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3,200 | Let $ABCD$ be a quadrilateral with side lengths $AB=2, BC=3, CD=5$, and $DA=4$. What is the maximum possible radius of a circle inscribed in quadrilateral $ABCD$? | \frac{2\sqrt{30}}{7} | 67.1875 |
3,201 | Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer $n<60$ (both Kelly and Jason know that $n<60$, but that they don't know what the value of $n$ is). The computer tells Kelly the unit digit of $n$, and it tells Jason the number of divisors of $n$. Then, Kelly and Jason have the following dialogue: Kelly: I don't know what $n$ is, and I'm sure that you don't know either. However, I know that $n$ is divisible by at least two different primes. Jason: Oh, then I know what the value of $n$ is. Kelly: Now I also know what $n$ is. Assuming that both Kelly and Jason speak truthfully and to the best of their knowledge, what are all the possible values of $n$? | 10 | 0.78125 |
3,202 | Compute the value of $\sqrt{105^{3}-104^{3}}$, given that it is a positive integer. | 181 | 71.09375 |
3,203 | Kelvin the Frog and 10 of his relatives are at a party. Every pair of frogs is either friendly or unfriendly. When 3 pairwise friendly frogs meet up, they will gossip about one another and end up in a fight (but stay friendly anyway). When 3 pairwise unfriendly frogs meet up, they will also end up in a fight. In all other cases, common ground is found and there is no fight. If all $\binom{11}{3}$ triples of frogs meet up exactly once, what is the minimum possible number of fights? | 28 | 0 |
3,204 | Cyclic pentagon $ABCDE$ has side lengths $AB=BC=5, CD=DE=12$, and $AE=14$. Determine the radius of its circumcircle. | \frac{225\sqrt{11}}{88} | 0 |
3,205 | For a positive integer $n$, let $\theta(n)$ denote the number of integers $0 \leq x<2010$ such that $x^{2}-n$ is divisible by 2010. Determine the remainder when $\sum_{n=0}^{2009} n \cdot \theta(n)$ is divided by 2010. | 335 | 48.4375 |
3,206 | The set of points $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ in $\mathbf{R}^{4}$ such that $x_{1} \geq x_{2} \geq x_{3} \geq x_{4}$ is a cone (or hypercone, if you insist). Into how many regions is this cone sliced by the hyperplanes $x_{i}-x_{j}=1$ for $1 \leq i<j \leq n$ ? | 14 | 10.15625 |
3,207 | A point in three-space has distances $2,6,7,8,9$ from five of the vertices of a regular octahedron. What is its distance from the sixth vertex? | \sqrt{21} | 0 |
3,208 | Find the largest integer that divides $m^{5}-5 m^{3}+4 m$ for all $m \geq 5$. | 120 | 53.90625 |
3,209 | How many lattice points are enclosed by the triangle with vertices $(0,99),(5,100)$, and $(2003,500) ?$ Don't count boundary points. | 0 | 14.84375 |
3,210 | Find the sum of the reciprocals of all the (positive) divisors of 144. | 403/144 | 50 |
3,211 | What is the smallest positive integer $x$ for which $x^{2}+x+41$ is not a prime? | 40 | 83.59375 |
3,212 | Tessa picks three real numbers $x, y, z$ and computes the values of the eight expressions of the form $\pm x \pm y \pm z$. She notices that the eight values are all distinct, so she writes the expressions down in increasing order. How many possible orders are there? | 96 | 0 |
3,213 | Alice, Bob, and Charlie roll a 4, 5, and 6-sided die, respectively. What is the probability that a number comes up exactly twice out of the three rolls? | \frac{13}{30} | 0 |
3,214 | Suppose that $A, B, C, D$ are four points in the plane, and let $Q, R, S, T, U, V$ be the respective midpoints of $A B, A C, A D, B C, B D, C D$. If $Q R=2001, S U=2002, T V=$ 2003, find the distance between the midpoints of $Q U$ and $R V$. | 2001 | 35.9375 |
