Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
4,100 | The number $$316990099009901=\frac{32016000000000001}{101}$$ is the product of two distinct prime numbers. Compute the smaller of these two primes. | 4002001 | 49.21875 |
4,101 | Circle $\Omega$ has radius 5. Points $A$ and $B$ lie on $\Omega$ such that chord $A B$ has length 6. A unit circle $\omega$ is tangent to chord $A B$ at point $T$. Given that $\omega$ is also internally tangent to $\Omega$, find $A T \cdot B T$. | 2 | 0 |
4,102 | If $a$ and $b$ are positive real numbers such that $a \cdot 2^{b}=8$ and $a^{b}=2$, compute $a^{\log _{2} a} 2^{b^{2}}$. | 128 | 64.84375 |
4,103 | Four points, $A, B, C$, and $D$, are chosen randomly on the circumference of a circle with independent uniform probability. What is the expected number of sides of triangle $A B C$ for which the projection of $D$ onto the line containing the side lies between the two vertices? | 3/2 | 76.5625 |
4,104 | In three-dimensional space, let $S$ be the region of points $(x, y, z)$ satisfying $-1 \leq z \leq 1$. Let $S_{1}, S_{2}, \ldots, S_{2022}$ be 2022 independent random rotations of $S$ about the origin ( $0,0,0$). The expected volume of the region $S_{1} \cap S_{2} \cap \cdots \cap S_{2022}$ can be expressed as $\frac{a \pi}{b}$, for relatively prime positive integers $a$ and $b$. Compute $100 a+b$. | 271619 | 0 |
4,105 | Let $A B C D$ be a convex quadrilateral such that $\angle A B D=\angle B C D=90^{\circ}$, and let $M$ be the midpoint of segment $B D$. Suppose that $C M=2$ and $A M=3$. Compute $A D$. | \sqrt{21} | 17.1875 |
4,106 | The vertices of a regular hexagon are labeled $\cos (\theta), \cos (2 \theta), \ldots, \cos (6 \theta)$. For every pair of vertices, Bob draws a blue line through the vertices if one of these functions can be expressed as a polynomial function of the other (that holds for all real $\theta$ ), and otherwise Roberta draws a red line through the vertices. In the resulting graph, how many triangles whose vertices lie on the hexagon have at least one red and at least one blue edge? | 14 | 3.90625 |
4,107 | Let $a_{1}=1$, and let $a_{n}=\left\lfloor n^{3} / a_{n-1}\right\rfloor$ for $n>1$. Determine the value of $a_{999}$. | 999 | 96.875 |
4,108 | In the diagram below, how many distinct paths are there from January 1 to December 31, moving from one adjacent dot to the next either to the right, down, or diagonally down to the right? | 372 | 0 |
4,109 | Let $f$ be a polynomial with integer coefficients such that the greatest common divisor of all its coefficients is 1. For any $n \in \mathbb{N}, f(n)$ is a multiple of 85. Find the smallest possible degree of $f$. | 17 | 3.90625 |
4,110 | Find the largest positive integer solution of the equation $\left\lfloor\frac{N}{3}\right\rfloor=\left\lfloor\frac{N}{5}\right\rfloor+\left\lfloor\frac{N}{7}\right\rfloor-\left\lfloor\frac{N}{35}\right\rfloor$. | 65 | 48.4375 |
4,111 | $A B C D$ is a cyclic quadrilateral in which $A B=3, B C=5, C D=6$, and $A D=10 . M, I$, and $T$ are the feet of the perpendiculars from $D$ to lines $A B, A C$, and $B C$ respectively. Determine the value of $M I / I T$. | \frac{25}{9} | 0 |
4,112 | Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition in at most one place are good enough. In other words, if $d_{i}$ denotes the $i$ th digit, then $d_{i} \leq d_{i+1}$ for at most one value of $i$. For example, Kelvin likes the numbers 43210, 132, and 3, but not the numbers 1337 and 123. How many 5-digit numbers does Kelvin like? | 14034 | 2.34375 |
