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3,600 | Let $A, B, C$, and $D$ be points randomly selected independently and uniformly within the unit square. What is the probability that the six lines \overline{A B}, \overline{A C}, \overline{A D}, \overline{B C}, \overline{B D}$, and \overline{C D}$ all have positive slope? | \frac{1}{24} | 43.75 |
3,601 | Let $k$ be the answer to this problem. The probability that an integer chosen uniformly at random from $\{1,2, \ldots, k\}$ is a multiple of 11 can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$. | 1000 | 3.125 |
3,602 | Let $E$ be a three-dimensional ellipsoid. For a plane $p$, let $E(p)$ be the projection of $E$ onto the plane $p$. The minimum and maximum areas of $E(p)$ are $9 \pi$ and $25 \pi$, and there exists a $p$ where $E(p)$ is a circle of area $16 \pi$. If $V$ is the volume of $E$, compute $V / \pi$. | 75 | 14.84375 |
3,603 | How many non-empty subsets of $\{1,2,3,4,5,6,7,8\}$ have exactly $k$ elements and do not contain the element $k$ for some $k=1,2, \ldots, 8$. | 127 | 66.40625 |
3,604 | Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\sqrt[3]{2}+\sqrt[3]{4})=0$. (A polynomial is called monic if its leading coefficient is 1.) | x^{3}-3x^{2}+9x-9 | 0 |
3,605 | Find $\log _{n}\left(\frac{1}{2}\right) \log _{n-1}\left(\frac{1}{3}\right) \cdots \log _{2}\left(\frac{1}{n}\right)$ in terms of $n$. | (-1)^{n-1} | 59.375 |
3,606 | A sequence of positive integers is given by $a_{1}=1$ and $a_{n}=\operatorname{gcd}\left(a_{n-1}, n\right)+1$ for $n>1$. Calculate $a_{2002}$. | 3 | 86.71875 |
3,607 | Compute the number of labelings $f:\{0,1\}^{3} \rightarrow\{0,1, \ldots, 7\}$ of the vertices of the unit cube such that $$\left|f\left(v_{i}\right)-f\left(v_{j}\right)\right| \geq d\left(v_{i}, v_{j}\right)^{2}$$ for all vertices $v_{i}, v_{j}$ of the unit cube, where $d\left(v_{i}, v_{j}\right)$ denotes the Euclidean distance between $v_{i}$ and $v_{j}$. | 144 | 55.46875 |
3,608 | A semicircle with radius 2021 has diameter $AB$ and center $O$. Points $C$ and $D$ lie on the semicircle such that $\angle AOC < \angle AOD = 90^{\circ}$. A circle of radius $r$ is inscribed in the sector bounded by $OA$ and $OC$ and is tangent to the semicircle at $E$. If $CD=CE$, compute $\lfloor r \rfloor$. | 673 | 10.9375 |
3,609 | What is the probability that a randomly selected set of 5 numbers from the set of the first 15 positive integers has a sum divisible by 3? | \frac{1}{3} | 31.25 |
3,610 | Natalie has a copy of the unit interval $[0,1]$ that is colored white. She also has a black marker, and she colors the interval in the following manner: at each step, she selects a value $x \in[0,1]$ uniformly at random, and (a) If $x \leq \frac{1}{2}$ she colors the interval $[x, x+\frac{1}{2}]$ with her marker. (b) If $x>\frac{1}{2}$ she colors the intervals $[x, 1]$ and $[0, x-\frac{1}{2}]$ with her marker. What is the expected value of the number of steps Natalie will need to color the entire interval black? | 5 | 0.78125 |
3,611 | Let $q(x)=q^{1}(x)=2x^{2}+2x-1$, and let $q^{n}(x)=q(q^{n-1}(x))$ for $n>1$. How many negative real roots does $q^{2016}(x)$ have? | \frac{2017+1}{3} | 0 |
3,612 | Find the sum $$\frac{2^{1}}{4^{1}-1}+\frac{2^{2}}{4^{2}-1}+\frac{2^{4}}{4^{4}-1}+\frac{2^{8}}{4^{8}-1}+\cdots$$ | 1 | 39.84375 |
3,613 | Define $a$ ? $=(a-1) /(a+1)$ for $a \neq-1$. Determine all real values $N$ for which $(N ?)$ ?=\tan 15. | -2-\sqrt{3} | 10.15625 |