3,215 | Let $\varphi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Let $S$ be the set of positive integers $n$ such that $\frac{2 n}{\varphi(n)}$ is an integer. Compute the sum $\sum_{n \in S} \frac{1}{n}$. | \frac{10}{3} | 0 |
3,216 | Find the real solution(s) to the equation $(x+y)^{2}=(x+1)(y-1)$. | (-1,1) | 57.03125 |
3,217 | Let $(x, y)$ be a pair of real numbers satisfying $$56x+33y=\frac{-y}{x^{2}+y^{2}}, \quad \text { and } \quad 33x-56y=\frac{x}{x^{2}+y^{2}}$$ Determine the value of $|x|+|y|$. | \frac{11}{65} | 17.96875 |
3,218 | The rational numbers $x$ and $y$, when written in lowest terms, have denominators 60 and 70 , respectively. What is the smallest possible denominator of $x+y$ ? | 84 | 0 |
3,219 | Let $A B C$ be a triangle and $D, E$, and $F$ be the midpoints of sides $B C, C A$, and $A B$ respectively. What is the maximum number of circles which pass through at least 3 of these 6 points? | 17 | 2.34375 |
3,220 | Let $a, b, c$ be nonzero real numbers such that $a+b+c=0$ and $a^{3}+b^{3}+c^{3}=a^{5}+b^{5}+c^{5}$. Find the value of $a^{2}+b^{2}+c^{2}$. | \frac{6}{5} | 24.21875 |
3,221 | Mrs. Toad has a class of 2017 students, with unhappiness levels $1,2, \ldots, 2017$ respectively. Today in class, there is a group project and Mrs. Toad wants to split the class in exactly 15 groups. The unhappiness level of a group is the average unhappiness of its members, and the unhappiness of the class is the sum of the unhappiness of all 15 groups. What's the minimum unhappiness of the class Mrs. Toad can achieve by splitting the class into 15 groups? | 1121 | 0 |
3,222 | You have 128 teams in a single elimination tournament. The Engineers and the Crimson are two of these teams. Each of the 128 teams in the tournament is equally strong, so during each match, each team has an equal probability of winning. Now, the 128 teams are randomly put into the bracket. What is the probability that the Engineers play the Crimson sometime during the tournament? | \frac{1}{64} | 23.4375 |
3,223 | For the sequence of numbers $n_{1}, n_{2}, n_{3}, \ldots$, the relation $n_{i}=2 n_{i-1}+a$ holds for all $i>1$. If $n_{2}=5$ and $n_{8}=257$, what is $n_{5}$ ? | 33 | 71.875 |
3,224 | In some squares of a $2012\times 2012$ grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle $B$ flies to the centre of the square on which it lands is called the [i]translation vector[/i] of beetle $B$.
For all possible starting and ending configurations, find the maximum length of the sum of the [i]translation vectors[/i] of all beetles. | \frac{2012^3}{4} | 0 |
3,225 | Let $z=1-2 i$. Find $\frac{1}{z}+\frac{2}{z^{2}}+\frac{3}{z^{3}}+\cdots$. | (2i-1)/4 | 0 |
3,226 | Let $P(x)$ be a polynomial with degree 2008 and leading coefficient 1 such that $P(0)=2007, P(1)=2006, P(2)=2005, \ldots, P(2007)=0$. Determine the value of $P(2008)$. You may use factorials in your answer. | 2008!-1 | 94.53125 |
3,227 | Let $P$ be a polyhedron where every face is a regular polygon, and every edge has length 1. Each vertex of $P$ is incident to two regular hexagons and one square. Choose a vertex $V$ of the polyhedron. Find the volume of the set of all points contained in $P$ that are closer to $V$ than to any other vertex. | \frac{\sqrt{2}}{3} | 25 |
3,228 | The sequences of real numbers $\left\{a_{i}\right\}_{i=1}^{\infty}$ and $\left\{b_{i}\right\}_{i=1}^{\infty}$ satisfy $a_{n+1}=\left(a_{n-1}-1\right)\left(b_{n}+1\right)$ and $b_{n+1}=a_{n} b_{n-1}-1$ for $n \geq 2$, with $a_{1}=a_{2}=2015$ and $b_{1}=b_{2}=2013$. Evaluate, with proof, the infinite sum $\sum_{n=1}^{\infty} b_{n}\left(\frac{1}{a_{n+1}}-\frac{1}{a_{n+3}}\right)$. | 1+\frac{1}{2014 \cdot 2015} | 0 |
