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3,700 | A nonempty set $S$ is called well-filled if for every $m \in S$, there are fewer than $\frac{1}{2}m$ elements of $S$ which are less than $m$. Determine the number of well-filled subsets of $\{1,2, \ldots, 42\}$. | \binom{43}{21}-1 | 0 |
3,701 | Let $A B C$ be a triangle with $A B=13, A C=14$, and $B C=15$. Let $G$ be the point on $A C$ such that the reflection of $B G$ over the angle bisector of $\angle B$ passes through the midpoint of $A C$. Let $Y$ be the midpoint of $G C$ and $X$ be a point on segment $A G$ such that $\frac{A X}{X G}=3$. Construct $F$ and $H$ on $A B$ and $B C$, respectively, such that $F X\|B G\| H Y$. If $A H$ and $C F$ concur at $Z$ and $W$ is on $A C$ such that $W Z \| B G$, find $W Z$. | \frac{1170 \sqrt{37}}{1379} | 0 |
3,702 | Compute the sum of all integers $1 \leq a \leq 10$ with the following property: there exist integers $p$ and $q$ such that $p, q, p^{2}+a$ and $q^{2}+a$ are all distinct prime numbers. | 20 | 88.28125 |
3,703 | Bob writes a random string of 5 letters, where each letter is either $A, B, C$, or $D$. The letter in each position is independently chosen, and each of the letters $A, B, C, D$ is chosen with equal probability. Given that there are at least two $A$ 's in the string, find the probability that there are at least three $A$ 's in the string. | 53/188 | 43.75 |
3,704 | There are 42 stepping stones in a pond, arranged along a circle. You are standing on one of the stones. You would like to jump among the stones so that you move counterclockwise by either 1 stone or 7 stones at each jump. Moreover, you would like to do this in such a way that you visit each stone (except for the starting spot) exactly once before returning to your initial stone for the first time. In how many ways can you do this? | 63 | 0 |
3,705 | How many nonempty subsets of $\{1,2,3, \ldots, 12\}$ have the property that the sum of the largest element and the smallest element is 13? | 1365 | 17.96875 |
3,706 | If the system of equations $$\begin{aligned} & |x+y|=99 \\ & |x-y|=c \end{aligned}$$ has exactly two real solutions $(x, y)$, find the value of $c$. | 0 | 60.15625 |
3,707 | Let $A=\{a_{1}, b_{1}, a_{2}, b_{2}, \ldots, a_{10}, b_{10}\}$, and consider the 2-configuration $C$ consisting of \( \{a_{i}, b_{i}\} \) for all \( 1 \leq i \leq 10, \{a_{i}, a_{i+1}\} \) for all \( 1 \leq i \leq 9 \), and \( \{b_{i}, b_{i+1}\} \) for all \( 1 \leq i \leq 9 \). Find the number of subsets of $C$ that are consistent of order 1. | 89 | 0 |
3,708 | Find the smallest possible area of an ellipse passing through $(2,0),(0,3),(0,7)$, and $(6,0)$. | \frac{56 \pi \sqrt{3}}{9} | 0 |
3,709 | P is a polynomial. When P is divided by $x-1$, the remainder is -4 . When P is divided by $x-2$, the remainder is -1 . When $P$ is divided by $x-3$, the remainder is 4 . Determine the remainder when $P$ is divided by $x^{3}-6 x^{2}+11 x-6$. | x^{2}-5 | 0 |
3,710 | For each positive integer $n$ and non-negative integer $k$, define $W(n, k)$ recursively by $$ W(n, k)= \begin{cases}n^{n} & k=0 \\ W(W(n, k-1), k-1) & k>0\end{cases} $$ Find the last three digits in the decimal representation of $W(555,2)$. | 875 | 60.9375 |
3,711 | There are 5 students on a team for a math competition. The math competition has 5 subject tests. Each student on the team must choose 2 distinct tests, and each test must be taken by exactly two people. In how many ways can this be done? | 2040 | 0 |
3,712 | How many functions $f:\{1,2,3,4,5\} \rightarrow\{1,2,3,4,5\}$ satisfy $f(f(x))=f(x)$ for all $x \in\{1,2,3,4,5\}$? | 196 | 28.90625 |
3,713 | Let $A=\{V, W, X, Y, Z, v, w, x, y, z\}$. Find the number of subsets of the 2-configuration \( \{\{V, W\}, \{W, X\}, \{X, Y\}, \{Y, Z\}, \{Z, V\}, \{v, x\}, \{v, y\}, \{w, y\}, \{w, z\}, \{x, z\}, \{V, v\}, \{W, w\}, \{X, x\}, \{Y, y\}, \{Z, z\}\} \) that are consistent of order 1. | 6 | 0 |
3,714 | Eli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the 4 people. The following constraints are imposed: - Every person must get a positive integer number of shares, and all 16 shares must be given out. - No one person can have more shares than the other three people combined. Assuming that shares are indistinguishable, but people are distinguishable, in how many ways can the shares be given out? | 315 | 26.5625 |
