prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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The real numbers $x$, $y$, $z$, and $t$ satisfy the following equation: \[2x^2 + 4xy + 3y^2 - 2xz -2 yz + z^2 + 1 = t + \sqrt{y + z - t} \] Find 100 times the maximum possible value for $t$. | 125 |
Taotao wants to buy a bracelet. The bracelets have 7 different beads on them, arranged in a circle. Two bracelets are the same if one can be rotated or flipped to get the other. If she can choose the colors and placement of the beads, and the beads come in orange, white, and black, how many possible bracelets can she b... | 198 |
Let $S$ be the sum of all real $x$ such that $4^x = x^4$. Find the nearest integer to $S$. | 5 |
Matt is asked to write the numbers from 1 to 10 in order, but he forgets how to count. He writes a permutation of the numbers $\{1, 2, 3\ldots , 10\}$ across his paper such that: [list] [*]The leftmost number is 1. [*]The rightmost number is 10. [*]Exactly one number (not including 1 or 10) is less than both the number... | 1636 |
$3n$ people take part in a chess tournament: $n$ girls and $2n$ boys. Each participant plays with each of the others exactly once. There were no ties and the number of games won by the girls is $\displaystyle\frac75$ the number of games won by the boys. How many people took part in the tournament? | 9 |
Let $N$ be the number of (positive) divisors of $2010^{2010}$ ending in the digit $2$. What is the remainder when $N$ is divided by 2010? | 503 |
A regular pentagon is drawn in the plane, along with all its diagonals. All its sides and diagonals are extended infinitely in both directions, dividing the plane into regions, some of which are unbounded. An ant starts in the center of the pentagon, and every second, the ant randomly chooses one of the edges of the re... | 200 |
Let $p$ be a polynomial with integer coefficients such that $p(15)=6$, $p(22)=1196$, and $p(35)=26$. Find an integer $n$ such that $p(n)=n+82$. | 28 |
In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$ say $C_1$, and then the midpoint of $A_1D$ which is $A_2$. He continues in this fashi... | 793 |
Let $N$ be the sum of all binomial coefficients $\binom{a}{b}$ such that $a$ and $b$ are nonnegative integers and $a+b$ is an even integer less than 100. Find the remainder when $N$ is divided by 144. (Note: $\binom{a}{b} = 0$ if $a<b$, and $\binom{0}{0} = 1$.) | 3 |
A cuboctahedron is a solid with 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle (see picture). Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is $r$. Find $100r^2$. [asy] // dragon96, replacing // [img]http://i.imgur... | 4 |
Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$. Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$. (Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".) | 2400 |
PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose one menu item at random, the probability that the 4th customer orders a previously unordered item is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 189 |
Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$, tens digit $y$, and units digit $z$, and similarly let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$. How many three-digit numbers $\underline{abc}$, none of whose digits are 0, are there such that ... | 120 |
Let $p(x) = x^2 + x + 1$. Find the fourth smallest prime $q$ such that $p(n)$ is divisble by $q$ for some integer $n$. | 19 |
In regular hexagon $ABCDEF$, $AC$, $CE$ are two diagonals. Points $M$, $N$ are on $AC$, $CE$ respectively and satisfy $AC: AM = CE: CN = r$. Suppose $B, M, N$ are collinear, find $100r^2$. [asy] size(120); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(pair P) { dot(P,linewidth(3)); return P; } pair A=dir(0), B=dir(6... | 300 |
We say that a rook is "attacking" another rook on a chessboard if the two rooks are in the same row or column of the chessboard and there is no piece directly between them. Let $n$ be the maximum number of rooks that can be placed on a $6\times 6$ chessboard such that each rook is attacking at most one other. How many ... | 8100 |
