prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
Find the rightmost non-zero digit of the expansion of $(20)(13!)$. | 6 |
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 $10$th term of $\{a_n+g_n\}$? | -54 |
Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers whose sum is $20$. Determine with proof the smallest possible value of \[ \displaystyle\sum_{1\le i \le j \le 5} \lfloor a_i + a_j \rfloor. \] | 72 |
Compute the number of ordered quintuples of nonnegative integers $(a_1,a_2,a_3,a_4,a_5)$ such that $0\leq a_1,a_2,a_3,a_4,a_5\leq 7$ and $5$ divides $2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$. | 6528 |
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 $x\leq\tfrac12$ she colors the interval $[x,x+\tfrac12]$ with her marker. (b) If $x>\tfr... | 5 |
Find the integer closest to \[\frac{1}{\sqrt[4]{5^4+1}-\sqrt[4]{5^4-1}}\] | 250 |
Find the number of triples of sets $(A, B, C)$ such that: (a) $A, B, C \subseteq \{1, 2, 3, \dots , 8 \}$. (b) $|A \cap B| = |B \cap C| = |C \cap A| = 2$. (c) $|A| = |B| = |C| = 4$. Here, $|S|$ denotes the number of elements in the set $S$. | 45360 |
The numbers $2^0, 2^1, \dots , 2{}^1{}^5, 2{}^1{}^6 = 65536$ are written on a blackboard. You repeatedly take two numbers on the blackboard, subtract one form the other, erase them both, and write the result of the subtraction on the blackboard. What is the largest possible number that can remain on the blackboard when... | 131069 |
Let $f:\mathbb{N}\to\mathbb{N}$ 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 |
Eli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the $4$ people. The following constraints are imposed: $\bullet$ Every person must get a positive integer number of shares, and all $16$ shares must be given out. $\bullet$ No one person can have more shares than the other th... | 315 |
We have a calculator with two buttons that displays and integer $x$. Pressing the first button replaces $x$ by $\lfloor \frac{x}{2} \rfloor$, and pressing the second button replaces $x$ by $4x+1$. Initially, the calculator displays $0$. How many integers less than or equal to $2014$ can be achieved through a sequence o... | 233 |
For an integer $n$, let $f_9(n)$ denote the number of positive integers $d\leq 9$ dividing $n$. Suppose that $m$ is a positive integer and $b_1,b_2,\ldots,b_m$ are real numbers such that $f_9(n)=\textstyle\sum_{j=1}^mb_jf_9(n-j)$ for all $n>m$. Find the smallest possible value of $m$. | 28 |
Let $a_1,a_2,\ldots$ be an infinite sequence of integers such that $a_i$ divides $a_{i+1}$ for all $i\geq 1$, and let $b_i$ be the remainder when $a_i$ is divided by $210$. What is the maximal number of distinct terms in the sequence $b_1,b_2,\ldots$? | 127 |
Let $A=\{a_1,a_2,\ldots,a_7\}$ be a set of distinct positive integers such that the mean of the elements of any nonempty subset of $A$ is an integer. Find the smallest possible value of the sum of the elements in $A$. | 1267 |
An [i]up-right path[/i] from $(a, b) \in \mathbb{R}^2$ to $(c, d) \in \mathbb{R}^2$ is a finite sequence $(x_1, y_z), \dots, (x_k, y_k)$ of points in $ \mathbb{R}^2 $ such that $(a, b)= (x_1, y_1), (c, d) = (x_k, y_k)$, and for each $1 \le i < k$ we have that either $(x_{i+1}, y_{y+1}) = (x_i+1, y_i)$ or $(x_{i+1}, y_{... | 1750 |
There is a heads up coin on every integer of the number line. Lucky is initially standing on the zero point of the number line facing in the positive direction. Lucky performs the following procedure: $\bullet$ Lucky looks at the coin (or lack thereof) underneath him. $\bullet \, - \, $ If the coin is heads, Lucky flip... | 6098 |
Suppose that $(a_1,\ldots,a_{20})$ and $(b_1,\ldots,b_{20})$ are two sequences of integers such that the sequence $(a_1,\ldots,a_{20},b_1,\ldots,b_{20})$ contains each of the numbers $1,\ldots,40$ exactly once. What is the maximum possible value of the sum \[\sum_{i=1}^{20}\sum_{j=1}^{20}\min(a_i,b_j)?\] | 5530 |