4,113 | Stan has a stack of 100 blocks and starts with a score of 0, and plays a game in which he iterates the following two-step procedure: (a) Stan picks a stack of blocks and splits it into 2 smaller stacks each with a positive number of blocks, say $a$ and $b$. (The order in which the new piles are placed does not matter.) (b) Stan adds the product of the two piles' sizes, $a b$, to his score. The game ends when there are only 1-block stacks left. What is the expected value of Stan's score at the end of the game? | 4950 | 68.75 |
4,114 | Let $a_{1}, a_{2}, \ldots$ be a sequence defined by $a_{1}=a_{2}=1$ and $a_{n+2}=a_{n+1}+a_{n}$ for $n \geq 1$. Find $$\sum_{n=1}^{\infty} \frac{a_{n}}{4^{n+1}}$$ | \frac{1}{11} | 67.96875 |
4,115 | A candy company makes 5 colors of jellybeans, which come in equal proportions. If I grab a random sample of 5 jellybeans, what is the probability that I get exactly 2 distinct colors? | \frac{12}{125} | 7.8125 |
4,116 | Forty two cards are labeled with the natural numbers 1 through 42 and randomly shuffled into a stack. One by one, cards are taken off of the top of the stack until a card labeled with a prime number is removed. How many cards are removed on average? | \frac{43}{14} | 0 |
4,117 | Suppose $a, b$, and $c$ are distinct positive integers such that $\sqrt{a \sqrt{b \sqrt{c}}}$ is an integer. Compute the least possible value of $a+b+c$. | 7 | 8.59375 |
4,118 | Let $A_{12}$ denote the answer to problem 12. There exists a unique triple of digits $(B, C, D)$ such that $10>A_{12}>B>C>D>0$ and $$\overline{A_{12} B C D}-\overline{D C B A_{12}}=\overline{B D A_{12} C}$$ where $\overline{A_{12} B C D}$ denotes the four digit base 10 integer. Compute $B+C+D$. | 11 | 10.15625 |
4,119 | Compute the circumradius of cyclic hexagon $A B C D E F$, which has side lengths $A B=B C=$ $2, C D=D E=9$, and $E F=F A=12$. | 8 | 0 |
4,120 | Convex quadrilateral $A B C D$ has right angles $\angle A$ and $\angle C$ and is such that $A B=B C$ and $A D=C D$. The diagonals $A C$ and $B D$ intersect at point $M$. Points $P$ and $Q$ lie on the circumcircle of triangle $A M B$ and segment $C D$, respectively, such that points $P, M$, and $Q$ are collinear. Suppose that $m \angle A B C=160^{\circ}$ and $m \angle Q M C=40^{\circ}$. Find $M P \cdot M Q$, given that $M C=6$. | 36 | 83.59375 |
4,121 | Suppose $m>n>1$ are positive integers such that there exist $n$ complex numbers $x_{1}, x_{2}, \ldots, x_{n}$ for which - $x_{1}^{k}+x_{2}^{k}+\cdots+x_{n}^{k}=1$ for $k=1,2, \ldots, n-1$ - $x_{1}^{n}+x_{2}^{n}+\cdots+x_{n}^{n}=2$; and - $x_{1}^{m}+x_{2}^{m}+\cdots+x_{n}^{m}=4$. Compute the smallest possible value of $m+n$. | 34 | 0 |
4,122 | Bob the bomb-defuser has stumbled upon an active bomb. He opens it up, and finds the red and green wires conveniently located for him to cut. Being a seasoned member of the bomb-squad, Bob quickly determines that it is the green wire that he should cut, and puts his wirecutters on the green wire. But just before he starts to cut, the bomb starts to count down, ticking every second. Each time the bomb ticks, starting at time $t=15$ seconds, Bob panics and has a certain chance to move his wirecutters to the other wire. However, he is a rational man even when panicking, and has a $\frac{1}{2 t^{2}}$ chance of switching wires at time $t$, regardless of which wire he is about to cut. When the bomb ticks at $t=1$, Bob cuts whatever wire his wirecutters are on, without switching wires. What is the probability that Bob cuts the green wire? | \frac{23}{30} | 0 |