3,614 | Calculate the probability of the Alphas winning given the probability of the Reals hitting 0, 1, 2, 3, or 4 singles. | \frac{224}{243} | 0 |
3,615 | What is the probability that exactly one person gets their hat back when 6 people randomly pick hats? | \frac{11}{30} | 92.96875 |
3,616 | Compute the product of all positive integers $b \geq 2$ for which the base $b$ number $111111_{b}$ has exactly $b$ distinct prime divisors. | 24 | 5.46875 |
3,617 | Evaluate $\sum_{n=2}^{17} \frac{n^{2}+n+1}{n^{4}+2 n^{3}-n^{2}-2 n}$. | \frac{592}{969} | 0 |
3,618 | Find all ordered pairs $(a, b)$ of complex numbers with $a^{2}+b^{2} \neq 0, a+\frac{10b}{a^{2}+b^{2}}=5$, and $b+\frac{10a}{a^{2}+b^{2}}=4$. | (1,2),(4,2),\left(\frac{5}{2}, 2 \pm \frac{3}{2} i\right) | 0 |
3,619 | Massachusetts Avenue is ten blocks long. One boy and one girl live on each block. They want to form friendships such that each boy is friends with exactly one girl and vice versa. Nobody wants a friend living more than one block away (but they may be on the same block). How many pairings are possible? | 89 | 35.15625 |
3,620 | Consider the eighth-sphere $\left\{(x, y, z) \mid x, y, z \geq 0, x^{2}+y^{2}+z^{2}=1\right\}$. What is the area of its projection onto the plane $x+y+z=1$ ? | \frac{\pi \sqrt{3}}{4} | 0 |
3,621 | There are two red, two black, two white, and a positive but unknown number of blue socks in a drawer. It is empirically determined that if two socks are taken from the drawer without replacement, the probability they are of the same color is $\frac{1}{5}$. How many blue socks are there in the drawer? | 4 | 71.875 |
3,622 | Find all the roots of $\left(x^{2}+3 x+2\right)\left(x^{2}-7 x+12\right)\left(x^{2}-2 x-1\right)+24=0$. | 0, 2, 1 \pm \sqrt{6}, 1 \pm 2 \sqrt{2} | 0 |
3,623 | Three points are chosen inside a unit cube uniformly and independently at random. What is the probability that there exists a cube with side length $\frac{1}{2}$ and edges parallel to those of the unit cube that contains all three points? | \frac{1}{8} | 11.71875 |
3,624 | Circles $C_{1}, C_{2}, C_{3}$ have radius 1 and centers $O, P, Q$ respectively. $C_{1}$ and $C_{2}$ intersect at $A, C_{2}$ and $C_{3}$ intersect at $B, C_{3}$ and $C_{1}$ intersect at $C$, in such a way that $\angle A P B=60^{\circ}, \angle B Q C=36^{\circ}$, and $\angle C O A=72^{\circ}$. Find angle $A B C$ (degrees). | 90 | 0 |
3,625 | Let $P$ be the set of points $$\{(x, y) \mid 0 \leq x, y \leq 25, x, y \in \mathbb{Z}\}$$ and let $T$ be the set of triangles formed by picking three distinct points in $P$ (rotations, reflections, and translations count as distinct triangles). Compute the number of triangles in $T$ that have area larger than 300. | 436 | 0 |
3,626 | A fair coin is flipped every second and the results are recorded with 1 meaning heads and 0 meaning tails. What is the probability that the sequence 10101 occurs before the first occurrence of the sequence 010101? | \frac{21}{32} | 0 |
3,627 | Given a regular pentagon of area 1, a pivot line is a line not passing through any of the pentagon's vertices such that there are 3 vertices of the pentagon on one side of the line and 2 on the other. A pivot point is a point inside the pentagon with only finitely many non-pivot lines passing through it. Find the area of the region of pivot points. | \frac{1}{2}(7-3 \sqrt{5}) | 0 |