3,229 | Let $S$ be the set of $3^{4}$ points in four-dimensional space where each coordinate is in $\{-1,0,1\}$. Let $N$ be the number of sequences of points $P_{1}, P_{2}, \ldots, P_{2020}$ in $S$ such that $P_{i} P_{i+1}=2$ for all $1 \leq i \leq 2020$ and $P_{1}=(0,0,0,0)$. (Here $P_{2021}=P_{1}$.) Find the largest integer $n$ such that $2^{n}$ divides $N$. | 4041 | 0.78125 |
3,230 | Let $S=\{(x, y) \mid x>0, y>0, x+y<200$, and $x, y \in \mathbb{Z}\}$. Find the number of parabolas $\mathcal{P}$ with vertex $V$ that satisfy the following conditions: - $\mathcal{P}$ goes through both $(100,100)$ and at least one point in $S$, - $V$ has integer coordinates, and - $\mathcal{P}$ is tangent to the line $x+y=0$ at $V$. | 264 | 0 |
3,231 | Let $\triangle A B C$ be a triangle with $A B=7, B C=1$, and $C A=4 \sqrt{3}$. The angle trisectors of $C$ intersect $\overline{A B}$ at $D$ and $E$, and lines $\overline{A C}$ and $\overline{B C}$ intersect the circumcircle of $\triangle C D E$ again at $X$ and $Y$, respectively. Find the length of $X Y$. | \frac{112}{65} | 0 |
3,232 | In a game show, Bob is faced with 7 doors, 2 of which hide prizes. After he chooses a door, the host opens three other doors, of which one is hiding a prize. Bob chooses to switch to another door. What is the probability that his new door is hiding a prize? | \frac{5}{21} | 9.375 |
3,233 | A collection $\mathcal{S}$ of 10000 points is formed by picking each point uniformly at random inside a circle of radius 1. Let $N$ be the expected number of points of $\mathcal{S}$ which are vertices of the convex hull of the $\mathcal{S}$. (The convex hull is the smallest convex polygon containing every point of $\mathcal{S}$.) Estimate $N$. | 72.8 | 0 |
3,234 | How many solutions in nonnegative integers $(a, b, c)$ are there to the equation $2^{a}+2^{b}=c!\quad ?$ | 5 | 3.90625 |
3,235 | A palindrome is a positive integer that reads the same backwards as forwards, such as 82328. What is the smallest 5 -digit palindrome that is a multiple of 99 ? | 54945 | 0 |
3,236 | We call a positive integer $t$ good if there is a sequence $a_{0}, a_{1}, \ldots$ of positive integers satisfying $a_{0}=15, a_{1}=t$, and $a_{n-1} a_{n+1}=\left(a_{n}-1\right)\left(a_{n}+1\right)$ for all positive integers $n$. Find the sum of all good numbers. | 296 | 0 |
3,237 | Anastasia is taking a walk in the plane, starting from $(1,0)$. Each second, if she is at $(x, y)$, she moves to one of the points $(x-1, y),(x+1, y),(x, y-1)$, and $(x, y+1)$, each with $\frac{1}{4}$ probability. She stops as soon as she hits a point of the form $(k, k)$. What is the probability that $k$ is divisible by 3 when she stops? | \frac{3-\sqrt{3}}{3} | 0 |
3,238 | Let $A B C$ be a triangle and $\omega$ be its circumcircle. The point $M$ is the midpoint of arc $B C$ not containing $A$ on $\omega$ and $D$ is chosen so that $D M$ is tangent to $\omega$ and is on the same side of $A M$ as $C$. It is given that $A M=A C$ and $\angle D M C=38^{\circ}$. Find the measure of angle $\angle A C B$. | 33^{\circ} | 3.125 |
3,239 | Square $A B C D$ has side length 1. A dilation is performed about point $A$, creating square $A B^{\prime} C^{\prime} D^{\prime}$. If $B C^{\prime}=29$, determine the area of triangle $B D C^{\prime}$. | 420 | 0 |