4,123 | Elisenda has a piece of paper in the shape of a triangle with vertices $A, B$, and $C$ such that $A B=42$. She chooses a point $D$ on segment $A C$, and she folds the paper along line $B D$ so that $A$ lands at a point $E$ on segment $B C$. Then, she folds the paper along line $D E$. When she does this, $B$ lands at the midpoint of segment $D C$. Compute the perimeter of the original unfolded triangle. | 168+48 \sqrt{7} | 0 |
4,124 | Given a positive integer $k$, let \|k\| denote the absolute difference between $k$ and the nearest perfect square. For example, \|13\|=3 since the nearest perfect square to 13 is 16. Compute the smallest positive integer $n$ such that $\frac{\|1\|+\|2\|+\cdots+\|n\|}{n}=100$. | 89800 | 95.3125 |
4,125 | The graph of the equation $x+y=\left\lfloor x^{2}+y^{2}\right\rfloor$ consists of several line segments. Compute the sum of their lengths. | 4+\sqrt{6}-\sqrt{2} | 0 |
4,126 | Find the number of 7 -tuples $\left(n_{1}, \ldots, n_{7}\right)$ of integers such that $$\sum_{i=1}^{7} n_{i}^{6}=96957$$ | 2688 | 0 |
4,127 | Circle $\Omega$ has radius 13. Circle $\omega$ has radius 14 and its center $P$ lies on the boundary of circle $\Omega$. Points $A$ and $B$ lie on $\Omega$ such that chord $A B$ has length 24 and is tangent to $\omega$ at point $T$. Find $A T \cdot B T$. | 56 | 1.5625 |
4,128 | The Fibonacci numbers are defined recursively by $F_{0}=0, F_{1}=1$, and $F_{i}=F_{i-1}+F_{i-2}$ for $i \geq 2$. Given 15 wooden blocks of weights $F_{2}, F_{3}, \ldots, F_{16}$, compute the number of ways to paint each block either red or blue such that the total weight of the red blocks equals the total weight of the blue blocks. | 32 | 2.34375 |
4,129 | Points $A, B$, and $C$ lie in that order on line $\ell$, such that $A B=3$ and $B C=2$. Point $H$ is such that $C H$ is perpendicular to $\ell$. Determine the length $C H$ such that $\angle A H B$ is as large as possible. | \sqrt{10} | 77.34375 |
4,130 | How many positive integers less than 1998 are relatively prime to 1547 ? (Two integers are relatively prime if they have no common factors besides 1.) | 1487 | 78.125 |
4,131 | Compute, in terms of $n$, $\sum_{k=0}^{n}\binom{n-k}{k} 2^{k}$. | \frac{2 \cdot 2^{n}+(-1)^{n}}{3} | 0 |
4,132 | Kelvin and 15 other frogs are in a meeting, for a total of 16 frogs. During the meeting, each pair of distinct frogs becomes friends with probability $\frac{1}{2}$. Kelvin thinks the situation after the meeting is cool if for each of the 16 frogs, the number of friends they made during the meeting is a multiple of 4. Say that the probability of the situation being cool can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime. Find $a$. | 1167 | 0 |
4,133 | $A B C D$ is a cyclic quadrilateral in which $A B=4, B C=3, C D=2$, and $A D=5$. Diagonals $A C$ and $B D$ intersect at $X$. A circle $\omega$ passes through $A$ and is tangent to $B D$ at $X . \omega$ intersects $A B$ and $A D$ at $Y$ and $Z$ respectively. Compute $Y Z / B D$. | \frac{115}{143} | 0 |
4,134 | The spikiness of a sequence $a_{1}, a_{2}, \ldots, a_{n}$ of at least two real numbers is the sum $\sum_{i=1}^{n-1}\left|a_{i+1}-a_{i}\right|$. Suppose $x_{1}, x_{2}, \ldots, x_{9}$ are chosen uniformly and randomly from the interval $[0,1]$. Let $M$ be the largest possible value of the spikiness of a permutation of $x_{1}, x_{2}, \ldots, x_{9}$. Compute the expected value of $M$. | \frac{79}{20} | 0 |