Square $ABCD$ is divided into four rectangles by $EF$ and $GH$. $EF$ is parallel to $AB$ and $GH$ parallel to $BC$. $\angle BAF = 18^\circ$. $EF$ and $GH$ meet at point $P$. The area of rectangle $PFCH$ is twice that of rectangle $AGPE$. Given that the value of $\angle FAH$ in degrees is $x$, find the nearest integer t... | 45 |
Assume that $f(a+b) = f(a) + f(b) + ab$, and that $f(75) - f(51) = 1230$. Find $f(100)$. | 3825 |
Let $f$ be a function such that $f(x)+f(x+1)=2^x$ and $f(0)=2010$. Find the last two digits of $f(2010)$. | 51 |
The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$, where $p$ is an integer. Find $p$. | 192 |
Let $n$ be the number of polynomial functions from the integers modulo $2010$ to the integers modulo $2010$. $n$ can be written as $n = p_1 p_2 \cdots p_k$, where the $p_i$s are (not necessarily distinct) primes. Find $p_1 + p_2 + \cdots + p_n$. | 4527 |
Triangle $ABC$ has $AB = 4$, $AC = 5$, and $BC = 6$. An angle bisector is drawn from angle $A$, and meets $BC$ at $M$. What is the nearest integer to $100 \frac{AM}{CM}$? | 100 |
On rectangular coordinates, point $A = (1,2)$, $B = (3,4)$. $P = (a, 0)$ is on $x$-axis. Given that $P$ is chosen such that $AP + PB$ is minimized, compute $60a$. | 100 |
Find the largest positive integer $n$ such that $n\varphi(n)$ is a perfect square. ($\varphi(n)$ is the number of integers $k$, $1 \leq k \leq n$ that are relatively prime to $n$) | 1 |
The Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play. If the Band can't play any two tiring songs in a row, how many ways can the band play its 8 songs? | 14400 |
Let $f(x)=3x^3-5x^2+2x-6$. If the roots of $f$ are given by $\alpha$, $\beta$, and $\gamma$, find \[ \left(\frac{1}{\alpha-2}\right)^2+\left(\frac{1}{\beta-2}\right)^2+\left(\frac{1}{\gamma-2}\right)^2. \] | 68 |
A sequence of real numbers $\{a_n\}_{n = 1}^\infty (n=1,2,...)$ has the following property: \begin{align*} 6a_n+5a_{n-2}=20+11a_{n-1}\ (\text{for }n\geq3). \end{align*} The first two elements are $a_1=0, a_2=1$. Find the integer closest to $a_{2011}$. | 40086 |
Let \[f_1(x) = \frac{1}{x}\quad\text{and}\quad f_2(x) = 1 - x\] Let $H$ be the set of all compositions of the form $h_1 \circ h_2 \circ \ldots \circ h_k$, where each $h_i$ is either $f_1$ or $f_2$. For all $h$ in $H$, let $h^{(n)}$ denote $h$ composed with itself $n$ times. Find the greatest integer $N$ such that $\pi,... | 2 |
Let $\sigma$ be a random permutation of $\{0, 1, \ldots, 6\}$. Let $L(\sigma)$ be the length of the longest initial monotonic consecutive subsequence of $\sigma$ not containing $0$; for example, \[L(\underline{2,3,4},6,5,1,0) = 3,\ L(\underline{3,2},4,5,6,1,0) = 2,\ L(0,1,2,3,4,5,6) = 0.\] If the expected value of $L(\... | 5399 |
Suppose the polynomial $x^3 - x^2 + bx + c$ has real roots $a, b, c$. What is the square of the minimum value of $abc$? | 1 |
Let $ABC$ be a triangle with $AB = 2, BC = 5, AC = 4$. Let $M$ be the projection of $C$ onto the external angle bisector at vertex $B$. Similarly, let $N$ be the projection of $B$ onto the external angle bisector at vertex $C$. If the ratio of the area of quadrilateral $BCNM$ to the area of triangle $ABC$ is $a/b$, whe... | 87 |
Two logs of length 10 are laying on the ground touching each other. Their radii are 3 and 1, and the smaller log is fastened to the ground. The bigger log rolls over the smaller log without slipping, and stops as soon as it touches the ground again. The volume of the set of points swept out by the larger log as it roll... | 250 |
Let $d(n)$ denote the number of divisors of $n$ (including itself). You are given that \[\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.\] Find $p(6)$, where $p(x)$ is the unique polynomial with rational coefficients satisfying \[p(\pi) = \sum_{n=1}^{\infty} \frac{d(n)}{n^2}.\] | 36 |