3,240 | How many ways are there to label the faces of a regular octahedron with the integers 18, using each exactly once, so that any two faces that share an edge have numbers that are relatively prime? Physically realizable rotations are considered indistinguishable, but physically unrealizable reflections are considered different. | 12 | 16.40625 |
3,241 | Cyclic pentagon $ABCDE$ has a right angle $\angle ABC=90^{\circ}$ and side lengths $AB=15$ and $BC=20$. Supposing that $AB=DE=EA$, find $CD$. | 7 | 0 |
3,242 | A vertex-induced subgraph is a subset of the vertices of a graph together with any edges whose endpoints are both in this subset. An undirected graph contains 10 nodes and $m$ edges, with no loops or multiple edges. What is the minimum possible value of $m$ such that this graph must contain a nonempty vertex-induced subgraph where all vertices have degree at least 5? | 31 | 1.5625 |
3,243 | There exist several solutions to the equation $1+\frac{\sin x}{\sin 4 x}=\frac{\sin 3 x}{\sin 2 x}$ where $x$ is expressed in degrees and $0^{\circ}<x<180^{\circ}$. Find the sum of all such solutions. | 320^{\circ} | 0 |
3,244 | Consider the graph in 3-space of $0=xyz(x+y)(y+z)(z+x)(x-y)(y-z)(z-x)$. This graph divides 3-space into $N$ connected regions. What is $N$? | 48 | 3.90625 |
3,245 | A snake of length $k$ is an animal which occupies an ordered $k$-tuple $\left(s_{1}, \ldots, s_{k}\right)$ of cells in a $n \times n$ grid of square unit cells. These cells must be pairwise distinct, and $s_{i}$ and $s_{i+1}$ must share a side for $i=1, \ldots, k-1$. If the snake is currently occupying $\left(s_{1}, \ldots, s_{k}\right)$ and $s$ is an unoccupied cell sharing a side with $s_{1}$, the snake can move to occupy $\left(s, s_{1}, \ldots, s_{k-1}\right)$ instead. Initially, a snake of length 4 is in the grid $\{1,2, \ldots, 30\}^{2}$ occupying the positions $(1,1),(1,2),(1,3),(1,4)$ with $(1,1)$ as its head. The snake repeatedly makes a move uniformly at random among moves it can legally make. Estimate $N$, the expected number of moves the snake makes before it has no legal moves remaining. | 4571.8706930 | 0 |
3,246 | Let $ABCD$ be a convex quadrilateral with $AC=7$ and $BD=17$. Let $M, P, N, Q$ be the midpoints of sides $AB, BC, CD, DA$ respectively. Compute $MN^{2}+PQ^{2}$. | 169 | 2.34375 |
3,247 | Find all integers $m$ such that $m^{2}+6 m+28$ is a perfect square. | 6,-12 | 20.3125 |
3,248 | At a nursery, 2006 babies sit in a circle. Suddenly each baby pokes the baby immediately to either its left or its right, with equal probability. What is the expected number of unpoked babies? | \frac{1003}{2} | 0 |
3,249 | Point $A$ lies at $(0,4)$ and point $B$ lies at $(3,8)$. Find the $x$-coordinate of the point $X$ on the $x$-axis maximizing $\angle AXB$. | 5 \sqrt{2}-3 | 0.78125 |
3,250 | Given right triangle $ABC$, with $AB=4, BC=3$, and $CA=5$. Circle $\omega$ passes through $A$ and is tangent to $BC$ at $C$. What is the radius of $\omega$? | \frac{25}{8} | 29.6875 |
3,251 | Alice and the Cheshire Cat play a game. At each step, Alice either (1) gives the cat a penny, which causes the cat to change the number of (magic) beans that Alice has from $n$ to $5n$ or (2) gives the cat a nickel, which causes the cat to give Alice another bean. Alice wins (and the cat disappears) as soon as the number of beans Alice has is greater than 2008 and has last two digits 42. What is the minimum number of cents Alice can spend to win the game, assuming she starts with 0 beans? | 35 | 72.65625 |
3,252 | Determine all \(\alpha \in \mathbb{R}\) such that for every continuous function \(f:[0,1] \rightarrow \mathbb{R}\), differentiable on \((0,1)\), with \(f(0)=0\) and \(f(1)=1\), there exists some \(\xi \in(0,1)\) such that \(f(\xi)+\alpha=f^{\prime}(\xi)\). | \(\alpha = \frac{1}{e-1}\) | 0 |
