w$ be the answer to number $13$, and $x$ be the answer to number $16$. Let $z$ be the number of integers $n$ in the set $S = \{w,w + 1, ... ,16x - 1, 16x\}$ such that $n^2 + n^3$ is a perfect square. Find $z$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Numbers $x,y,z$ are positive integers and satisfy the equation $x+y+z=2013$. (E)
a) Find the number of the triplets $(x,y,z)$ that are solutions of the equation (E).
b) Find the number of the solutions of the equation (E) for which $x=y$.
c) Find the solution $(x,y,z)$ of the equation (E) for which the product $xyz$ becomes maximum.
Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$. How many passcodes satisfy these conditions?
Consider a round-robin tournament with $2n+1$ teams, where each team plays each other team exactly one. We say that three teams $X,Y$ and $Z$, form a [i]cycle triplet [/i] if $X$ beats $Y$, $Y$ beats $Z$ and $Z$ beats $X$. There are no ties.
a)Determine the minimum number of cycle triplets possible.
b)Determine the maximum number of cycle triplets possible.
[b]p21[/b]. If $f = \cos(\sin (x))$. Calculate the sum $\sum^{2021}_{n=0} f'' (n \pi)$.
[b]p22.[/b] Find all real values of $A$ that minimize the difference between the local maximum and local minimum of $f(x) = \left(3x^2 - 4\right)\left(x - A + \frac{1}{A}\right)$.
[b]p23.[/b] Bessie is playing a game. She labels a square with vertices labeled $A, B, C, D$ in clockwise order. There are $7$ possible moves: she can rotate her square $90$ degrees about the center, $180$ degrees about the center, $270$ degrees about the center; or she can flip across diagonal $AC$, flip across diagonal $BD$, flip the square horizontally (flip the square so that vertices A and B are switched and vertices $C$ and $D$ are switched), or flip the square vertically (vertices $B$ and $C$ are switched, vertices $A$ and $D$ are switched). In how many ways can Bessie arrive back at the original square for the first time in $3$ moves?
[b]p24.[/b] A positive integer is called [i]happy [/i] if the sum of its digits equals the two-digit integer formed by its two leftmost digits. Find the number of $5$-digit happy integers.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Find the number of five-digit positive integers, $n$, that satisfy the following conditions:
(a) the number $n$ is divisible by $5$,
(b) the first and last digits of $n$ are equal, and
(c) the sum of the digits of $n$ is divisible by $5$.
Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that $p$ does not exceed 1 percent.
[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]
PS. You should use hide for answers. Rest rounds have been posted [url=https://artofproblemsolving.com/community/c4h2786958p24498425]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Everyday at school, Jo climbs a flight of $6$ stairs. Joe can take the stairs $1,2$, or $3$ at a time. For example, Jo could climb $3$, then $1$, then $2$. In how many ways can Jo climb the stairs?
$ \textbf{(A)}\ 13 \qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24 $