3,715 | Let $B_{k}(n)$ be the largest possible number of elements in a 2-separable $k$-configuration of a set with $2n$ elements $(2 \leq k \leq n)$. Find a closed-form expression (i.e. an expression not involving any sums or products with a variable number of terms) for $B_{k}(n)$. | \binom{2n}{k} - 2\binom{n}{k} | 0 |
3,716 | A $4 \times 4$ window is made out of 16 square windowpanes. How many ways are there to stain each of the windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of its neighbors? | 24 | 0 |
3,717 | Almondine has a bag with $N$ balls, each of which is red, white, or blue. If Almondine picks three balls from the bag without replacement, the probability that she picks one ball of each color is larger than 23 percent. Compute the largest possible value of $\left\lfloor\frac{N}{3}\right\rfloor$. | 29 | 9.375 |
3,718 | Let $s(n)$ denote the number of 1's in the binary representation of $n$. Compute $$\frac{1}{255} \sum_{0 \leq n<16} 2^{n}(-1)^{s(n)}$$ | 45 | 46.09375 |
3,719 | Find the real solutions of $(2 x+1)(3 x+1)(5 x+1)(30 x+1)=10$. | \frac{-4 \pm \sqrt{31}}{15} | 0 |
3,720 | Every second, Andrea writes down a random digit uniformly chosen from the set $\{1,2,3,4\}$. She stops when the last two numbers she has written sum to a prime number. What is the probability that the last number she writes down is 1? | 15/44 | 0 |
3,721 | The squares of a $3 \times 3$ grid are filled with positive integers such that 1 is the label of the upperleftmost square, 2009 is the label of the lower-rightmost square, and the label of each square divides the one directly to the right of it and the one directly below it. How many such labelings are possible? | 2448 | 0 |
3,722 | How many times does the letter "e" occur in all problem statements in this year's HMMT February competition? | 1661 | 0 |
3,723 | Six distinguishable players are participating in a tennis tournament. Each player plays one match of tennis against every other player. There are no ties in this tournament; each tennis match results in a win for one player and a loss for the other. Suppose that whenever $A$ and $B$ are players in the tournament such that $A$ wins strictly more matches than $B$ over the course of the tournament, it is also true that $A$ wins the match against $B$ in the tournament. In how many ways could the tournament have gone? | 2048 | 0 |
3,724 | Let $$ A=\lim _{n \rightarrow \infty} \sum_{i=0}^{2016}(-1)^{i} \cdot \frac{\binom{n}{i}\binom{n}{i+2}}{\binom{n}{i+1}^{2}} $$ Find the largest integer less than or equal to $\frac{1}{A}$. | 1 | 28.125 |
3,725 | How many sequences of 5 positive integers $(a, b, c, d, e)$ satisfy $a b c d e \leq a+b+c+d+e \leq 10$? | 116 | 0.78125 |
3,726 | Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $I_{A}, I_{B}, I_{C}$ be the $A, B, C$ excenters of this triangle, and let $O$ be the circumcenter of the triangle. Let $\gamma_{A}, \gamma_{B}, \gamma_{C}$ be the corresponding excircles and $\omega$ be the circumcircle. $X$ is one of the intersections between $\gamma_{A}$ and $\omega$. Likewise, $Y$ is an intersection of $\gamma_{B}$ and $\omega$, and $Z$ is an intersection of $\gamma_{C}$ and $\omega$. Compute $$\cos \angle O X I_{A}+\cos \angle O Y I_{B}+\cos \angle O Z I_{C}$$ | -\frac{49}{65} | 0 |
3,727 | Find the number of triples of sets $(A, B, C)$ such that: (a) $A, B, C \subseteq\{1,2,3, \ldots, 8\}$. (b) $|A \cap B|=|B \cap C|=|C \cap A|=2$. (c) $|A|=|B|=|C|=4$. Here, $|S|$ denotes the number of elements in the set $S$. | 45360 | 1.5625 |
3,728 | An up-right path from $(a, b) \in \mathbb{R}^{2}$ to $(c, d) \in \mathbb{R}^{2}$ is a finite sequence $\left(x_{1}, y_{1}\right), \ldots,\left(x_{k}, y_{k}\right)$ of points in $\mathbb{R}^{2}$ such that $(a, b)=\left(x_{1}, y_{1}\right),(c, d)=\left(x_{k}, y_{k}\right)$, and for each $1 \leq i<k$ we have that either $\left(x_{i+1}, y_{i+1}\right)=\left(x_{i}+1, y_{i}\right)$ or $\left(x_{i+1}, y_{i+1}\right)=\left(x_{i}, y_{i}+1\right)$. Two up-right paths are said to intersect if they share any point. Find the number of pairs $(A, B)$ where $A$ is an up-right path from $(0,0)$ to $(4,4), B$ is an up-right path from $(2,0)$ to $(6,4)$, and $A$ and $B$ do not intersect. | 1750 | 1.5625 |