Let $ABCD$ be a trapezoid with $AD$ parallel to $BC$, $AD = 2$, and $BC = 1$. Let $M$ be the midpoint of $AD$, and let $P$ be the intersection of $BD$ with $CM$. Extend $AP$ to meet segment $CD$ at point $Q$. If the ratio $CQ/QD = a/b$, where $a$ and $b$ are positive integers and $\text{gcd}(a, b) = 1$, find $a + b$. | 3 |
Let $\ell_1$ and $\ell_2$ be two parallel lines, a distance of 15 apart. Points $A$ and $B$ lie on $\ell_1$ while points $C$ and $D$ lie on $\ell_2$ such that $\angle BAC = 30^\circ$ and $\angle ABD = 60^\circ$. The minimum value of $AD + BC$ is $a\sqrt b$, where $a$ and $b$ are integers and $b$ is squarefree. Find $a ... | 23 |
A function $S(m, n)$ satisfies the initial conditions $S(1, n) = n$, $S(m, 1) = 1$, and the recurrence $S(m, n) = S(m - 1, n)S(m, n - 1)$ for $m\geq 2, n\geq 2$. Find the largest integer $k$ such that $2^k$ divides $S(7, 7)$. | 370 |
Let $1,\alpha_1,\alpha_2,...,\alpha_{10}$ be the roots of the polynomial $x^{11}-1$. It is a fact that there exists a unique polynomial of the form $f(x) = x^{10}+c_9x^9+ \dots + c_1x$ such that each $c_i$ is an integer, $f(0) = f(1) = 0$, and for any $1 \leq i \leq 10$ we have $(f(\alpha_i))^2 = -11$. Find $\left|c_1+... | 15 |
Shirley has a magical machine. If she inputs a positive even integer $n$, the machine will output $n/2$, but if she inputs a positive odd integer $m$, the machine will output $m+3$. The machine keeps going by automatically using its output as a new input, stopping immediately before it obtains a number already processe... | 67 |
Let $N$ be the number of ways to place $4$ bishops on a $5 \times 5$ chessboard such that no $3$ are on the same diagonal. Find the remainder when $N$ is divided by $100$. (Note: the length of a diagonal on a $5 \times 5$ chessboard can be 2, 3, 4, or 5.) | 12 |
Let $f(x) = x^3-7x^2+16x-10$. As $x$ ranges over all integers, find the sum of distinct prime values taken on by $f(x)$. | 2 |
Let $ABC$ be a triangle with $AB = 15, BC = 17$, $CA = 21$, and incenter $I$. If the circumcircle of triangle $IBC$ intersects side $AC$ again at $P$, find $CP$. | 6 |
Consider the sum $\overline{a b} + \overline{ c d e}$, where each of the letters is a distinct digit between $1$ and $5$. How many values are possible for this sum? | 27 |
Let $\omega_1$ be a circle of radius 6, and let $\omega_2$ be a circle of radius 5 that passes through the center $O$ of $\omega_1$. Let $A$ and $B$ be the points of intersection of the two circles, and let $P$ be a point on major arc $AB$ of $\omega_2$. Let $M$ and $N$ be the second intersections of $PA$ and $PB$ with... | 17 |
Calculate the sum of the coordinates of all pairs of positive integers $(n, k)$ such that $k\equiv 0, 3\pmod 4$, $n > k$, and $\displaystyle\sum^n_{i = k + 1} i^3 = (96^2\cdot3 - 1)\displaystyle\left(\sum^k_{i = 1} i\right)^2 + 48^2$ | 103 |
A set of $n$ dominoes, each colored with one white square and one black square, is used to cover a $2 \times n$ board of squares. For $n = 6$, how many different patterns of colors can the board have? (For $n = 2$, this number is $6$.) | 328 |
A function $f:\{1,2, \ldots, n\} \to \{1, \ldots, m\}$ is [i]multiplication-preserving[/i] if $f(i)f(j) = f(ij)$ for all $1 \le i \le j \le ij \le n$, and [i]injective[/i] if $f(i) = f(j)$ only when $i = j$. For $n = 9, m = 88$, the number of injective, multiplication-preserving functions is $N$. Find the sum of the pr... | 33 |
Let $\omega$ be a circle of radius $6$ with center $O$. Let $AB$ be a chord of $\omega$ having length $5$. For any real constant $c$, consider the locus $\mathcal{L}(c) $ of all points $P$ such that $PA^2 - PB^2 = c$. Find the largest value of $c$ for which the intersection of $\mathcal{L}(c)$ and $\omega$ consists of ... | 60 |
Let $a$ and $b$ be positive integers such that $a + bz = x^3 + y^4$ has no solutions for any integers $x, y, z$, with $b$ as small as possible, and $a$ as small as possible for the minimum $b$. Find $ab$. | 91 |
A polynomial $p$ can be written as \begin{align*} p(x) = x^6+3x^5-3x^4+ax^3+bx^2+cx+d. \end{align*} Given that all roots of $p(x)$ are equal to either $m$ or $n$ where $m$ and $n$ are integers, compute $p(2)$. | 64 |