3,628 | Let $n$ be a positive integer. Claudio has $n$ cards, each labeled with a different number from 1 to n. He takes a subset of these cards, and multiplies together the numbers on the cards. He remarks that, given any positive integer $m$, it is possible to select some subset of the cards so that the difference between their product and $m$ is divisible by 100. Compute the smallest possible value of $n$. | 17 | 33.59375 |
3,629 | If $a, b$, and $c$ are random real numbers from 0 to 1, independently and uniformly chosen, what is the average (expected) value of the smallest of $a, b$, and $c$? | 1/4 | 69.53125 |
3,630 | Boris was given a Connect Four game set for his birthday, but his color-blindness makes it hard to play the game. Still, he enjoys the shapes he can make by dropping checkers into the set. If the number of shapes possible modulo (horizontal) flips about the vertical axis of symmetry is expressed as $9(1+2+\cdots+n)$, find $n$. | 729 | 0 |
3,631 | Triangle $A B C$ has $A B=1, B C=\sqrt{7}$, and $C A=\sqrt{3}$. Let $\ell_{1}$ be the line through $A$ perpendicular to $A B, \ell_{2}$ the line through $B$ perpendicular to $A C$, and $P$ the point of intersection of $\ell_{1}$ and $\ell_{2}$. Find $P C$. | 3 | 19.53125 |
3,632 | In the base 10 arithmetic problem $H M M T+G U T S=R O U N D$, each distinct letter represents a different digit, and leading zeroes are not allowed. What is the maximum possible value of $R O U N D$? | 16352 | 0 |
3,633 | For positive integers $n$, let $c_{n}$ be the smallest positive integer for which $n^{c_{n}}-1$ is divisible by 210, if such a positive integer exists, and $c_{n}=0$ otherwise. What is $c_{1}+c_{2}+\cdots+c_{210}$? | 329 | 39.84375 |
3,634 | Find the 6-digit number beginning and ending in the digit 2 that is the product of three consecutive even integers. | 287232 | 75.78125 |
3,635 | Let $A=H_{1}, B=H_{6}+1$. A real number $x$ is chosen randomly and uniformly in the interval $[A, B]$. Find the probability that $x^{2}>x^{3}>x$. | \frac{1}{4} | 0 |
3,636 | If 5 points are placed in the plane at lattice points (i.e. points $(x, y)$ where $x$ and $y$ are both integers) such that no three are collinear, then there are 10 triangles whose vertices are among these points. What is the minimum possible number of these triangles that have area greater than $1 / 2$ ? | 4 | 17.1875 |
3,637 | Let $m \circ n=(m+n) /(m n+4)$. Compute $((\cdots((2005 \circ 2004) \circ 2003) \circ \cdots \circ 1) \circ 0)$. | 1/12 | 0 |
3,638 | In a town of $n$ people, a governing council is elected as follows: each person casts one vote for some person in the town, and anyone that receives at least five votes is elected to council. Let $c(n)$ denote the average number of people elected to council if everyone votes randomly. Find \lim _{n \rightarrow \infty} c(n) / n. | 1-65 / 24 e | 0 |
3,639 | In a group of 50 children, each of the children in the group have all of their siblings in the group. Each child with no older siblings announces how many siblings they have; however, each child with an older sibling is too embarrassed, and says they have 0 siblings. If the average of the numbers everyone says is $\frac{12}{25}$, compute the number of different sets of siblings represented in the group. | 26 | 9.375 |
3,640 | The Dingoberry Farm is a 10 mile by 10 mile square, broken up into 1 mile by 1 mile patches. Each patch is farmed either by Farmer Keith or by Farmer Ann. Whenever Ann farms a patch, she also farms all the patches due west of it and all the patches due south of it. Ann puts up a scarecrow on each of her patches that is adjacent to exactly two of Keith's patches (and nowhere else). If Ann farms a total of 30 patches, what is the largest number of scarecrows she could put up? | 7 | 0 |