3,253 | In how many ways can one fill a \(4 \times 4\) grid with a 0 or 1 in each square such that the sum of the entries in each row, column, and long diagonal is even? | 256 | 82.8125 |
3,254 | Determine the number of four-digit integers $n$ such that $n$ and $2n$ are both palindromes. | 20 | 83.59375 |
3,255 | Let $A B C D$ be a cyclic quadrilateral, and let segments $A C$ and $B D$ intersect at $E$. Let $W$ and $Y$ be the feet of the altitudes from $E$ to sides $D A$ and $B C$, respectively, and let $X$ and $Z$ be the midpoints of sides $A B$ and $C D$, respectively. Given that the area of $A E D$ is 9, the area of $B E C$ is 25, and $\angle E B C-\angle E C B=30^{\circ}$, then compute the area of $W X Y Z$. | 17+\frac{15}{2} \sqrt{3} | 0 |
3,256 | Consider an equilateral triangular grid $G$ with 20 points on a side, where each row consists of points spaced 1 unit apart. More specifically, there is a single point in the first row, two points in the second row, ..., and 20 points in the last row, for a total of 210 points. Let $S$ be a closed non-selfintersecting polygon which has 210 vertices, using each point in $G$ exactly once. Find the sum of all possible values of the area of $S$. | 52 \sqrt{3} | 0 |
3,257 | A Vandal and a Moderator are editing a Wikipedia article. The article originally is error-free. Each day, the Vandal introduces one new error into the Wikipedia article. At the end of the day, the moderator checks the article and has a $2 / 3$ chance of catching each individual error still in the article. After 3 days, what is the probability that the article is error-free? | \frac{416}{729} | 0.78125 |
3,258 | For odd primes $p$, let $f(p)$ denote the smallest positive integer $a$ for which there does not exist an integer $n$ satisfying $p \mid n^{2}-a$. Estimate $N$, the sum of $f(p)^{2}$ over the first $10^{5}$ odd primes $p$. An estimate of $E>0$ will receive $\left\lfloor 22 \min (N / E, E / N)^{3}\right\rfloor$ points. | 2266067 | 0 |
3,259 | In the alphametic $W E \times E Y E=S C E N E$, each different letter stands for a different digit, and no word begins with a 0. The $W$ in this problem has the same value as the $W$ in problem 31. Find $S$. | 5 | 0.78125 |
3,260 | Two sides of a regular $n$-gon are extended to meet at a $28^{\circ}$ angle. What is the smallest possible value for $n$? | 45 | 0 |
3,261 | Let $A B C$ be a triangle with $A B=2, C A=3, B C=4$. Let $D$ be the point diametrically opposite $A$ on the circumcircle of $A B C$, and let $E$ lie on line $A D$ such that $D$ is the midpoint of $\overline{A E}$. Line $l$ passes through $E$ perpendicular to $\overline{A E}$, and $F$ and $G$ are the intersections of the extensions of $\overline{A B}$ and $\overline{A C}$ with $l$. Compute $F G$. | \frac{1024}{45} | 0 |
3,262 | Compute the positive integer less than 1000 which has exactly 29 positive proper divisors. | 720 | 75 |
3,263 | Let $\triangle A B C$ be a triangle inscribed in a unit circle with center $O$. Let $I$ be the incenter of $\triangle A B C$, and let $D$ be the intersection of $B C$ and the angle bisector of $\angle B A C$. Suppose that the circumcircle of $\triangle A D O$ intersects $B C$ again at a point $E$ such that $E$ lies on $I O$. If $\cos A=\frac{12}{13}$, find the area of $\triangle A B C$. | \frac{15}{169} | 0 |
3,264 | Find all positive integers $n>1$ for which $\frac{n^{2}+7 n+136}{n-1}$ is the square of a positive integer. | 5, 37 | 0.78125 |