At the start of the PUMaC opening ceremony in McCosh auditorium, the speaker counts $90$ people in the audience. Every minute afterwards, either one person enters the auditorium (due to waking up late) or leaves (in order to take a dreadful math contest). The speaker observes that in this time, exactly $100$ people ent... | 6 |
Let $f$ be an invertible function defined on the complex numbers such that \[z^2 = f(z + f(iz + f(-z + f(-iz + f(z + \ldots)))))\] for all complex numbers $z$. Suppose $z_0 \neq 0$ satisfies $f(z_0) = z_0$. Find $1/z_0$. (Note: an invertible function is one that has an inverse). | 4 |
A road company is trying to build a system of highways in a country with $21$ cities. Each highway runs between two cities. A trip is a sequence of distinct cities $C_1,\dots, C_n$, for which there is a highway between $C_i$ and $C_{i+1}$. The company wants to fulfill the following two constraints: (1) for any ordered ... | 192 |
For every integer $n$ from $0$ to $6$, we have $3$ identical weights with weight $2^n$. How many ways are there to form a total weight of 263 grams using only these given weights? | 60 |
What is the sum of all primes $p$ such that $7^p - 6^p + 2$ is divisible by 43? | 3 |
Four circles are situated in the plane so that each is tangent to the other three. If three of the radii are $5$, $5$, and $8$, the largest possible radius of the fourth circle is $a/b$, where $a$ and $b$ are positive integers and gcd$(a, b) = 1$. Find $a + b$. | 43 |
Let $\{g_i\}_{i=0}^{\infty}$ be a sequence of positive integers such that $g_0=g_1=1$ and the following recursions hold for every positive integer $n$: \begin{align*} g_{2n+1} &= g_{2n-1}^2+g_{2n-2}^2 \\ g_{2n} &= 2g_{2n-1}g_{2n-2}-g_{2n-2}^2 \end{align*} Compute the remainder when $g_{2011}$ is divided by $216$. | 34 |
A rectangular piece of paper has corners labeled $A, B, C$, and $D$, with $BC = 80$ and $CD = 120$. Let $M$ be the midpoint of side $AB$. The corner labeled $A$ is folded along line $MD$ and the corner labeled $B$ is folded along line $MC$ until the segments $AM$ and $MB$ coincide. Let $S$ denote the point in space whe... | 45 |
Let $ABC$ be a triangle with $\angle BAC = 60^\circ, BA = 2$, and $CA = 3$. A point $M$ is located inside $ABC$ such that $MB = 1$ and $MC = 2$. A semicircle tangent to $MB$ and $MC$ has its center $O$ on $BC$. Let $P$ be the intersection of the angle bisector of $\angle BAC$ and the perpendicular bisector of $AC$. If ... | 3 |
What is the largest positive integer $n < 1000$ for which there is a positive integer $m$ satisfying \[\text{lcm}(m,n) = 3m \times \gcd(m,n)?\] | 972 |
$5$ people stand in a line facing one direction. In every round, the person at the front moves randomly to any position in the line, including the front or the end. Suppose that $\frac{m}{n}$ is the expected number of rounds needed for the last person of the initial line to appear at the front of the line, where $m$ an... | 137 |
If $n$ is an integer such that $n \ge 2^k$ and $n < 2^{k+1}$, where $k = 1000$, compute the following: $$n - \left( \lfloor \frac{n -2^0}{2^1} \rfloor + \lfloor \frac{n -2^1}{2^2} \rfloor + ...+ \lfloor \frac{n -2^{k-1}}{2^k} \rfloor \right)$$ | 2037 |
Let $a, b$, and $c$ be positive integers satisfying $a^4 + a^2b^2 + b^4 = 9633$ $2a^2 + a^2b^2 + 2b^2 + c^5 = 3605$. What is the sum of all distinct values of $a + b + c$? | 18 |
Three circles, with radii of $1, 1$, and $2$, are externally tangent to each other. The minimum possible area of a quadrilateral that contains and is tangent to all three circles can be written as $a + b\sqrt{c}$ where $c$ is not divisible by any perfect square larger than $1$. Find $a + b + c$ | 22 |
Find the largest possible sum $ m + n$ for positive integers $m, n \le 100$ such that $m + 1 \equiv 3$ (mod $4$) and there exists a prime number $p$ and nonnegative integer $a$ such $\frac{m^{2n-1}-1}{m-1} = m^n+p^a$ . | 84 |