Find the sum of all real numbers $x$ such that $5x^4-10x^3+10x^2-5x-11=0$. | 1 |
Let \[ A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3) \]. Compute $2^A$. | 72 |
Find the number of ordered quadruples of positive integers $(a,b,c,d)$ such that $a,b,c,$ and $d$ are all (not necessarily distinct) factors of $30$ and $abcd>900$. | 1940 |
Let $a,b,c,d,e$ be nonnegative integers such that $625a+250b+100c+40d+16e=15^3$. What is the maximum possible value of $a+b+c+d+e$? | 153 |
Let $Q$ be a polynomial \[Q(x)=a_0+a_1x+\cdots+a_nx^n,\] where $a_0,\ldots,a_n$ are nonnegative integers. Given that $Q(1)=4$ and $Q(5)=152$, find $Q(6)$. | 254 |
Find the number of ordered pairs of integers $(a,b)\in\{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a "quadratic residue modulo $x^2+1$ and $35$", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following $\textit{equivalent}$ conditions holds: [list] [*] th... | 225 |
In triangle $ABC$, $AB=2$, $AC=1+\sqrt{5}$, and $\angle CAB=54^{\circ}$. Suppose $D$ lies on the extension of $AC$ through $C$ such that $CD=\sqrt{5}-1$. If $M$ is the midpoint of $BD$, determine the measure of $\angle ACM$, in degrees. | 63 |
Let $N=30^{2015}$. Find the number of ordered 4-tuples of integers $(A,B,C,D)\in\{1,2,\ldots,N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3+Bn^2+2Cn+D$ is divisible by $N$. | 24 |
Compute the number of sequences of integers $(a_1,\ldots,a_{200})$ such that the following conditions hold. [list] [*] $0\leq a_1<a_2<\cdots<a_{200}\leq 202.$ [*] There exists a positive integer $N$ with the following property: for every index $i\in\{1,\ldots,200\}$ there exists an index $j\in\{1,\ldots,200\}$ such tha... | 20503 |
The fraction $\tfrac1{2015}$ has a unique "(restricted) partial fraction decomposition'' of the form \[\dfrac1{2015}=\dfrac a5+\dfrac b{13}+\dfrac c{31},\] where $a$, $b$, and $c$ are integers with $0\leq a<5$ and $0\leq b<13$. Find $a+b$. | 14 |
For each positive integer $n$ and non-negative integer $k$, define $W(n,k)$ recursively by \[ W(n,k) = \begin{cases} n^n & k = 0 \\ W(W(n,k-1), k-1) & k > 0. \end{cases} \] Find the last three digits in the decimal representation of $W(555,2)$. | 875 |
Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0), (2,0), (2,1),$ and $(0,1)$. $R$ can be divided into two unit squares, as shown; the resulting figure has seven edges. [asy] size(3cm); draw((0,0)--(2,0)--(2,1)--(0,1)--cycle); draw((1,0)--(1,1)); [/asy] Compute the number of ways to choose one or ... | 81 |
Compute the sum of all integers $1 \le a \le 10$ with the following property: there exist integers $p$ and $q$ such that $p$, $q$, $p^2+a$ and $q^2+a$ are all distinct prime numbers. | 20 |
Determine the number of triples $0 \le k,m,n \le 100$ of integers such that \[ 2^mn - 2^nm = 2^k. \] | 22 |
An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies the recurrence \[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \] for every positive integer $n$. Given that $a_1 = a_3 = 1$ and $a_{98} = a_{99}$, compute $a_1 + a_2 + \dots + a_{100}$. | 3 |
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 \[ \sum_{\substack{2 \le n \le 50 \\ \gcd(n,50)=1}} \phi^!(n) \] is divided by $50$. | 30 |
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 $n$ $\emph{good}$ if $\varphi (n) + 4\tau (n) = n$. For example, the number $44$... | 172 |
Nine pairwise noncongruent circles are drawn in the plane such that any two circles intersect twice. For each pair of circles, we draw the line through these two points, for a total of $\binom 92 = 36$ lines. Assume that all $36$ lines drawn are distinct. What is the maximum possible number of points which lie on at le... | 462 |
Let $a,b$ and $c$ be real numbers such that \begin{align*} a^2+ab+b^2&=9 \\ b^2+bc+c^2&=52 \\ c^2+ca+a^2&=49. \end{align*} Compute the value of $\dfrac{49b^2+39bc+9c^2}{a^2}$. | 52 |