4,135 | Von Neumann's Poker: The first step in Von Neumann's game is selecting a random number on $[0,1]$. To generate this number, Chebby uses the factorial base: the number $0 . A_{1} A_{2} A_{3} A_{4} \ldots$ stands for $\sum_{n=0}^{\infty} \frac{A_{n}}{(n+1)!}$, where each $A_{n}$ is an integer between 0 and $n$, inclusive. Chebby has an infinite number of cards labeled $0, 1, 2, \ldots$. He begins by putting cards $0$ and $1$ into a hat and drawing randomly to determine $A_{1}$. The card assigned $A_{1}$ does not get reused. Chebby then adds in card 2 and draws for $A_{2}$, and continues in this manner to determine the random number. At each step, he only draws one card from two in the hat. Unfortunately, this method does not result in a uniform distribution. What is the expected value of Chebby's final number? | 0.57196 | 0 |
4,136 | For $a$ a positive real number, let $x_{1}, x_{2}, x_{3}$ be the roots of the equation $x^{3}-a x^{2}+a x-a=0$. Determine the smallest possible value of $x_{1}^{3}+x_{2}^{3}+x_{3}^{3}-3 x_{1} x_{2} x_{3}$. | -4 | 92.96875 |
4,137 | There are three video game systems: the Paystation, the WHAT, and the ZBoz2 \pi, and none of these systems will play games for the other systems. Uncle Riemann has three nephews: Bernoulli, Galois, and Dirac. Bernoulli owns a Paystation and a WHAT, Galois owns a WHAT and a ZBoz2 \pi, and Dirac owns a ZBoz2 \pi and a Paystation. A store sells 4 different games for the Paystation, 6 different games for the WHAT, and 10 different games for the ZBoz2 \pi. Uncle Riemann does not understand the difference between the systems, so he walks into the store and buys 3 random games (not necessarily distinct) and randomly hands them to his nephews. What is the probability that each nephew receives a game he can play? | \frac{7}{25} | 7.03125 |
4,138 | Determine all pairs $(a, b)$ of integers with the property that the numbers $a^{2}+4 b$ and $b^{2}+4 a$ are both perfect squares. | (-4,-4),(-5,-6),(-6,-5),(0, k^{2}),(k^{2}, 0),(k, 1-k) | 0 |
4,139 | Suppose a real number \(x>1\) satisfies \(\log _{2}\left(\log _{4} x\right)+\log _{4}\left(\log _{16} x\right)+\log _{16}\left(\log _{2} x\right)=0\). Compute \(\log _{2}\left(\log _{16} x\right)+\log _{16}\left(\log _{4} x\right)+\log _{4}\left(\log _{2} x\right)\). | -\frac{1}{4} | 76.5625 |
4,140 | For positive integers $n$ and $k$, let $\mho(n, k)$ be the number of distinct prime divisors of $n$ that are at least $k$. Find the closest integer to $$\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} \frac{\mho(n, k)}{3^{n+k-7}}$$ | 167 | 59.375 |
4,141 | In a game, \(N\) people are in a room. Each of them simultaneously writes down an integer between 0 and 100 inclusive. A person wins the game if their number is exactly two-thirds of the average of all the numbers written down. There can be multiple winners or no winners in this game. Let \(m\) be the maximum possible number such that it is possible to win the game by writing down \(m\). Find the smallest possible value of \(N\) for which it is possible to win the game by writing down \(m\) in a room of \(N\) people. | 34 | 2.34375 |
4,142 | Descartes's Blackjack: How many integer lattice points (points of the form $(m, n)$ for integers $m$ and $n$) lie inside or on the boundary of the disk of radius 2009 centered at the origin? | 12679605 | 0 |
4,143 | A square can be divided into four congruent figures as shown: If each of the congruent figures has area 1, what is the area of the square? | 4 | 87.5 |
4,144 | Let $n$ be a positive integer, and let $s$ be the sum of the digits of the base-four representation of $2^{n}-1$. If $s=2023$ (in base ten), compute $n$ (in base ten). | 1349 | 23.4375 |
4,145 | Compute $$\sum_{n=0}^{\infty} \frac{n}{n^{4}+n^{2}+1}$$ | 1/2 | 63.28125 |
4,146 | Given that \(x\) is a positive real, find the maximum possible value of \(\sin \left(\tan ^{-1}\left(\frac{x}{9}\right)-\tan ^{-1}\left(\frac{x}{16}\right)\right)\). | \frac{7}{25} | 43.75 |