3,265 | There are 100 houses in a row on a street. A painter comes and paints every house red. Then, another painter comes and paints every third house (starting with house number 3) blue. Another painter comes and paints every fifth house red (even if it is already red), then another painter paints every seventh house blue, and so forth, alternating between red and blue, until 50 painters have been by. After this is finished, how many houses will be red? | 52 | 0 |
3,266 | Points $A, C$, and $B$ lie on a line in that order such that $A C=4$ and $B C=2$. Circles $\omega_{1}, \omega_{2}$, and $\omega_{3}$ have $\overline{B C}, \overline{A C}$, and $\overline{A B}$ as diameters. Circle $\Gamma$ is externally tangent to $\omega_{1}$ and $\omega_{2}$ at $D$ and $E$ respectively, and is internally tangent to $\omega_{3}$. Compute the circumradius of triangle $C D E$. | \frac{2}{3} | 0.78125 |
3,267 | There are \(n\) girls \(G_{1}, \ldots, G_{n}\) and \(n\) boys \(B_{1}, \ldots, B_{n}\). A pair \((G_{i}, B_{j})\) is called suitable if and only if girl \(G_{i}\) is willing to marry boy \(B_{j}\). Given that there is exactly one way to pair each girl with a distinct boy that she is willing to marry, what is the maximal possible number of suitable pairs? | \frac{n(n+1)}{2} | 0 |
3,268 | Find the maximum possible number of diagonals of equal length in a convex hexagon. | 7 | 0.78125 |
3,269 | Let $Y$ be as in problem 14. Find the maximum $Z$ such that three circles of radius $\sqrt{Z}$ can simultaneously fit inside an equilateral triangle of area $Y$ without overlapping each other. | 10 \sqrt{3}-15 | 0 |
3,270 | Spencer is making burritos, each of which consists of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos; in how many orders can he make exactly five burritos? | 25 | 49.21875 |
3,271 | Let $a \geq b \geq c$ be real numbers such that $$\begin{aligned} a^{2} b c+a b^{2} c+a b c^{2}+8 & =a+b+c \\ a^{2} b+a^{2} c+b^{2} c+b^{2} a+c^{2} a+c^{2} b+3 a b c & =-4 \\ a^{2} b^{2} c+a b^{2} c^{2}+a^{2} b c^{2} & =2+a b+b c+c a \end{aligned}$$ If $a+b+c>0$, then compute the integer nearest to $a^{5}$. | 1279 | 0 |
3,272 | A sequence is defined by $A_{0}=0, A_{1}=1, A_{2}=2$, and, for integers $n \geq 3$, $$A_{n}=\frac{A_{n-1}+A_{n-2}+A_{n-3}}{3}+\frac{1}{n^{4}-n^{2}}$$ Compute $\lim _{N \rightarrow \infty} A_{N}$. | \frac{13}{6}-\frac{\pi^{2}}{12} | 0 |
3,273 | In a game, there are three indistinguishable boxes; one box contains two red balls, one contains two blue balls, and the last contains one ball of each color. To play, Raj first predicts whether he will draw two balls of the same color or two of different colors. Then, he picks a box, draws a ball at random, looks at the color, and replaces the ball in the same box. Finally, he repeats this; however, the boxes are not shuffled between draws, so he can determine whether he wants to draw again from the same box. Raj wins if he predicts correctly; if he plays optimally, what is the probability that he will win? | \frac{5}{6} | 38.28125 |
3,274 | Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers defined by $a_{0}=21, a_{1}=35$, and $a_{n+2}=4 a_{n+1}-4 a_{n}+n^{2}$ for $n \geq 2$. Compute the remainder obtained when $a_{2006}$ is divided by 100. | 0 | 0 |
3,275 | Wesyu is a farmer, and she's building a cao (a relative of the cow) pasture. She starts with a triangle $A_{0} A_{1} A_{2}$ where angle $A_{0}$ is $90^{\circ}$, angle $A_{1}$ is $60^{\circ}$, and $A_{0} A_{1}$ is 1. She then extends the pasture. First, she extends $A_{2} A_{0}$ to $A_{3}$ such that $A_{3} A_{0}=\frac{1}{2} A_{2} A_{0}$ and the new pasture is triangle $A_{1} A_{2} A_{3}$. Next, she extends $A_{3} A_{1}$ to $A_{4}$ such that $A_{4} A_{1}=\frac{1}{6} A_{3} A_{1}$. She continues, each time extending $A_{n} A_{n-2}$ to $A_{n+1}$ such that $A_{n+1} A_{n-2}=\frac{1}{2^{n}-2} A_{n} A_{n-2}$. What is the smallest $K$ such that her pasture never exceeds an area of $K$? | \sqrt{3} | 5.46875 |