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 |
Starting with an empty string, we create a string by repeatedly appending one of the letters $H$, $M$, $T$ with probabilities $\frac 14$, $\frac 12$, $\frac 14$, respectively, until the letter $M$ appears twice consecutively. What is the expected value of the length of the resulting string? | 6 |
Let $V = \left\{ 1, \dots, 8 \right\}$. How many permutations $\sigma : V \to V$ are automorphisms of some tree? (A $\emph{graph}$ consists of some set of vertices and some edges between pairs of distinct vertices. It is $\emph{connected}$ if every two vertices in it are connected by some path of one or more edges. A $... | 30212 |
Determine the smallest positive integer $n \ge 3$ for which \[ A \equiv 2^{10n} \pmod{2^{170}} \] where $A$ denotes the result when the numbers $2^{10}$, $2^{20}$, $\dots$, $2^{10n}$ are written in decimal notation and concatenated (for example, if $n=2$ we have $A = 10241048576$). | 14 |
For positive integers $n$, let $S_n$ be the set of integers $x$ such that $n$ distinct lines, no three concurrent, can divide a plane into $x$ regions (for example, $S_2=\{3,4\}$, because the plane is divided into 3 regions if the two lines are parallel, and 4 regions otherwise). What is the minimum $i$ such that $S_i$... | 4 |
Find the number of ordered pairs of integers $(a, b)$ such that $a, b$ are divisors of 720 but $ab$ is not. | 2520 |
Determine the number of integers $2 \le n \le 2016$ such that $n^n-1$ is divisible by $2$, $3$, $5$, $7$. | 9 |
How many equilateral hexagons of side length $\sqrt{13}$ have one vertex at $(0,0)$ and the other five vertices at lattice points? (A lattice point is a point whose Cartesian coordinates are both integers. A hexagon may be concave but not self-intersecting.) | 216 |
Let \[ A = \lim_{n \rightarrow \infty} \sum_{i=0}^{2016} (-1)^i \cdot \frac{\binom{n}{i}\binom{n}{i+2}}{\binom{n}{i+1}^2} \] Find the largest integer less than or equal to $\frac{1}{A}$. The following decimal approximation might be useful: $ 0.6931 < \ln(2) < 0.6932$, where $\ln$ denotes the natural logarithm function. | 1 |
Let $\Delta A_1B_1C$ be a triangle with $\angle A_1B_1C = 90^{\circ}$ and $\frac{CA_1}{CB_1} = \sqrt{5}+2$. For any $i \ge 2$, define $A_i$ to be the point on the line $A_1C$ such that $A_iB_{i-1} \perp A_1C$ and define $B_i$ to be the point on the line $B_1C$ such that $A_iB_i \perp B_1C$. Let $\Gamma_1$ be the incirc... | 4030 |
Kristoff is planning to transport a number of indivisible ice blocks with positive integer weights from the north mountain to Arendelle. He knows that when he reaches Arendelle, Princess Anna and Queen Elsa will name an ordered pair $(p,q)$ of nonnegative integers satisfying $p + q \le 2016$. Kristoff must then give Pr... | 18 |
Let $A$ denote the set of all integers $n$ such that $1 \le n \le 10000$, and moreover the sum of the decimal digits of $n$ is $2$. Find the sum of the squares of the elements of $A$. | 7294927 |
Define $\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the number of integers $2 \le n \le 50$ such that $n$ divides $\phi^{!}(n)+1$. | 12 |
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 $c_1 + c_2 + \dots + c_{210}$? | 329 |
Determine the remainder when $$\sum_{i=0}^{2015} \left\lfloor \frac{2^i}{25} \right\rfloor$$ is divided by 100, where $\lfloor x \rfloor$ denotes the largest integer not greater than $x$. | 14 |
Call a positive integer $N \ge 2$ ``special'' if for every $k$ such that $2 \leq k \leq N$, $N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). How many special integers are there less than $100$? | 50 |
On the Cartesian plane $\mathbb{R}^2$, a circle is said to be \textit{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 $AB$? For refer... | 10 |