3,729 | (Lucas Numbers) The Lucas numbers are defined by $L_{0}=2, L_{1}=1$, and $L_{n+2}=L_{n+1}+L_{n}$ for every $n \geq 0$. There are $N$ integers $1 \leq n \leq 2016$ such that $L_{n}$ contains the digit 1 . Estimate $N$. | 1984 | 7.8125 |
3,730 | The integers $1,2, \ldots, 64$ are written in the squares of a $8 \times 8$ chess board, such that for each $1 \leq i<64$, the numbers $i$ and $i+1$ are in squares that share an edge. What is the largest possible sum that can appear along one of the diagonals? | 432 | 0 |
3,731 | Given a rearrangement of the numbers from 1 to $n$, each pair of consecutive elements $a$ and $b$ of the sequence can be either increasing (if $a<b$ ) or decreasing (if $b<a$ ). How many rearrangements of the numbers from 1 to $n$ have exactly two increasing pairs of consecutive elements? | 3^{n}-(n+1) \cdot 2^{n}+n(n+1) / 2 | 0 |
3,732 | What is the radius of the smallest sphere in which 4 spheres of radius 1 will fit? | \frac{2+\sqrt{6}}{2} | 0 |
3,733 | In an election for the Peer Pressure High School student council president, there are 2019 voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice's boyfriend Bob votes for Alice as well. Then one by one, each of the remaining 2016 voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate's votes. For example, the first undecided voter David has a $\frac{2}{3}$ probability of voting for Alice and a $\frac{1}{3}$ probability of voting for Celia. What is the probability that Alice wins the election (by having more votes than Celia)? | \frac{1513}{2017} | 0 |
3,734 | Let $S$ be the set of lattice points inside the circle $x^{2}+y^{2}=11$. Let $M$ be the greatest area of any triangle with vertices in $S$. How many triangles with vertices in $S$ have area $M$? | 16 | 47.65625 |
3,735 | How many ways can one fill a $3 \times 3$ square grid with nonnegative integers such that no nonzero integer appears more than once in the same row or column and the sum of the numbers in every row and column equals 7 ? | 216 | 1.5625 |
3,736 | Regular tetrahedron $A B C D$ is projected onto a plane sending $A, B, C$, and $D$ to $A^{\prime}, B^{\prime}, C^{\prime}$, and $D^{\prime}$ respectively. Suppose $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ is a convex quadrilateral with $A^{\prime} B^{\prime}=A^{\prime} D^{\prime}$ and $C^{\prime} B^{\prime}=C^{\prime} D^{\prime}$, and suppose that the area of $A^{\prime} B^{\prime} C^{\prime} D^{\prime}=4$. Given these conditions, the set of possible lengths of $A B$ consists of all real numbers in the interval $[a, b)$. Compute $b$. | 2 \sqrt[4]{6} | 0.78125 |
3,737 | Fred the Four-Dimensional Fluffy Sheep is walking in 4 -dimensional space. He starts at the origin. Each minute, he walks from his current position $\left(a_{1}, a_{2}, a_{3}, a_{4}\right)$ to some position $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ with integer coordinates satisfying $\left(x_{1}-a_{1}\right)^{2}+\left(x_{2}-a_{2}\right)^{2}+\left(x_{3}-a_{3}\right)^{2}+\left(x_{4}-a_{4}\right)^{2}=4$ and $\left|\left(x_{1}+x_{2}+x_{3}+x_{4}\right)-\left(a_{1}+a_{2}+a_{3}+a_{4}\right)\right|=2$. In how many ways can Fred reach $(10,10,10,10)$ after exactly 40 minutes, if he is allowed to pass through this point during his walk? | \binom{40}{10}\binom{40}{20}^{3} | 0 |
3,738 | Triangle $A B C$ has incircle $\omega$ which touches $A B$ at $C_{1}, B C$ at $A_{1}$, and $C A$ at $B_{1}$. Let $A_{2}$ be the reflection of $A_{1}$ over the midpoint of $B C$, and define $B_{2}$ and $C_{2}$ similarly. Let $A_{3}$ be the intersection of $A A_{2}$ with $\omega$ that is closer to $A$, and define $B_{3}$ and $C_{3}$ similarly. If $A B=9, B C=10$, and $C A=13$, find \left[A_{3} B_{3} C_{3}\right] /[A B C]. | 14/65 | 0 |
3,739 | Let $C$ be a circle with two diameters intersecting at an angle of 30 degrees. A circle $S$ is tangent to both diameters and to $C$, and has radius 1. Find the largest possible radius of $C$. | 1+\sqrt{2}+\sqrt{6} | 5.46875 |