3,276 | A function $f: A \rightarrow A$ is called idempotent if $f(f(x))=f(x)$ for all $x \in A$. Let $I_{n}$ be the number of idempotent functions from $\{1,2, \ldots, n\}$ to itself. Compute $\sum_{n=1}^{\infty} \frac{I_{n}}{n!}$. | e^{e}-1 | 0 |
3,277 | Determine all triplets of real numbers $(x, y, z)$ satisfying the system of equations $x^{2} y+y^{2} z =1040$, $x^{2} z+z^{2} y =260$, $(x-y)(y-z)(z-x) =-540$. | (16,4,1),(1,16,4) | 0 |
3,278 | Find, as a function of $\, n, \,$ the sum of the digits of \[9 \times 99 \times 9999 \times \cdots \times \left( 10^{2^n} - 1 \right),\] where each factor has twice as many digits as the previous one. | \[ 9 \cdot 2^n \] | 0 |
3,279 | Let $ABC$ be a right triangle with $\angle A=90^{\circ}$. Let $D$ be the midpoint of $AB$ and let $E$ be a point on segment $AC$ such that $AD=AE$. Let $BE$ meet $CD$ at $F$. If $\angle BFC=135^{\circ}$, determine $BC/AB$. | \frac{\sqrt{13}}{2} | 6.25 |
3,280 | Find the smallest positive integer $n$ such that $\frac{5^{n+1}+2^{n+1}}{5^{n}+2^{n}}>4.99$. | 7 | 30.46875 |
3,281 | Let $X$ be as in problem 13. Let $Y$ be the number of ways to order $X$ crimson flowers, $X$ scarlet flowers, and $X$ vermillion flowers in a row so that no two flowers of the same hue are adjacent. (Flowers of the same hue are mutually indistinguishable.) Find $Y$. | 30 | 3.125 |
3,282 | Let $ABCD$ be a regular tetrahedron, and let $O$ be the centroid of triangle $BCD$. Consider the point $P$ on $AO$ such that $P$ minimizes $PA+2(PB+PC+PD)$. Find $\sin \angle PBO$. | \frac{1}{6} | 2.34375 |
3,283 | When a single number is added to each member of the sequence 20, 50, 100, the sequence becomes expressible as $x, a x, a^{2} x$. Find $a$. | \frac{5}{3} | 49.21875 |
3,284 | In bridge, a standard 52-card deck is dealt in the usual way to 4 players. By convention, each hand is assigned a number of "points" based on the formula $$4 \times(\# \mathrm{~A} \text { 's })+3 \times(\# \mathrm{~K} \text { 's })+2 \times(\# \mathrm{Q} \text { 's })+1 \times(\# \mathrm{~J} \text { 's })$$ Given that a particular hand has exactly 4 cards that are A, K, Q, or J, find the probability that its point value is 13 or higher. | \frac{197}{1820} | 0 |
3,285 | An isosceles trapezoid $A B C D$ with bases $A B$ and $C D$ has $A B=13, C D=17$, and height 3. Let $E$ be the intersection of $A C$ and $B D$. Circles $\Omega$ and $\omega$ are circumscribed about triangles $A B E$ and $C D E$. Compute the sum of the radii of $\Omega$ and $\omega$. | 39 | 0.78125 |
3,286 | Suppose $A B C$ is a triangle with incircle $\omega$, and $\omega$ is tangent to $\overline{B C}$ and $\overline{C A}$ at $D$ and $E$ respectively. The bisectors of $\angle A$ and $\angle B$ intersect line $D E$ at $F$ and $G$ respectively, such that $B F=1$ and $F G=G A=6$. Compute the radius of $\omega$. | \frac{2 \sqrt{5}}{5} | 0 |
3,287 | Rahul has ten cards face-down, which consist of five distinct pairs of matching cards. During each move of his game, Rahul chooses one card to turn face-up, looks at it, and then chooses another to turn face-up and looks at it. If the two face-up cards match, the game ends. If not, Rahul flips both cards face-down and keeps repeating this process. Initially, Rahul doesn't know which cards are which. Assuming that he has perfect memory, find the smallest number of moves after which he can guarantee that the game has ended. | 4 | 0 |