Considering all numbers of the form $n = \lfloor \frac{k^3}{2012} \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $k$ ranges from $1$ to $2012$, how many of these $n$’s are distinct? | 1995 |
Find the sum of all possible sums $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is a perfect square. | 9 |
Two circles centered at $O$ and $P$ have radii of length $5$ and $6$ respectively. Circle $O$ passes through point $P$. Let the intersection points of circles $O$ and $P$ be $M$ and $N$. The area of triangle $\vartriangle MNP$ can be written in simplest form as $a/b$. Find $a + b$. | 457 |
When some number $a^2$ is written in base $b$, the result is $144_b$. $a$ and $b$ also happen to be integer side lengths of a right triangle. If $a$ and $b$ are both less than $20$, find the sum of all possible values of $a$. | 169 |
Cyclic quadrilateral $ABCD$ has side lengths $AB = 2, BC = 3, CD = 5, AD = 4$. Find $\sin A \sin B(\cot A/2 + \cot B/2 + \cot C/2 + \cot D/2)^2$. Your answer can be written in simplest form as $a/b$. Find $a + b$. | 5101 |
Two white pentagonal pyramids, with side lengths all the same, are glued to each other at their regular pentagon bases. Some of the resulting $10$ faces are colored black. How many rotationally distinguishable colorings may result? | 120 |
Compute the smallest positive integer $a$ for which $$\sqrt{a +\sqrt{a +...}} - \frac{1}{a +\frac{1}{a+...}}> 7$$ | 43 |
A PUMaC grader is grading the submissions of forty students $s_1, s_2, ..., s_{40}$ for the individual finals round, which has three problems. After grading a problem of student $s_i$, the grader either: $\bullet$ grades another problem of the same student, or $\bullet$ grades the same problem of the student $s_{i-1}$ ... | 78 |
Let f be a polynomial of degree $3$ with integer coefficients such that $f(0) = 3$ and $f(1) = 11$. If f has exactly $2$ integer roots, how many such polynomials $f$ exist? | 0 |
How many factors of $(20^{12})^2$ less than $20^{12}$ are not factors of $20^{12}$ ? | 288 |
Call a positive integer $x$ a leader if there exists a positive integer $n$ such that the decimal representation of $x^n$ starts ([u]not ends[/u]) with $2012$. For example, $586$ is a leader since $586^3 =201230056$. How many leaders are there in the set $\{1, 2, 3, ..., 2012\}$? | 2008 |
An octahedron (a solid with 8 triangular faces) has a volume of $1040$. Two of the spatial diagonals intersect, and their plane of intersection contains four edges that form a cyclic quadrilateral. The third spatial diagonal is perpendicularly bisected by this plane and intersects the plane at the circumcenter of the c... | 541 |
If the probability that the sum of three distinct integers between $16$ and $30$ (inclusive) is even can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m + n$. | 97 |
How many (possibly empty) sets of lattice points $\{P_1, P_2, ... , P_M\}$, where each point $P_i =(x_i, y_i)$ for $x_i , y_i \in \{0, 1, 2, 3, 4, 5, 6\}$, satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$ ? An infinite slope, e.g. $P_i$ is vertically above $P_j$ , does not count as p... | 3432 |
Six ants are placed on the vertices of a regular hexagon with an area of $12$. At each point in time, each ant looks at the next ant in the hexagon (in counterclockwise order), and measures the distance, $s$, to the next ant. Each ant then proceeds towards the next ant at a speed of $\frac{s}{100}$ units per year. Afte... | 103 |
Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors. Suppose $M$ can be written as $\Pi_{k=1}^{n}p_k^{a_k}$ where the $p_k$’s are distinct primes and the $a_k$’s are positive integers. Find $\Sigma_{k=1}^{n}(p_k + a_k)$ | 1012 |
Let $a_n$ be a sequence such that $a_1 = 1$ and $a_{n+1} = \lfloor a_n +\sqrt{a_n} +\frac12 \rfloor $, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. What are the last four digits of $a_{2012}$? | 2037 |
Let $\vartriangle ABC$ be a triangle with $\angle BAC = 45^o, \angle BCA = 30^o$, and $AB = 1$. Point $D$ lies on segment $AC$ such that $AB = BD$. Find the square of the length of the common external tangent to the circumcircles of triangles $\vartriangle BDC$ and $\vartriangle ABC$. | 2 |