$\textbf{(Lucas Numbers)}$ The Lucas numbers are defined by $L_0 = 2$, $L_1 = 1$, and $L_{n+2} = L_{n+1} + L_n$ for every $n \ge 0$. There are $N$ integers $1 \le n \le 2016$ such that $L_n$ contains the digit $1$. Estimate $N$. An estimate of $E$ earns $\left\lfloor 20 - 2|N-E| \right\rfloor$ or $0$ points, whichever ... | 1984 |
For positive integers $a,b$, $a\uparrow\uparrow b$ is defined as follows: $a\uparrow\uparrow 1=a$, and $a\uparrow\uparrow b=a^{a\uparrow\uparrow (b-1)}$ if $b>1$. Find the smallest positive integer $n$ for which there exists a positive integer $a$ such that $a\uparrow\uparrow 6\not \equiv a\uparrow\uparrow 7$ mod $n$. | 329 |
Define the sequence $a_1, a_2 \dots$ as follows: $a_1=1$ and for every $n\ge 2$, \[ a_n = \begin{cases} n-2 & \text{if } a_{n-1} =0 \\ a_{n-1} -1 & \text{if } a_{n-1} \neq 0 \end{cases} \] A non-negative integer $d$ is said to be {\em jet-lagged} if there are non-negative integers $r,s$ and a positive integer $n$ such ... | 51 |
Kelvin the Frog was bored in math class one day, so he wrote all ordered triples $(a, b, c)$ of positive integers such that $abc=2310$ on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute $$\sum_{\stackrel{abc=2310}{a,b,c\in \mathbb{N}}} (a+b+c),$$ where $\mathbb{N}$ denotes the ... | 49140 |
Find the number of ordered triples of positive integers $(a, b, c)$ such that \[6a + 10b + 15c = 3000.\] | 4851 |
Let $P$ and $A$ denote the perimeter and area respectively of a right triangle with relatively prime integer side-lengths. Find the largest possible integral value of $\frac{P^2}{A}$ [color = red]The official statement does not have the final period.[/color] | 45 |
Sam spends his days walking around the following $2\times 2$ grid of squares. \begin{tabular}[t]{|c|c|}\hline 1&2\\ \hline 4&3 \\ \hline \end{tabular} Say that two squares are adjacent if they share a side. He starts at the square labeled $1$ and every second walks to an adjacent square. How many paths can Sam take so ... | 167 |
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 |
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 [I]cool[/I] if for each of the $16$ frogs, the number of friends they made during the meeting is a mul... | 1167 |
Sean is a biologist, and is looking at a strng of length $66$ composed of the letters $A$, $T$, $C$, $G$. A [i]substring[/i] of a string is a contiguous sequence of letters in the string. For example, the string $AGTC$ has $10$ substrings: $A$, $G$, $T$, $C$, $AG$, $GT$, $TC$, $AGT$, $GTC$, $AGTC$. What is the maximum ... | 2100 |
Kelvin the Frog and $10$ of his relatives are at a party. Every pair of frogs is either [i]friendly[/i] or [i]unfriendly[/i]. When $3$ pairwise friendly frogs meet up, they will gossip about one another and end up in a [i]fight[/i] (but stay [i]friendly[/i] anyway). When $3$ pairwise unfriendly frogs meet up, they will... | 28 |
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 $d_i\le d_{i+1}$ for at most one value of $i$. For example, Kelvin likes the numbers $43210$, $132$, and $3$, but not the num... | 14034 |
Let $ABCD$ be a convex quadrilateral with $AB=5$, $BC=6$, $CD=7$, and $DA=8$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Compute $MN^2-PQ^2$. | 13 |
Start by writing the integers $1, 2, 4, 6$ on the blackboard. At each step, write the smallest positive integer $n$ that satisfies both of the following properties on the board. [list] [*] $n$ is larger than any integer on the board currently. [*] $n$ cannot be written as the sum of $2$ distinct integers on the board. ... | 388 |
Let $Q(x)=a_0+a_1x+\dots+a_nx^n$ be a polynomial with integer coefficients, and $0\le a_i<3$ for all $0\le i\le n$. Given that $Q(\sqrt{3})=20+17\sqrt{3}$, compute $Q(2)$. | 86 |