3,740 | Let $\Delta A_{1} B_{1} C$ be a triangle with $\angle A_{1} B_{1} C=90^{\circ}$ and $\frac{C A_{1}}{C B_{1}}=\sqrt{5}+2$. For any $i \geq 2$, define $A_{i}$ to be the point on the line $A_{1} C$ such that $A_{i} B_{i-1} \perp A_{1} C$ and define $B_{i}$ to be the point on the line $B_{1} C$ such that $A_{i} B_{i} \perp B_{1} C$. Let $\Gamma_{1}$ be the incircle of $\Delta A_{1} B_{1} C$ and for $i \geq 2, \Gamma_{i}$ be the circle tangent to $\Gamma_{i-1}, A_{1} C, B_{1} C$ which is smaller than $\Gamma_{i-1}$. How many integers $k$ are there such that the line $A_{1} B_{2016}$ intersects $\Gamma_{k}$ ? | 4030 | 0 |
3,741 | How many ways, without taking order into consideration, can 2002 be expressed as the sum of 3 positive integers (for instance, $1000+1000+2$ and $1000+2+1000$ are considered to be the same way)? | 334000 | 43.75 |
3,742 | Farmer Bill's 1000 animals - ducks, cows, and rabbits - are standing in a circle. In order to feel safe, every duck must either be standing next to at least one cow or between two rabbits. If there are 600 ducks, what is the least number of cows there can be for this to be possible? | 201 | 0 |
3,743 | For a positive integer $N$, we color the positive divisors of $N$ (including 1 and $N$ ) with four colors. A coloring is called multichromatic if whenever $a, b$ and $\operatorname{gcd}(a, b)$ are pairwise distinct divisors of $N$, then they have pairwise distinct colors. What is the maximum possible number of multichromatic colorings a positive integer can have if it is not the power of any prime? | 192 | 0 |
3,744 | Let $a$ and $b$ be five-digit palindromes (without leading zeroes) such that $a<b$ and there are no other five-digit palindromes strictly between $a$ and $b$. What are all possible values of $b-a$? | 100, 110, 11 | 0 |
3,745 | Contessa is taking a random lattice walk in the plane, starting at $(1,1)$. (In a random lattice walk, one moves up, down, left, or right 1 unit with equal probability at each step.) If she lands on a point of the form $(6 m, 6 n)$ for $m, n \in \mathbb{Z}$, she ascends to heaven, but if she lands on a point of the form $(6 m+3,6 n+3)$ for $m, n \in \mathbb{Z}$, she descends to hell. What is the probability that she ascends to heaven? | \frac{13}{22} | 0 |
3,746 | A certain cafeteria serves ham and cheese sandwiches, ham and tomato sandwiches, and tomato and cheese sandwiches. It is common for one meal to include multiple types of sandwiches. On a certain day, it was found that 80 customers had meals which contained both ham and cheese; 90 had meals containing both ham and tomatoes; 100 had meals containing both tomatoes and cheese. 20 customers' meals included all three ingredients. How many customers were there? | 230 | 61.71875 |
3,747 | Triangle $A B C$ has perimeter 1. Its three altitudes form the side lengths of a triangle. Find the set of all possible values of $\min (A B, B C, C A)$. | \left(\frac{3-\sqrt{5}}{4}, \frac{1}{3}\right] | 0 |
3,748 | Alice Czarina is bored and is playing a game with a pile of rocks. The pile initially contains 2015 rocks. At each round, if the pile has $N$ rocks, she removes $k$ of them, where $1 \leq k \leq N$, with each possible $k$ having equal probability. Alice Czarina continues until there are no more rocks in the pile. Let $p$ be the probability that the number of rocks left in the pile after each round is a multiple of 5. If $p$ is of the form $5^{a} \cdot 31^{b} \cdot \frac{c}{d}$, where $a, b$ are integers and $c, d$ are positive integers relatively prime to $5 \cdot 31$, find $a+b$. | -501 | 0 |
3,749 | A permutation of \{1,2, \ldots, 7\} is chosen uniformly at random. A partition of the permutation into contiguous blocks is correct if, when each block is sorted independently, the entire permutation becomes sorted. For example, the permutation $(3,4,2,1,6,5,7)$ can be partitioned correctly into the blocks $[3,4,2,1]$ and $[6,5,7]$, since when these blocks are sorted, the permutation becomes $(1,2,3,4,5,6,7)$. Find the expected value of the maximum number of blocks into which the permutation can be partitioned correctly. | \frac{151}{105} | 0 |
3,750 | Sherry is waiting for a train. Every minute, there is a $75 \%$ chance that a train will arrive. However, she is engrossed in her game of sudoku, so even if a train arrives she has a $75 \%$ chance of not noticing it (and hence missing the train). What is the probability that Sherry catches the train in the next five minutes? | 1-\left(\frac{13}{16}\right)^{5} | 0 |