3,288 | Triangle $ABC$ obeys $AB=2AC$ and $\angle BAC=120^{\circ}$. Points $P$ and $Q$ lie on segment $BC$ such that $$\begin{aligned} AB^{2}+BC \cdot CP & =BC^{2} \\ 3AC^{2}+2BC \cdot CQ & =BC^{2} \end{aligned}$$ Find $\angle PAQ$ in degrees. | 40^{\circ} | 0 |
3,289 | Find the number of positive divisors $d$ of $15!=15 \cdot 14 \cdots 2 \cdot 1$ such that $\operatorname{gcd}(d, 60)=5$. | 36 | 27.34375 |
3,290 | I have 8 unit cubes of different colors, which I want to glue together into a $2 \times 2 \times 2$ cube. How many distinct $2 \times 2 \times 2$ cubes can I make? Rotations of the same cube are not considered distinct, but reflections are. | 1680 | 42.96875 |
3,291 | Let $f(r)=\sum_{j=2}^{2008} \frac{1}{j^{r}}=\frac{1}{2^{r}}+\frac{1}{3^{r}}+\cdots+\frac{1}{2008^{r}}$. Find $\sum_{k=2}^{\infty} f(k)$. | \frac{2007}{2008} | 92.1875 |
3,292 | Given a point $p$ and a line segment $l$, let $d(p, l)$ be the distance between them. Let $A, B$, and $C$ be points in the plane such that $A B=6, B C=8, A C=10$. What is the area of the region in the $(x, y)$-plane formed by the ordered pairs $(x, y)$ such that there exists a point $P$ inside triangle $A B C$ with $d(P, A B)+x=d(P, B C)+y=d(P, A C)$? | \frac{288}{5} | 0 |
3,293 | Let $\mathcal{H}$ be the unit hypercube of dimension 4 with a vertex at $(x, y, z, w)$ for each choice of $x, y, z, w \in \{0,1\}$. A bug starts at the vertex $(0,0,0,0)$. In how many ways can the bug move to $(1,1,1,1)$ by taking exactly 4 steps along the edges of $\mathcal{H}$? | 24 | 98.4375 |
3,294 | Let $A B C D$ be a parallelogram with $A B=8, A D=11$, and $\angle B A D=60^{\circ}$. Let $X$ be on segment $C D$ with $C X / X D=1 / 3$ and $Y$ be on segment $A D$ with $A Y / Y D=1 / 2$. Let $Z$ be on segment $A B$ such that $A X, B Y$, and $D Z$ are concurrent. Determine the area of triangle $X Y Z$. | \frac{19 \sqrt{3}}{2} | 0 |
3,295 | For how many unordered sets $\{a, b, c, d\}$ of positive integers, none of which exceed 168, do there exist integers $w, x, y, z$ such that $(-1)^{w} a+(-1)^{x} b+(-1)^{y} c+(-1)^{z} d=168$? If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor 25 e^{\left.-3 \frac{|C-A|}{C}\right\rfloor}\right.$. | 761474 | 0 |
3,296 | Find the smallest real constant $\alpha$ such that for all positive integers $n$ and real numbers $0=y_{0}<$ $y_{1}<\cdots<y_{n}$, the following inequality holds: $\alpha \sum_{k=1}^{n} \frac{(k+1)^{3 / 2}}{\sqrt{y_{k}^{2}-y_{k-1}^{2}}} \geq \sum_{k=1}^{n} \frac{k^{2}+3 k+3}{y_{k}}$. | \frac{16 \sqrt{2}}{9} | 0 |
3,297 | Let $W$ be the hypercube $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \mid 0 \leq x_{1}, x_{2}, x_{3}, x_{4} \leq 1\right\}$. The intersection of $W$ and a hyperplane parallel to $x_{1}+x_{2}+x_{3}+x_{4}=0$ is a non-degenerate 3-dimensional polyhedron. What is the maximum number of faces of this polyhedron? | 8 | 54.6875 |
3,298 | For how many integers $1 \leq k \leq 2013$ does the decimal representation of $k^{k}$ end with a 1? | 202 | 5.46875 |
3,299 | Tim and Allen are playing a match of tenus. In a match of tenus, the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won more games wins the match. In odd-numbered games, Tim wins with probability $3 / 4$, and in the even-numbered games, Allen wins with probability $3 / 4$. What is the expected number of games in a match? | \frac{16}{3} | 0.78125 |
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