3,641 | Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions: (a) $f(1)=1$ (b) $f(a) \leq f(b)$ whenever $a$ and $b$ are positive integers with $a \leq b$. (c) $f(2a)=f(a)+1$ for all positive integers $a$. How many possible values can the 2014-tuple $(f(1), f(2), \ldots, f(2014))$ take? | 1007 | 0.78125 |
3,642 | Five people are at a party. Each pair of them are friends, enemies, or frenemies (which is equivalent to being both friends and enemies). It is known that given any three people $A, B, C$ : - If $A$ and $B$ are friends and $B$ and $C$ are friends, then $A$ and $C$ are friends; - If $A$ and $B$ are enemies and $B$ and $C$ are enemies, then $A$ and $C$ are friends; - If $A$ and $B$ are friends and $B$ and $C$ are enemies, then $A$ and $C$ are enemies. How many possible relationship configurations are there among the five people? | 17 | 0 |
3,643 | If $n$ is a positive integer, let $s(n)$ denote the sum of the digits of $n$. We say that $n$ is zesty if there exist positive integers $x$ and $y$ greater than 1 such that $x y=n$ and $s(x) s(y)=s(n)$. How many zesty two-digit numbers are there? | 34 | 0 |
3,644 | If $a$ and $b$ are randomly selected real numbers between 0 and 1, find the probability that the nearest integer to $\frac{a-b}{a+b}$ is odd. | \frac{1}{3} | 21.09375 |
3,645 | The sides of a regular hexagon are trisected, resulting in 18 points, including vertices. These points, starting with a vertex, are numbered clockwise as $A_{1}, A_{2}, \ldots, A_{18}$. The line segment $A_{k} A_{k+4}$ is drawn for $k=1,4,7,10,13,16$, where indices are taken modulo 18. These segments define a region containing the center of the hexagon. Find the ratio of the area of this region to the area of the large hexagon. | 9/13 | 0 |
3,646 | Find the number of positive integer solutions to $n^{x}+n^{y}=n^{z}$ with $n^{z}<2001$. | 10 | 7.03125 |
3,647 | Find all integers $n$ for which $\frac{n^{3}+8}{n^{2}-4}$ is an integer. | 0,1,3,4,6 | 11.71875 |
3,648 | Let $A B C D$ be a convex quadrilateral inscribed in a circle with shortest side $A B$. The ratio $[B C D] /[A B D]$ is an integer (where $[X Y Z]$ denotes the area of triangle $X Y Z$.) If the lengths of $A B, B C, C D$, and $D A$ are distinct integers no greater than 10, find the largest possible value of $A B$. | 5 | 19.53125 |
3,649 | In a chess-playing club, some of the players take lessons from other players. It is possible (but not necessary) for two players both to take lessons from each other. It so happens that for any three distinct members of the club, $A, B$, and $C$, exactly one of the following three statements is true: $A$ takes lessons from $B ; B$ takes lessons from $C ; C$ takes lessons from $A$. What is the largest number of players there can be? | 4 | 10.9375 |
3,650 | Let $P R O B L E M Z$ be a regular octagon inscribed in a circle of unit radius. Diagonals $M R, O Z$ meet at $I$. Compute $L I$. | \sqrt{2} | 14.0625 |
3,651 | For how many integers $n$ between 1 and 2005, inclusive, is $2 \cdot 6 \cdot 10 \cdots(4 n-2)$ divisible by $n!$? | 2005 | 67.96875 |
3,652 | Compute $\sum_{i=1}^{\infty} \frac{a i}{a^{i}}$ for $a>1$. | \left(\frac{a}{1-a}\right)^{2} | 0 |
3,653 | Suppose $x^{3}-a x^{2}+b x-48$ is a polynomial with three positive roots $p, q$, and $r$ such that $p<q<r$. What is the minimum possible value of $1 / p+2 / q+3 / r$ ? | 3 / 2 | 32.03125 |