3,751 | For each positive real number $\alpha$, define $$ \lfloor\alpha \mathbb{N}\rfloor:=\{\lfloor\alpha m\rfloor \mid m \in \mathbb{N}\} $$ Let $n$ be a positive integer. A set $S \subseteq\{1,2, \ldots, n\}$ has the property that: for each real $\beta>0$, $$ \text { if } S \subseteq\lfloor\beta \mathbb{N}\rfloor \text {, then }\{1,2, \ldots, n\} \subseteq\lfloor\beta \mathbb{N}\rfloor $$ Determine, with proof, the smallest possible size of $S$. | \lfloor n / 2\rfloor+1 | 0 |
3,752 | Determine the number of triples $0 \leq k, m, n \leq 100$ of integers such that $$ 2^{m} n-2^{n} m=2^{k} $$ | 22 | 0 |
3,753 | If Alex does not sing on Saturday, then she has a $70 \%$ chance of singing on Sunday; however, to rest her voice, she never sings on both days. If Alex has a $50 \%$ chance of singing on Sunday, find the probability that she sings on Saturday. | \frac{2}{7} | 89.84375 |
3,754 | Given a permutation $\sigma$ of $\{1,2, \ldots, 2013\}$, let $f(\sigma)$ to be the number of fixed points of $\sigma$ - that is, the number of $k \in\{1,2, \ldots, 2013\}$ such that $\sigma(k)=k$. If $S$ is the set of all possible permutations $\sigma$, compute $$\sum_{\sigma \in S} f(\sigma)^{4}$$ (Here, a permutation $\sigma$ is a bijective mapping from $\{1,2, \ldots, 2013\}$ to $\{1,2, \ldots, 2013\}$.) | 15(2013!) | 0 |
3,755 | For positive integers $a, b, a \uparrow \uparrow b$ is defined as follows: $a \uparrow \uparrow 1=a$, and $a \uparrow \uparrow b=a^{a \uparrow \uparrow(b-1)}$ if $b>1$. Find the smallest positive integer $n$ for which there exists a positive integer $a$ such that $a \uparrow \uparrow 6 \not \equiv a \uparrow \uparrow 7$ $\bmod n$. | 283 | 0 |
3,756 | How many equilateral hexagons of side length $\sqrt{13}$ have one vertex at $(0,0)$ and the other five vertices at lattice points? (A lattice point is a point whose Cartesian coordinates are both integers. A hexagon may be concave but not self-intersecting.) | 216 | 0 |
3,757 | Alice and Bob play a game on a circle with 8 marked points. Alice places an apple beneath one of the points, then picks five of the other seven points and reveals that none of them are hiding the apple. Bob then drops a bomb on any of the points, and destroys the apple if he drops the bomb either on the point containing the apple or on an adjacent point. Bob wins if he destroys the apple, and Alice wins if he fails. If both players play optimally, what is the probability that Bob destroys the apple? | \frac{1}{2} | 10.15625 |
3,758 | A bag contains nine blue marbles, ten ugly marbles, and one special marble. Ryan picks marbles randomly from this bag with replacement until he draws the special marble. He notices that none of the marbles he drew were ugly. Given this information, what is the expected value of the number of total marbles he drew? | \frac{20}{11} | 2.34375 |
3,759 | Compute the side length of the largest cube contained in the region $\{(x, y, z): x^{2}+y^{2}+z^{2} \leq 25 \text{ and } x \geq 0\}$ of three-dimensional space. | \frac{5 \sqrt{6}}{3} | 10.9375 |
3,760 | All subscripts in this problem are to be considered modulo 6 , that means for example that $\omega_{7}$ is the same as $\omega_{1}$. Let $\omega_{1}, \ldots \omega_{6}$ be circles of radius $r$, whose centers lie on a regular hexagon of side length 1 . Let $P_{i}$ be the intersection of $\omega_{i}$ and $\omega_{i+1}$ that lies further from the center of the hexagon, for $i=1, \ldots 6$. Let $Q_{i}, i=1 \ldots 6$, lie on $\omega_{i}$ such that $Q_{i}, P_{i}, Q_{i+1}$ are colinear. Find the number of possible values of $r$. | 5 | 0 |
3,761 | A tourist is learning an incorrect way to sort a permutation $(p_{1}, \ldots, p_{n})$ of the integers $(1, \ldots, n)$. We define a fix on two adjacent elements $p_{i}$ and $p_{i+1}$, to be an operation which swaps the two elements if $p_{i}>p_{i+1}$, and does nothing otherwise. The tourist performs $n-1$ rounds of fixes, numbered $a=1,2, \ldots, n-1$. In round $a$ of fixes, the tourist fixes $p_{a}$ and $p_{a+1}$, then $p_{a+1}$ and $p_{a+2}$, and so on, up to $p_{n-1}$ and $p_{n}$. In this process, there are $(n-1)+(n-2)+\cdots+1=\frac{n(n-1)}{2}$ total fixes performed. How many permutations of $(1, \ldots, 2018)$ can the tourist start with to obtain $(1, \ldots, 2018)$ after performing these steps? | 1009! \cdot 1010! | 0 |