4,147 | Compute the number of integers \(n \in\{1,2, \ldots, 300\}\) such that \(n\) is the product of two distinct primes, and is also the length of the longest leg of some nondegenerate right triangle with integer side lengths. | 13 | 3.90625 |
4,148 | Find the maximum number of points $X_{i}$ such that for each $i$, $\triangle A B X_{i} \cong \triangle C D X_{i}$. | 4 | 0 |
4,149 | Two real numbers $x$ and $y$ are such that $8 y^{4}+4 x^{2} y^{2}+4 x y^{2}+2 x^{3}+2 y^{2}+2 x=x^{2}+1$. Find all possible values of $x+2 y^{2}$. | \frac{1}{2} | 61.71875 |
4,150 | Find the maximum value of $m$ for a sequence $P_{0}, P_{1}, \cdots, P_{m+1}$ of points on a grid satisfying certain conditions. | n(n-1) | 0 |
4,151 | On a board the following six vectors are written: \((1,0,0), \quad(-1,0,0), \quad(0,1,0), \quad(0,-1,0), \quad(0,0,1), \quad(0,0,-1)\). Given two vectors \(v\) and \(w\) on the board, a move consists of erasing \(v\) and \(w\) and replacing them with \(\frac{1}{\sqrt{2}}(v+w)\) and \(\frac{1}{\sqrt{2}}(v-w)\). After some number of moves, the sum of the six vectors on the board is \(u\). Find, with proof, the maximum possible length of \(u\). | 2 \sqrt{3} | 0.78125 |
4,152 | Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying $f(x) f(y)=f(x-y)$. Find all possible values of $f(2017)$. | 0, 1 | 20.3125 |
4,153 | The Houson Association of Mathematics Educators decides to hold a grand forum on mathematics education and invites a number of politicians from the United States to participate. Around lunch time the politicians decide to play a game. In this game, players can score 19 points for pegging the coordinator of the gathering with a spit ball, 9 points for downing an entire cup of the forum's interpretation of coffee, or 8 points for quoting more than three consecutive words from the speech Senator Bobbo delivered before lunch. What is the product of the two greatest scores that a player cannot score in this game? | 1209 | 28.90625 |
4,154 | The number 770 is written on a blackboard. Melody repeatedly performs moves, where a move consists of subtracting either 40 or 41 from the number on the board. She performs moves until the number is not positive, and then she stops. Let $N$ be the number of sequences of moves that Melody could perform. Suppose $N=a \cdot 2^{b}$ where $a$ is an odd positive integer and $b$ is a nonnegative integer. Compute $100 a+b$. | 318 | 0 |
4,155 | How many perfect squares divide $2^{3} \cdot 3^{5} \cdot 5^{7} \cdot 7^{9}$? | 120 | 71.09375 |
4,156 | Find the value of \(\sum_{k=1}^{60} \sum_{n=1}^{k} \frac{n^{2}}{61-2 n}\). | -18910 | 0 |
4,157 | Kevin starts with the vectors \((1,0)\) and \((0,1)\) and at each time step, he replaces one of the vectors with their sum. Find the cotangent of the minimum possible angle between the vectors after 8 time steps. | 987 | 0.78125 |
4,158 | Suppose \(\triangle A B C\) has lengths \(A B=5, B C=8\), and \(C A=7\), and let \(\omega\) be the circumcircle of \(\triangle A B C\). Let \(X\) be the second intersection of the external angle bisector of \(\angle B\) with \(\omega\), and let \(Y\) be the foot of the perpendicular from \(X\) to \(B C\). Find the length of \(Y C\). | \frac{13}{2} | 0 |
4,159 | Michael picks a random subset of the complex numbers \(\left\{1, \omega, \omega^{2}, \ldots, \omega^{2017}\right\}\) where \(\omega\) is a primitive \(2018^{\text {th }}\) root of unity and all subsets are equally likely to be chosen. If the sum of the elements in his subset is \(S\), what is the expected value of \(|S|^{2}\)? (The sum of the elements of the empty set is 0.) | \frac{1009}{2} | 7.8125 |