3,654 | A right triangle has side lengths $a, b$, and $\sqrt{2016}$ in some order, where $a$ and $b$ are positive integers. Determine the smallest possible perimeter of the triangle. | 48+\sqrt{2016} | 0.78125 |
3,655 | Let $\mathbb{R}$ be the set of real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that for all real numbers $x$ and $y$, we have $$f\left(x^{2}\right)+f\left(y^{2}\right)=f(x+y)^{2}-2 x y$$ Let $S=\sum_{n=-2019}^{2019} f(n)$. Determine the number of possible values of $S$. | 2039191 | 0 |
3,656 | Victor has a drawer with two red socks, two green socks, two blue socks, two magenta socks, two lavender socks, two neon socks, two mauve socks, two wisteria socks, and 2000 copper socks, for a total of 2016 socks. He repeatedly draws two socks at a time from the drawer at random, and stops if the socks are of the same color. However, Victor is red-green colorblind, so he also stops if he sees a red and green sock. What is the probability that Victor stops with two socks of the same color? Assume Victor returns both socks to the drawer at each step. | \frac{1999008}{1999012} | 0 |
3,657 | Find all prime numbers $p$ such that $y^{2}=x^{3}+4x$ has exactly $p$ solutions in integers modulo $p$. In other words, determine all prime numbers $p$ with the following property: there exist exactly $p$ ordered pairs of integers $(x, y)$ such that $x, y \in\{0,1, \ldots, p-1\}$ and $p \text{ divides } y^{2}-x^{3}-4x$. | p=2 \text{ and } p \equiv 3(\bmod 4) | 21.875 |
3,658 | Suppose there exists a convex $n$-gon such that each of its angle measures, in degrees, is an odd prime number. Compute the difference between the largest and smallest possible values of $n$. | 356 | 11.71875 |
3,659 | A regular octahedron $A B C D E F$ is given such that $A D, B E$, and $C F$ are perpendicular. Let $G, H$, and $I$ lie on edges $A B, B C$, and $C A$ respectively such that \frac{A G}{G B}=\frac{B H}{H C}=\frac{C I}{I A}=\rho. For some choice of $\rho>1, G H, H I$, and $I G$ are three edges of a regular icosahedron, eight of whose faces are inscribed in the faces of $A B C D E F$. Find $\rho$. | (1+\sqrt{5}) / 2 | 0 |
3,660 | You are given a set of cards labeled from 1 to 100. You wish to make piles of three cards such that in any pile, the number on one of the cards is the product of the numbers on the other two cards. However, no card can be in more than one pile. What is the maximum number of piles you can form at once? | 8 | 80.46875 |
3,661 | In how many ways can 6 purple balls and 6 green balls be placed into a $4 \times 4$ grid of boxes such that every row and column contains two balls of one color and one ball of the other color? Only one ball may be placed in each box, and rotations and reflections of a single configuration are considered different. | 5184 | 0 |
3,662 | How many regions of the plane are bounded by the graph of $$x^{6}-x^{5}+3 x^{4} y^{2}+10 x^{3} y^{2}+3 x^{2} y^{4}-5 x y^{4}+y^{6}=0 ?$$ | 5 | 0 |
3,663 | Compute $$2 \sqrt{2 \sqrt[3]{2 \sqrt[4]{2 \sqrt[5]{2 \cdots}}}}$$ | 2^{e-1} | 3.90625 |
3,664 | A tournament among 2021 ranked teams is played over 2020 rounds. In each round, two teams are selected uniformly at random among all remaining teams to play against each other. The better ranked team always wins, and the worse ranked team is eliminated. Let $p$ be the probability that the second best ranked team is eliminated in the last round. Compute $\lfloor 2021 p \rfloor$. | 674 | 0 |