3,762 | Let $p>2$ be a prime number. $\mathbb{F}_{p}[x]$ is defined as the set of all polynomials in $x$ with coefficients in $\mathbb{F}_{p}$ (the integers modulo $p$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^{k}$ are equal in $\mathbb{F}_{p}$ for each nonnegative integer $k$. For example, $(x+2)(2 x+3)=2 x^{2}+2 x+1$ in $\mathbb{F}_{5}[x]$ because the corresponding coefficients are equal modulo 5 . Let $f, g \in \mathbb{F}_{p}[x]$. The pair $(f, g)$ is called compositional if $$f(g(x)) \equiv x^{p^{2}}-x$$ in $\mathbb{F}_{p}[x]$. Find, with proof, the number of compositional pairs (in terms of $p$ ). | 4 p(p-1) | 0 |
3,763 | A single-elimination ping-pong tournament has $2^{2013}$ players, seeded in order of ability. If the player with seed $x$ plays the player with seed $y$, then it is possible for $x$ to win if and only if $x \leq y+3$. For how many players $P$ it is possible for $P$ to win? (In each round of a single elimination tournament, the remaining players are randomly paired up; each player plays against the other player in his pair, with the winner from each pair progressing to the next round and the loser eliminated. This is repeated until there is only one player remaining.) | 6038 | 0 |
3,764 | Your math friend Steven rolls five fair icosahedral dice (each of which is labelled $1,2, \ldots, 20$ on its sides). He conceals the results but tells you that at least half of the rolls are 20. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show $20 ?$ | \frac{1}{58} | 0 |
3,765 | We have a calculator with two buttons that displays an integer $x$. Pressing the first button replaces $x$ by $\left\lfloor\frac{x}{2}\right\rfloor$, and pressing the second button replaces $x$ by $4 x+1$. Initially, the calculator displays 0. How many integers less than or equal to 2014 can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, $\lfloor y\rfloor$ denotes the greatest integer less than or equal to the real number $y$.) | 233 | 1.5625 |
3,766 | A peacock is a ten-digit positive integer that uses each digit exactly once. Compute the number of peacocks that are exactly twice another peacock. | 184320 | 71.09375 |
3,767 | How many ways can you mark 8 squares of an $8 \times 8$ chessboard so that no two marked squares are in the same row or column, and none of the four corner squares is marked? (Rotations and reflections are considered different.) | 21600 | 35.15625 |
3,768 | It is known that exactly one of the three (distinguishable) musketeers stole the truffles. Each musketeer makes one statement, in which he either claims that one of the three is guilty, or claims that one of the three is innocent. It is possible for two or more of the musketeers to make the same statement. After hearing their claims, and knowing that exactly one musketeer lied, the inspector is able to deduce who stole the truffles. How many ordered triplets of statements could have been made? | 99 | 0 |
3,769 | Andrea flips a fair coin repeatedly, continuing until she either flips two heads in a row (the sequence $H H$ ) or flips tails followed by heads (the sequence $T H$ ). What is the probability that she will stop after flipping $H H$ ? | 1/4 | 4.6875 |
3,770 | One hundred people are in line to see a movie. Each person wants to sit in the front row, which contains one hundred seats, and each has a favorite seat, chosen randomly and independently. They enter the row one at a time from the far right. As they walk, if they reach their favorite seat, they sit, but to avoid stepping over people, if they encounter a person already seated, they sit to that person's right. If the seat furthest to the right is already taken, they sit in a different row. What is the most likely number of people that will get to sit in the first row? | 10 | 0 |
3,771 | A best-of-9 series is to be played between two teams; that is, the first team to win 5 games is the winner. The Mathletes have a chance of $2 / 3$ of winning any given game. What is the probability that exactly 7 games will need to be played to determine a winner? | 20/81 | 3.90625 |