Let $\mathbb{N}$ denote the natural numbers. Compute the number of functions $f:\mathbb{N}\rightarrow \{0, 1, \dots, 16\}$ such that $$f(x+17)=f(x)\qquad \text{and} \qquad f(x^2)\equiv f(x)^2+15 \pmod {17}$$ for all integers $x\ge 1$. | 12066 |
There are $2017$ jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that $N$ of the jars all contain the same positive integer number of coins (i.e. there is an integer $d>0$ such that $N$ of ... | 2014 |
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ and let $P$ be the intersection of its diagonals $AC$ and $BD$. Let $R_1$, $R_2$, $R_3$, $R_4$ be the circumradii of triangles $APB$, $BPC$, $CPD$, $DPA$ respectively. If $R_1=31$ and $R_2=24$ and $R_3=12$, find $R_4$. | 19 |
A polynomial $P$ of degree $2015$ satisfies the equation $P(n)=\frac{1}{n^2}$ for $n=1, 2, \dots, 2016$. Find $\lfloor 2017P(2017)\rfloor$. | -9 |
For positive integers $a$ and $N$, let $r(a, N) \in \{0, 1, \dots, N - 1\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \le 1000000$ for which \[r(n, 1000) > r(n, 1001).\] | 499500 |
Jeffrey writes the numbers $1$ and $100000000 = 10^8$ on the blackboard. Every minute, if $x, y$ are on the board, Jeffery replaces them with \[\frac{x + y}{2} \quad \text{and} \quad 2 \left(\frac{1}{x} + \frac{1}{y}\right)^{-1}.\] After $2017$ minutes the two numbers are $a$ and $b$. Find $\min(a, b)$ to the nearest i... | 10000 |
Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy \[(ab + 1)(bc + 1)(ca + 1) = 84.\] | 12 |
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 $MN^2 + PQ^2$ [color = red]The official problem statement does not have the final period.[/color] | 169 |
At a recent math contest, Evan was asked to find $2^{2016} \pmod{p}$ for a given prime number $p$ with $100 < p < 500$. Evan has forgotten what the prime $p$ was, but still remembers how he solved it: [list] [*] Evan first tried taking $2016$ modulo $p - 1$, but got a value $e$ larger than $100$. [*] However, Evan note... | 211 |
Let $m$ be a positive integer, and let $T$ denote the set of all subsets of $\{1, 2, \dots, m\}$. Call a subset $S$ of $T$ $\delta$-[I]good[/I] if for all $s_1, s_2\in S$, $s_1\neq s_2$, $|\Delta (s_1, s_2)|\ge \delta m$, where $\Delta$ denotes the symmetric difference (the symmetric difference of two sets is the set o... | 2048 |
In convex quadrilateral $ABCD$ we have $AB=15$, $BC=16$, $CD=12$, $DA=25$, and $BD=20$. Let $M$ and $\gamma$ denote the circumcenter and circumcircle of $\triangle ABD$. Line $CB$ meets $\gamma$ again at $F$, line $AF$ meets $MC$ at $G$, and line $GD$ meets $\gamma$ again at $E$. Determine the area of pentagon $ABCDE$. | 396 |
There are two prime numbers $p$ so that $5p$ can be expressed in the form $\left\lfloor \dfrac{n^2}{5}\right\rfloor$ for some positive integer $n.$ What is the sum of these two prime numbers? | 52 |
For some real number $c,$ the graphs of the equation $y=|x-20|+|x+18|$ and the line $y=x+c$ intersect at exactly one point. What is $c$? | 18 |
Lily has a $300\times 300$ grid of squares. She now removes $100 \times 100$ squares from each of the four corners and colors each of the remaining $50000$ squares black and white. Given that no $2\times 2$ square is colored in a checkerboard pattern, find the maximum possible number of (unordered) pairs of squares suc... | 49998 |
Consider a $2\times 3$ grid where each entry is either $0$, $1$, or $2$. For how many such grids is the sum of the numbers in every row and in every column a multiple of $3$? One valid grid is shown below: $$\begin{bmatrix} 1 & 2 & 0 \\ 2 & 1 & 0 \end{bmatrix}$$ | 9 |
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 tail... | 13 |
Let $\{\omega_1,\omega_2,\cdots,\omega_{100}\}$ be the roots of $\frac{x^{101}-1}{x-1}$ (in some order). Consider the set $$S=\{\omega_1^1,\omega_2^2,\omega_3^3,\cdots,\omega_{100}^{100}\}.$$ Let $M$ be the maximum possible number of unique values in $S,$ and let $N$ be the minimum possible number of unique values in $... | 98 |