4,160 | A polynomial $P$ has four roots, $\frac{1}{4}, \frac{1}{2}, 2,4$. The product of the roots is 1, and $P(1)=1$. Find $P(0)$. | \frac{8}{9} | 54.6875 |
4,161 | In a wooden block shaped like a cube, all the vertices and edge midpoints are marked. The cube is cut along all possible planes that pass through at least four marked points. Let \(N\) be the number of pieces the cube is cut into. Estimate \(N\). An estimate of \(E>0\) earns \(\lfloor 20 \min (N / E, E / N)\rfloor\) points. | 15600 | 0 |
4,162 | Let $a_{1}, a_{2}, \ldots$ be an arithmetic sequence and $b_{1}, b_{2}, \ldots$ be a geometric sequence. Suppose that $a_{1} b_{1}=20$, $a_{2} b_{2}=19$, and $a_{3} b_{3}=14$. Find the greatest possible value of $a_{4} b_{4}$. | \frac{37}{4} | 14.0625 |
4,163 | A triple of integers \((a, b, c)\) satisfies \(a+b c=2017\) and \(b+c a=8\). Find all possible values of \(c\). | -6,0,2,8 | 25.78125 |
4,164 | Find the largest positive integer \(n\) for which there exist \(n\) finite sets \(X_{1}, X_{2}, \ldots, X_{n}\) with the property that for every \(1 \leq a<b<c \leq n\), the equation \(\left|X_{a} \cup X_{b} \cup X_{c}\right|=\lceil\sqrt{a b c}\rceil\) holds. | 4 | 9.375 |
4,165 | The sequence $\left\{a_{n}\right\}_{n \geq 1}$ is defined by $a_{n+2}=7 a_{n+1}-a_{n}$ for positive integers $n$ with initial values $a_{1}=1$ and $a_{2}=8$. Another sequence, $\left\{b_{n}\right\}$, is defined by the rule $b_{n+2}=3 b_{n+1}-b_{n}$ for positive integers $n$ together with the values $b_{1}=1$ and $b_{2}=2$. Find \operatorname{gcd}\left(a_{5000}, b_{501}\right). | 89 | 77.34375 |
4,166 | Suppose there are 100 cookies arranged in a circle, and 53 of them are chocolate chip, with the remainder being oatmeal. Pearl wants to choose a contiguous subsegment of exactly 67 cookies and wants this subsegment to have exactly \(k\) chocolate chip cookies. Find the sum of the \(k\) for which Pearl is guaranteed to succeed regardless of how the cookies are arranged. | 71 | 0.78125 |
4,167 | Five people take a true-or-false test with five questions. Each person randomly guesses on every question. Given that, for each question, a majority of test-takers answered it correctly, let $p$ be the probability that every person answers exactly three questions correctly. Suppose that $p=\frac{a}{2^{b}}$ where $a$ is an odd positive integer and $b$ is a nonnegative integer. Compute 100a+b. | 25517 | 0 |
4,168 | How many graphs are there on 10 vertices labeled \(1,2, \ldots, 10\) such that there are exactly 23 edges and no triangles? | 42840 | 0 |
4,169 | Find all positive integers $a$ and $b$ such that $\frac{a^{2}+b}{b^{2}-a}$ and $\frac{b^{2}+a}{a^{2}-b}$ are both integers. | (2,2),(3,3),(1,2),(2,3),(2,1),(3,2) | 0 |
4,170 | Find the sum of the digits of \(11 \cdot 101 \cdot 111 \cdot 110011\). | 48 | 0 |
4,171 | Triangle \(\triangle A B C\) has \(A B=21, B C=55\), and \(C A=56\). There are two points \(P\) in the plane of \(\triangle A B C\) for which \(\angle B A P=\angle C A P\) and \(\angle B P C=90^{\circ}\). Find the distance between them. | \frac{5}{2} \sqrt{409} | 0 |
4,172 | Find the value of $$\sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \sum_{c=1}^{\infty} \frac{a b(3 a+c)}{4^{a+b+c}(a+b)(b+c)(c+a)}$$ | \frac{1}{54} | 0 |
4,173 | Allen and Yang want to share the numbers \(1,2,3,4,5,6,7,8,9,10\). How many ways are there to split all ten numbers among Allen and Yang so that each person gets at least one number, and either Allen's numbers or Yang's numbers sum to an even number? | 1022 | 7.8125 |