3,772 | A moth starts at vertex $A$ of a certain cube and is trying to get to vertex $B$, which is opposite $A$, in five or fewer "steps," where a step consists in traveling along an edge from one vertex to another. The moth will stop as soon as it reaches $B$. How many ways can the moth achieve its objective? | 48 | 4.6875 |
3,773 | A sequence is defined by $a_{0}=1$ and $a_{n}=2^{a_{n-1}}$ for $n \geq 1$. What is the last digit (in base 10) of $a_{15}$? | 6 | 78.90625 |
3,774 | An unfair coin has the property that when flipped four times, it has the same probability of turning up 2 heads and 2 tails (in any order) as 3 heads and 1 tail (in any order). What is the probability of getting a head in any one flip? | \frac{3}{5} | 79.6875 |
3,775 | Given a $9 \times 9$ chess board, we consider all the rectangles whose edges lie along grid lines (the board consists of 81 unit squares, and the grid lines lie on the borders of the unit squares). For each such rectangle, we put a mark in every one of the unit squares inside it. When this process is completed, how many unit squares will contain an even number of marks? | 56 | 32.03125 |
3,776 | Fifteen freshmen are sitting in a circle around a table, but the course assistant (who remains standing) has made only six copies of today's handout. No freshman should get more than one handout, and any freshman who does not get one should be able to read a neighbor's. If the freshmen are distinguishable but the handouts are not, how many ways are there to distribute the six handouts subject to the above conditions? | 125 | 17.1875 |
3,777 | For positive integers $x$, let $g(x)$ be the number of blocks of consecutive 1's in the binary expansion of $x$. For example, $g(19)=2$ because $19=10011_{2}$ has a block of one 1 at the beginning and a block of two 1's at the end, and $g(7)=1$ because $7=111_{2}$ only has a single block of three 1's. Compute $g(1)+g(2)+g(3)+\cdots+g(256)$. | 577 | 0 |
3,778 | Compute $$\sum_{n_{60}=0}^{2} \sum_{n_{59}=0}^{n_{60}} \cdots \sum_{n_{2}=0}^{n_{3}} \sum_{n_{1}=0}^{n_{2}} \sum_{n_{0}=0}^{n_{1}} 1$$ | 1953 | 58.59375 |
3,779 | Each unit square of a $4 \times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color? | 18 | 0 |
3,780 | Calvin has a bag containing 50 red balls, 50 blue balls, and 30 yellow balls. Given that after pulling out 65 balls at random (without replacement), he has pulled out 5 more red balls than blue balls, what is the probability that the next ball he pulls out is red? | \frac{9}{26} | 0 |
3,781 | What is the probability that in a randomly chosen arrangement of the numbers and letters in "HMMT2005," one can read either "HMMT" or "2005" from left to right? | 23/144 | 0 |
3,782 | Compute the number of ways there are to assemble 2 red unit cubes and 25 white unit cubes into a $3 \times 3 \times 3$ cube such that red is visible on exactly 4 faces of the larger cube. (Rotations and reflections are considered distinct.) | 114 | 1.5625 |
3,783 | Let $n$ be a positive integer, and let Pushover be a game played by two players, standing squarely facing each other, pushing each other, where the first person to lose balance loses. At the HMPT, $2^{n+1}$ competitors, numbered 1 through $2^{n+1}$ clockwise, stand in a circle. They are equals in Pushover: whenever two of them face off, each has a $50 \%$ probability of victory. The tournament unfolds in $n+1$ rounds. In each round, the referee randomly chooses one of the surviving players, and the players pair off going clockwise, starting from the chosen one. Each pair faces off in Pushover, and the losers leave the circle. What is the probability that players 1 and $2^{n}$ face each other in the last round? Express your answer in terms of $n$. | \frac{2^{n}-1}{8^{n}} | 0 |
3,784 | The country of HMMTLand has 8 cities. Its government decides to construct several two-way roads between pairs of distinct cities. After they finish construction, it turns out that each city can reach exactly 3 other cities via a single road, and from any pair of distinct cities, either exactly 0 or 2 other cities can be reached from both cities by a single road. Compute the number of ways HMMTLand could have constructed the roads. | 875 | 0 |
3,785 | Let $S=\{1,2, \ldots, 9\}$. Compute the number of functions $f: S \rightarrow S$ such that, for all $s \in S, f(f(f(s)))=s$ and $f(s)-s$ is not divisible by 3. | 288 | 0 |