3,665 | Let $x, y$, and $z$ be positive real numbers such that $(x \cdot y)+z=(x+z) \cdot(y+z)$. What is the maximum possible value of $x y z$? | 1/27 | 78.90625 |
3,666 | Let $p=2^{24036583}-1$, the largest prime currently known. For how many positive integers $c$ do the quadratics \pm x^{2} \pm p x \pm c all have rational roots? | 0 | 48.4375 |
3,667 | A cuboctahedron is a polyhedron whose faces are squares and equilateral triangles such that two squares and two triangles alternate around each vertex. What is the volume of a cuboctahedron of side length 1? | 5 \sqrt{2} / 3 | 0 |
3,668 | Alex picks his favorite point $(x, y)$ in the first quadrant on the unit circle $x^{2}+y^{2}=1$, such that a ray from the origin through $(x, y)$ is $\theta$ radians counterclockwise from the positive $x$-axis. He then computes $\cos ^{-1}\left(\frac{4 x+3 y}{5}\right)$ and is surprised to get $\theta$. What is $\tan (\theta)$? | \frac{1}{3} | 56.25 |
3,669 | Consider a three-person game involving the following three types of fair six-sided dice. - Dice of type $A$ have faces labelled $2,2,4,4,9,9$. - Dice of type $B$ have faces labelled $1,1,6,6,8,8$. - Dice of type $C$ have faces labelled $3,3,5,5,7,7$. All three players simultaneously choose a die (more than one person can choose the same type of die, and the players don't know one another's choices) and roll it. Then the score of a player $P$ is the number of players whose roll is less than $P$ 's roll (and hence is either 0,1 , or 2 ). Assuming all three players play optimally, what is the expected score of a particular player? | \frac{8}{9} | 0 |
3,670 | How many different graphs with 9 vertices exist where each vertex is connected to 2 others? | 4 | 0 |
3,671 | For any real number $\alpha$, define $$\operatorname{sign}(\alpha)= \begin{cases}+1 & \text { if } \alpha>0 \\ 0 & \text { if } \alpha=0 \\ -1 & \text { if } \alpha<0\end{cases}$$ How many triples $(x, y, z) \in \mathbb{R}^{3}$ satisfy the following system of equations $$\begin{aligned} & x=2018-2019 \cdot \operatorname{sign}(y+z) \\ & y=2018-2019 \cdot \operatorname{sign}(z+x) \\ & z=2018-2019 \cdot \operatorname{sign}(x+y) \end{aligned}$$ | 3 | 47.65625 |
3,672 | On the Cartesian plane $\mathbb{R}^{2}$, a circle is said to be nice if its center is at the origin $(0,0)$ and it passes through at least one lattice point (i.e. a point with integer coordinates). Define the points $A=(20,15)$ and $B=(20,16)$. How many nice circles intersect the open segment $A B$ ? | 10 | 1.5625 |
3,673 | Let $n>0$ be an integer. Each face of a regular tetrahedron is painted in one of $n$ colors (the faces are not necessarily painted different colors.) Suppose there are $n^{3}$ possible colorings, where rotations, but not reflections, of the same coloring are considered the same. Find all possible values of $n$. | 1,11 | 28.125 |
3,674 | Patrick and Anderson are having a snowball fight. Patrick throws a snowball at Anderson which is shaped like a sphere with a radius of 10 centimeters. Anderson catches the snowball and uses the snow from the snowball to construct snowballs with radii of 4 centimeters. Given that the total volume of the snowballs that Anderson constructs cannot exceed the volume of the snowball that Patrick threw, how many snowballs can Anderson construct? | 15 | 45.3125 |
3,675 | Consider the equation $F O R T Y+T E N+T E N=S I X T Y$, where each of the ten letters represents a distinct digit from 0 to 9. Find all possible values of $S I X T Y$. | 31486 | 1.5625 |