4,174 | Find the total number of different integer values the function $$f(x)=[x]+[2 x]+\left[\frac{5 x}{3}\right]+[3 x]+[4 x]$$ takes for real numbers $x$ with $0 \leq x \leq 100$. Note: $[t]$ is the largest integer that does not exceed $t$. | 734 | 57.03125 |
4,175 | Let $\omega_{1}, \omega_{2}, \ldots, \omega_{100}$ be the roots of $\frac{x^{101}-1}{x-1}$ (in some order). Consider the set $S=\left\{\omega_{1}^{1}, \omega_{2}^{2}, \omega_{3}^{3}, \ldots, \omega_{100}^{100}\right\}$. Let $M$ be the maximum possible number of unique values in $S$, and let $N$ be the minimum possible number of unique values in $S$. Find $M-N$. | 98 | 2.34375 |
4,176 | Let $P A B C$ be a tetrahedron such that $\angle A P B=\angle A P C=\angle B P C=90^{\circ}, \angle A B C=30^{\circ}$, and $A P^{2}$ equals the area of triangle $A B C$. Compute $\tan \angle A C B$. | 8+5 \sqrt{3} | 0 |
4,177 | Let $a_{0}, a_{1}, a_{2}, \ldots$ be an infinite sequence where each term is independently and uniformly random in the set $\{1,2,3,4\}$. Define an infinite sequence $b_{0}, b_{1}, b_{2}, \ldots$ recursively by $b_{0}=1$ and $b_{i+1}=a_{i}^{b_{i}}$. Compute the expected value of the smallest positive integer $k$ such that $b_{k} \equiv 1(\bmod 5)$. | \frac{35}{16} | 0 |
4,178 | 679 contestants participated in HMMT February 2017. Let \(N\) be the number of these contestants who performed at or above the median score in at least one of the three individual tests. Estimate \(N\). An estimate of \(E\) earns \(\left\lfloor 20-\frac{|E-N|}{2}\right\rfloor\) or 0 points, whichever is greater. | 516 | 0 |
4,179 | Determine all positive integers $n$ for which the equation $$x^{n}+(2+x)^{n}+(2-x)^{n}=0$$ has an integer as a solution. | n=1 | 25.78125 |
4,180 | How many positive integers $n \leq 2009$ have the property that $\left\lfloor\log _{2}(n)\right\rfloor$ is odd? | 682 | 37.5 |
4,181 | Let $A X B Y$ be a cyclic quadrilateral, and let line $A B$ and line $X Y$ intersect at $C$. Suppose $A X \cdot A Y=6, B X \cdot B Y=5$, and $C X \cdot C Y=4$. Compute $A B^{2}$. | \frac{242}{15} | 0 |
4,182 | Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying $f(1)=1$ and for $n \in \mathbb{N}, f(2 n)=2 f(n)$ and $f(2 n+1)=2 f(n)-1$. Determine the sum of all positive integer solutions to $f(x)=19$ that do not exceed 2019. | 1889 | 0 |
4,183 | Randall proposes a new temperature system called Felsius temperature with the following conversion between Felsius \(^{\circ} \mathrm{E}\), Celsius \(^{\circ} \mathrm{C}\), and Fahrenheit \(^{\circ} \mathrm{F}\): \(^{\circ} E=\frac{7 \times{ }^{\circ} \mathrm{C}}{5}+16=\frac{7 \times{ }^{\circ} \mathrm{F}-80}{9}\). For example, \(0^{\circ} \mathrm{C}=16^{\circ} \mathrm{E}\). Let \(x, y, z\) be real numbers such that \(x^{\circ} \mathrm{C}=x^{\circ} \mathrm{E}, y^{\circ} E=y^{\circ} \mathrm{F}, z^{\circ} \mathrm{C}=z^{\circ} F\). Find \(x+y+z\). | -120 | 26.5625 |
4,184 | A sequence consists of the digits $122333444455555 \ldots$ such that each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the 4501st and 4052nd digits of this sequence. | 13 | 9.375 |
4,185 | Let \{a_{n}\}_{n \geq 1}$ be an arithmetic sequence and \{g_{n}\}_{n \geq 1}$ be a geometric sequence such that the first four terms of \{a_{n}+g_{n}\}$ are $0,0,1$, and 0 , in that order. What is the 10th term of \{a_{n}+g_{n}\}$ ? | -54 | 18.75 |
4,186 | Let \(a, b, c\) be positive integers. All the roots of each of the quadratics \(a x^{2}+b x+c, a x^{2}+b x-c, a x^{2}-b x+c, a x^{2}-b x-c\) are integers. Over all triples \((a, b, c)\), find the triple with the third smallest value of \(a+b+c\). | (1,10,24) | 0 |