3,786 | Sally the snail sits on the $3 \times 24$ lattice of points $(i, j)$ for all $1 \leq i \leq 3$ and $1 \leq j \leq 24$. She wants to visit every point in the lattice exactly once. In a move, Sally can move to a point in the lattice exactly one unit away. Given that Sally starts at $(2,1)$, compute the number of possible paths Sally can take. | 4096 | 0 |
3,787 | We have a polyhedron such that an ant can walk from one vertex to another, traveling only along edges, and traversing every edge exactly once. What is the smallest possible total number of vertices, edges, and faces of this polyhedron? | 20 | 19.53125 |
3,788 | Compute the number of ways to fill each cell in a $8 \times 8$ square grid with one of the letters $H, M$, or $T$ such that every $2 \times 2$ square in the grid contains the letters $H, M, M, T$ in some order. | 1076 | 0 |
3,789 | There are three pairs of real numbers \left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), and \left(x_{3}, y_{3}\right) that satisfy both $x^{3}-3 x y^{2}=2005$ and $y^{3}-3 x^{2} y=2004$. Compute \left(1-\frac{x_{1}}{y_{1}}\right)\left(1-\frac{x_{2}}{y_{2}}\right)\left(1-\frac{x_{3}}{y_{3}}\right). | 1/1002 | 34.375 |
3,790 | A dot is marked at each vertex of a triangle $A B C$. Then, 2,3 , and 7 more dots are marked on the sides $A B, B C$, and $C A$, respectively. How many triangles have their vertices at these dots? | 357 | 4.6875 |
3,791 | Doug and Ryan are competing in the 2005 Wiffle Ball Home Run Derby. In each round, each player takes a series of swings. Each swing results in either a home run or an out, and an out ends the series. When Doug swings, the probability that he will hit a home run is $1 / 3$. When Ryan swings, the probability that he will hit a home run is $1 / 2$. In one round, what is the probability that Doug will hit more home runs than Ryan hits? | 1/5 | 12.5 |
3,792 | Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a strictly increasing function such that $f(1)=1$ and $f(2n)f(2n+1)=9f(n)^{2}+3f(n)$ for all $n \in \mathbb{N}$. Compute $f(137)$. | 2215 | 2.34375 |
3,793 | Anne-Marie has a deck of 16 cards, each with a distinct positive factor of 2002 written on it. She shuffles the deck and begins to draw cards from the deck without replacement. She stops when there exists a nonempty subset of the cards in her hand whose numbers multiply to a perfect square. What is the expected number of cards in her hand when she stops? | \frac{837}{208} | 0 |
3,794 | Let $f(n)$ be the largest prime factor of $n$. Estimate $$N=\left\lfloor 10^{4} \cdot \frac{\sum_{n=2}^{10^{6}} f\left(n^{2}-1\right)}{\sum_{n=2}^{10^{6}} f(n)}\right\rfloor$$ An estimate of $E$ will receive $\max \left(0,\left\lfloor 20-20\left(\frac{|E-N|}{10^{3}}\right)^{1 / 3}\right\rfloor\right)$ points. | 18215 | 0 |
3,795 | How many positive integers less than or equal to 240 can be expressed as a sum of distinct factorials? Consider 0 ! and 1 ! to be distinct. | 39 | 17.96875 |
3,796 | Our next object up for bid is an arithmetic progression of primes. For example, the primes 3,5, and 7 form an arithmetic progression of length 3. What is the largest possible length of an arithmetic progression formed of positive primes less than 1,000,000? Be prepared to justify your answer. | 12 | 0 |
3,797 | Let $A B C$ be a triangle such that $A B=13, B C=14, C A=15$ and let $E, F$ be the feet of the altitudes from $B$ and $C$, respectively. Let the circumcircle of triangle $A E F$ be $\omega$. We draw three lines, tangent to the circumcircle of triangle $A E F$ at $A, E$, and $F$. Compute the area of the triangle these three lines determine. | \frac{462}{5} | 0 |
3,798 | Compute the number of triples $(f, g, h)$ of permutations on $\{1,2,3,4,5\}$ such that $$ \begin{aligned} & f(g(h(x)))=h(g(f(x)))=g(x), \\ & g(h(f(x)))=f(h(g(x)))=h(x), \text { and } \\ & h(f(g(x)))=g(f(h(x)))=f(x) \end{aligned} $$ for all $x \in\{1,2,3,4,5\}$. | 146 | 0 |
3,799 | In each cell of a $4 \times 4$ grid, one of the two diagonals is drawn uniformly at random. Compute the probability that the resulting 32 triangular regions can be colored red and blue so that any two regions sharing an edge have different colors. | \frac{1}{512} | 0 |
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