3,676 | Two vertices of a cube are given in space. The locus of points that could be a third vertex of the cube is the union of $n$ circles. Find $n$. | 10 | 0 |
3,677 | For positive integers $a$ and $b$ such that $a$ is coprime to $b$, define $\operatorname{ord}_{b}(a)$ as the least positive integer $k$ such that $b \mid a^{k}-1$, and define $\varphi(a)$ to be the number of positive integers less than or equal to $a$ which are coprime to $a$. Find the least positive integer $n$ such that $$\operatorname{ord}_{n}(m)<\frac{\varphi(n)}{10}$$ for all positive integers $m$ coprime to $n$. | 240 | 0 |
3,678 | Consider a $2 \times n$ grid of points and a path consisting of $2 n-1$ straight line segments connecting all these $2 n$ points, starting from the bottom left corner and ending at the upper right corner. Such a path is called efficient if each point is only passed through once and no two line segments intersect. How many efficient paths are there when $n=2016$ ? | \binom{4030}{2015} | 0 |
3,679 | Find all positive integers $n$ for which there do not exist $n$ consecutive composite positive integers less than $n$ !. | 1, 2, 3, 4 | 10.9375 |
3,680 | Two fair octahedral dice, each with the numbers 1 through 8 on their faces, are rolled. Let $N$ be the remainder when the product of the numbers showing on the two dice is divided by 8. Find the expected value of $N$. | \frac{11}{4} | 0.78125 |
3,681 | A regular octahedron has a side length of 1. What is the distance between two opposite faces? | \sqrt{6} / 3 | 0 |
3,682 | In how many ways can the set of ordered pairs of integers be colored red and blue such that for all $a$ and $b$, the points $(a, b),(-1-b, a+1)$, and $(1-b, a-1)$ are all the same color? | 16 | 1.5625 |
3,683 | The Red Sox play the Yankees in a best-of-seven series that ends as soon as one team wins four games. Suppose that the probability that the Red Sox win Game $n$ is $\frac{n-1}{6}$. What is the probability that the Red Sox will win the series? | 1/2 | 0 |
3,684 | What is the 18 th digit after the decimal point of $\frac{10000}{9899}$ ? | 5 | 3.90625 |
3,685 | For a positive integer $n$, denote by $\tau(n)$ the number of positive integer divisors of $n$, and denote by $\phi(n)$ the number of positive integers that are less than or equal to $n$ and relatively prime to $n$. Call a positive integer $n$ good if $\varphi(n)+4 \tau(n)=n$. For example, the number 44 is good because $\varphi(44)+4 \tau(44)=44$. Find the sum of all good positive integers $n$. | 172 | 19.53125 |
3,686 | $A B C$ is an acute triangle with incircle $\omega$. $\omega$ is tangent to sides $\overline{B C}, \overline{C A}$, and $\overline{A B}$ at $D, E$, and $F$ respectively. $P$ is a point on the altitude from $A$ such that $\Gamma$, the circle with diameter $\overline{A P}$, is tangent to $\omega$. $\Gamma$ intersects $\overline{A C}$ and $\overline{A B}$ at $X$ and $Y$ respectively. Given $X Y=8, A E=15$, and that the radius of $\Gamma$ is 5, compute $B D \cdot D C$. | \frac{675}{4} | 0 |
3,687 | Three fair six-sided dice, each numbered 1 through 6 , are rolled. What is the probability that the three numbers that come up can form the sides of a triangle? | 37/72 | 23.4375 |
3,688 | Determine the number of integers $2 \leq n \leq 2016$ such that $n^{n}-1$ is divisible by $2,3,5,7$. | 9 | 12.5 |