4,187 | Compute the value of \(\frac{\cos 30.5^{\circ}+\cos 31.5^{\circ}+\ldots+\cos 44.5^{\circ}}{\sin 30.5^{\circ}+\sin 31.5^{\circ}+\ldots+\sin 44.5^{\circ}}\). | (\sqrt{2}-1)(\sqrt{3}+\sqrt{2})=2-\sqrt{2}-\sqrt{3}+\sqrt{6} | 0 |
4,188 | Let $\alpha, \beta$, and $\gamma$ be three real numbers. Suppose that $\cos \alpha+\cos \beta+\cos \gamma =1$ and $\sin \alpha+\sin \beta+\sin \gamma =1$. Find the smallest possible value of $\cos \alpha$. | \frac{-1-\sqrt{7}}{4} | 0 |
4,189 | Find the number of integers $n$ such that $$ 1+\left\lfloor\frac{100 n}{101}\right\rfloor=\left\lceil\frac{99 n}{100}\right\rceil $$ | 10100 | 6.25 |
4,190 | Let $S$ be the set of integers of the form $2^{x}+2^{y}+2^{z}$, where $x, y, z$ are pairwise distinct non-negative integers. Determine the 100th smallest element of $S$. | 577 | 99.21875 |
4,191 | Determine the form of $n$ such that $2^n + 2$ is divisible by $n$ where $n$ is less than 100. | n=6, 66, 946 | 0 |
4,192 | Let $S_{0}$ be a unit square in the Cartesian plane with horizontal and vertical sides. For any $n>0$, the shape $S_{n}$ is formed by adjoining 9 copies of $S_{n-1}$ in a $3 \times 3$ grid, and then removing the center copy. Let $a_{n}$ be the expected value of $\left|x-x^{\prime}\right|+\left|y-y^{\prime}\right|$, where $(x, y)$ and $\left(x^{\prime}, y^{\prime}\right)$ are two points chosen randomly within $S_{n}$. There exist relatively prime positive integers $a$ and $b$ such that $$\lim _{n \rightarrow \infty} \frac{a_{n}}{3^{n}}=\frac{a}{b}$$ Compute $100 a+b$. | 1217 | 0 |
4,193 | Determine all pairs $(h, s)$ of positive integers with the following property: If one draws $h$ horizontal lines and another $s$ lines which satisfy (i) they are not horizontal, (ii) no two of them are parallel, (iii) no three of the $h+s$ lines are concurrent, then the number of regions formed by these $h+s$ lines is 1992. | (995,1),(176,10),(80,21) | 0 |
4,194 | Rachel has the number 1000 in her hands. When she puts the number $x$ in her left pocket, the number changes to $x+1$. When she puts the number $x$ in her right pocket, the number changes to $x^{-1}$. Each minute, she flips a fair coin. If it lands heads, she puts the number into her left pocket, and if it lands tails, she puts it into her right pocket. She then takes the new number out of her pocket. If the expected value of the number in Rachel's hands after eight minutes is $E$, then compute $\left\lfloor\frac{E}{10}\right\rfloor$. | 13 | 0 |
4,195 | Given that $\sin A+\sin B=1$ and $\cos A+\cos B=3 / 2$, what is the value of $\cos (A-B)$? | 5/8 | 80.46875 |
4,196 | Compute the number of ways to select 99 cells of a $19 \times 19$ square grid such that no two selected cells share an edge or vertex. | 1000 | 0 |
4,197 | Suppose $A B C D$ is a convex quadrilateral with $\angle A B D=105^{\circ}, \angle A D B=15^{\circ}, A C=7$, and $B C=C D=5$. Compute the sum of all possible values of $B D$. | \sqrt{291} | 0 |
4,198 | A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied: (i) All the squares are congruent. (ii) If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares. (iii) Each square touches exactly three other squares. How many positive integers $n$ are there with $2018 \leq n \leq 3018$, such that there exists a collection of $n$ squares that is tri-connected? | 501 | 75 |
4,199 | Let $x, y$ be complex numbers such that \frac{x^{2}+y^{2}}{x+y}=4$ and \frac{x^{4}+y^{4}}{x^{3}+y^{3}}=2$. Find all possible values of \frac{x^{6}+y^{6}}{x^{5}+y^{5}}$. | 10 \pm 2 \sqrt{17} | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.