Let $p_1 = 2012$ and $p_n = 2012^{p_{n-1}}$ for $n > 1$. Find the largest integer $k$ such that $p_{2012}- p_{2011}$ is divisible by $2011^k$. | 2012 |
Compute $\Sigma_{n=1}^{\infty}\frac{n + 1}{n^2(n + 2)^2}$ . Your answer in simplest form can be written as $a/b$, where $a, b$ are relatively-prime positive integers. Find $a + b$. | 21 |
Let A be a regular $12$-sided polygon. A new $12$-gon B is constructed by connecting the midpoints of the sides of A. The ratio of the area of B to the area of A can be written in simplest form as $(a +\sqrt{b})/c$, where $a, b, c$ are integers. Find $a + b + c$. | 9 |
Let the sequence $\{x_n\}$ be defined by $x_1 \in \{5, 7\}$ and, for $k \ge 1, x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$. For example, the possible values of $x_3$ are $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$, and $7^{7^7}$. Determine the sum of all possible values for the last two digits of $x_{2012}$... | 75 |
Let $f_n(x) = n + x^2$. Evaluate the product $gcd\{f_{2001}(2002), f_{2001}(2003)\} \times gcd\{f_{2011}(2012), f_{2011}(2013)\} \times gcd\{f_{2021}(2022), f_{2021}(2023)\}$, where $gcd\{x, y\}$ is the greatest common divisor of $x$ and $y$ | 125 |
What is the smallest natural number $n$ greater than $2012$ such that the polynomial $f(x) =(x^6 + x^4)^n - x^{4n} - x^6$ is divisible by $g(x) = x^4 + x^2 + 1$? | 2015 |
Let an be a sequence such that $a_0 = 0$ and: $a_{3n+1} = a_{3n} + 1 = a_n + 1$ $a_{3n+2} = a_{3n} + 2 = a_n + 2$ for all natural numbers $n$. How many $n$ less than $2012$ have the property that $a_n = 7$? | 379 |
Jim has two fair $6$-sided dice, one whose faces are labelled from $1$ to $6$, and the second whose faces are labelled from $3$ to $8$. Twice, he randomly picks one of the dice (each die equally likely) and rolls it. Given the sum of the resulting two rolls is $9$, if $\frac{m}{n}$ is the probability he rolled the same... | 7 |
A square is inscribed in an ellipse such that two sides of the square respectively pass through the two foci of the ellipse. The square has a side length of $4$. The square of the length of the minor axis of the ellipse can be written in the form $a + b\sqrt{c}$ where $a, b$, and $c$ are integers, and $c$ is not divisi... | 21 |
Find the largest $n$ such that the last nonzero digit of $n!$ is $1$. | 1 |
Proctors Andy and Kristin have a PUMaC team of eight students labelled $s_1, s_2, ... , s_8$ (the PUMaC staff being awful with names). The following occurs: $1$. Andy tells the students to arrange themselves in a line in arbitrary order. $2$. Kristin tells each student $s_i$ to move to the current spot of student $s_j$... | 1260 |
During chemistry labs, we oftentimes fold a disk-shaped filter paper twice, and then open up a flap of the quartercircle to form a cone shape, as in the diagram. What is the angle $\theta$, in degrees, of the bottom of the cone when we look at it from the side? [img]https://cdn.artofproblemsolving.com/attachments/d/2/f... | 60 |
If $x, y$, and $z$ are real numbers with $\frac{x - y}{z}+\frac{y - z}{x}+\frac{z - x}{y}= 36$, find $2012 +\frac{x - y}{z}\cdot \frac{y - z}{x}\cdot\frac{z - x}{y}$ . | 1976 |
A cyclic quadrilateral $ABCD$ has side lengths $AB = 3, BC = AD = 5$, and $CD = 8$. The radius of its circumcircle can be written in the form $a\sqrt{b}/c$, where $a, b, c$ are positive integers, $a, c$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a + b + c$. | 16 |
How many ways can $2^{2012}$ be expressed as the sum of four (not necessarily distinct) positive squares? | 1 |
Find the number of pairs $(n,C)$ of positive integers such that $C\leq 100$ and $n^2+n+C$ is a perfect square. | 108 |
Suppose $P(x)$ is a degree $n$ monic polynomial with integer coefficients such that $2013$ divides $P(r)$ for exactly $1000$ values of $r$ between $1$ and $2013$ inclusive. Find the minimum value of $n$. | 50 |
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