3,689 | (Self-Isogonal Cubics) Let $A B C$ be a triangle with $A B=2, A C=3, B C=4$. The isogonal conjugate of a point $P$, denoted $P^{*}$, is the point obtained by intersecting the reflection of lines $P A$, $P B, P C$ across the angle bisectors of $\angle A, \angle B$, and $\angle C$, respectively. Given a point $Q$, let $\mathfrak{K}(Q)$ denote the unique cubic plane curve which passes through all points $P$ such that line $P P^{*}$ contains $Q$. Consider: (a) the M'Cay cubic $\mathfrak{K}(O)$, where $O$ is the circumcenter of $\triangle A B C$, (b) the Thomson cubic $\mathfrak{K}(G)$, where $G$ is the centroid of $\triangle A B C$, (c) the Napoleon-Feurerbach cubic $\mathfrak{K}(N)$, where $N$ is the nine-point center of $\triangle A B C$, (d) the Darboux cubic $\mathfrak{K}(L)$, where $L$ is the de Longchamps point (the reflection of the orthocenter across point $O)$ (e) the Neuberg cubic $\mathfrak{K}\left(X_{30}\right)$, where $X_{30}$ is the point at infinity along line $O G$, (f) the nine-point circle of $\triangle A B C$, (g) the incircle of $\triangle A B C$, and (h) the circumcircle of $\triangle A B C$. Estimate $N$, the number of points lying on at least two of these eight curves. | 49 | 0 |
3,690 | Let $a_{1}=3$, and for $n \geq 1$, let $a_{n+1}=(n+1) a_{n}-n$. Find the smallest $m \geq 2005$ such that $a_{m+1}-1 \mid a_{m}^{2}-1$. | 2010 | 49.21875 |
3,691 | Define $\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the remainder when $$ \sum_{\substack{2 \leq n \leq 50 \\ \operatorname{gcd}(n, 50)=1}} \phi^{!}(n) $$ is divided by 50 . | 12 | 1.5625 |
3,692 | Let $ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. Assume that $\angle OIA=90^{\circ}$. Given that $AI=97$ and $BC=144$, compute the area of $\triangle ABC$. | 14040 | 0 |
3,693 | Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $O$ be the circumcenter of $A B C$. Find the distance between the circumcenters of triangles $A O B$ and $A O C$. | \frac{91}{6} | 0 |
3,694 | In how many ways can 4 purple balls and 4 green balls be placed into a $4 \times 4$ grid such that every row and column contains one purple ball and one green ball? Only one ball may be placed in each box, and rotations and reflections of a single configuration are considered different. | 216 | 0.78125 |
3,695 | Let $m, n > 2$ be integers. One of the angles of a regular $n$-gon is dissected into $m$ angles of equal size by $(m-1)$ rays. If each of these rays intersects the polygon again at one of its vertices, we say $n$ is $m$-cut. Compute the smallest positive integer $n$ that is both 3-cut and 4-cut. | 14 | 29.6875 |
3,696 | Eight coins are arranged in a circle heads up. A move consists of flipping over two adjacent coins. How many different sequences of six moves leave the coins alternating heads up and tails up? | 7680 | 46.09375 |
3,697 | (Caos) A cao [sic] has 6 legs, 3 on each side. A walking pattern for the cao is defined as an ordered sequence of raising and lowering each of the legs exactly once (altogether 12 actions), starting and ending with all legs on the ground. The pattern is safe if at any point, he has at least 3 legs on the ground and not all three legs are on the same side. Estimate $N$, the number of safe patterns. | 1416528 | 0 |
3,698 | Two jokers are added to a 52 card deck and the entire stack of 54 cards is shuffled randomly. What is the expected number of cards that will be between the two jokers? | 52 / 3 | 8.59375 |
3,699 | A contest has six problems worth seven points each. On any given problem, a contestant can score either 0,1 , or 7 points. How many possible total scores can a contestant achieve over all six problems? | 28 | 66.40625 |
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