A $4\times 4$ window is made out of $16$ square windowpanes. How many ways are there to stain each of the windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of its neighbors? Two different windowpanes are neighbors if they share a side. | 24 |
Distinct prime numbers $p,q,r$ satisfy the equation $$2pqr+50pq=7pqr+55pr=8pqr+12qr=A$$ for some positive integer $A.$ What is $A$? | 1980 |
Let $ABC$ be an equilateral triangle with side length $8.$ Let $X$ be on side $AB$ so that $AX=5$ and $Y$ be on side $AC$ so that $AY=3.$ Let $Z$ be on side $BC$ so that $AZ,BY,CX$ are concurrent. Let $ZX,ZY$ intersect the circumcircle of $AXY$ again at $P,Q$ respectively. Let $XQ$ and $YP$ intersect at $K.$ Compute $K... | 304 |
How many ways can you fill a $3 \times 3$ square grid with nonnegative integers such that no [i]nonzero[/i] integer appears more than once in the same row or column and the sum of the numbers in every row and column equals 7? | 216 |
Convex hexagon $ABCDEF$ is drawn in the plane such that $ACDF$ and $ABDE$ are parallelograms with area 168. $AC$ and $BD$ intersect at $G$. Given that the area of $AGB$ is 10 more than the area of $CGB$, find the smallest possible area of hexagon $ABCDEF$. | 196 |
Let $N = 2^{\left(2^2\right)}$ and $x$ be a real number such that $N^{\left(N^N\right)} = 2^{(2^x)}$. Find $x$. | 66 |
Let $x$ and $y$ be positive real numbers. Define $a = 1 + \tfrac{x}{y}$ and $b = 1 + \tfrac{y}{x}$. If $a^2 + b^2 = 15$, compute $a^3 + b^3$. | 50 |
In triangle $ABC$, $AB = 13$, $BC = 14$, $CA = 15$. Squares $ABB_1A_2$, $BCC_1B_2$, $CAA_1B_2$ are constructed outside the triangle. Squares $A_1A_2A_3A_4$, $B_1B_2B_3B_4$ are constructed outside the hexagon $A_1A_2B_1B_2C_1C_2$. Squares $A_3B_4B_5A_6$, $B_3C_4C_5B_6$, $C_3A_4A_5C_6$ are constructed outside the hexagon... | 19444 |
Tessa the hyper-ant has a 2019-dimensional hypercube. For a real number $k$, she calls a placement of nonzero real numbers on the $2^{2019}$ vertices of the hypercube [i]$k$-harmonic[/i] if for any vertex, the sum of all 2019 numbers that are edge-adjacent to this vertex is equal to $k$ times the number on this vertex.... | 2040200 |
Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \to \mathbb{N}$ be a function satisfying [list] [*] $f(1) = 1$, [*] for $n \in \mathbb{N}$, $f(2n) = 2f(n)$ and $f(2n+1) = 2f(n) - 1$. [/list] Determine the sum of all positive integer solutions to $f(x) = 19$ that do not exceed 2019. | 1899 |
What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer is [i]palindromic[/i] if the sequence of decimal digits are the same when read backwards.) | 21 |
For a positive integer $N$, we color the positive divisors of $N$ (including 1 and $N$) with four colors. A coloring is called [i]multichromatic[/i] if whenever $a$, $b$ and $\gcd(a, b)$ are pairwise distinct divisors of $N$, then they have pairwise distinct colors. What is the maximum possible number of multichromatic... | 192 |
For positive reals $p$ and $q$, define the [i]remainder[/i] when $p$ and $q$ as the smallest nonnegative real $r$ such that $\tfrac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_1$ and $r_2$ be the remainder when $a\sqrt{2} + b\sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respect... | 16 |
How many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring of length at least two? (A [i]substring[/i] is a continuous block of letters that is part of the string. A string is [i]palindromic[/i] if it is the same when read backwards.) | 6 |
A polynomial $P(x)$ is a \emph{base-$n$ polynomial} if it is of the form $a_dx^d+a_{d-1}x^{d-1}+\cdots + a_1x+a_0$, where each $a_i$ is an integer between $0$ and $n-1$ inclusive and $a_d>0$. Find the largest positive integer $n$ such that for any real number $c$, there exists at most one base-$n$ polynomial $P(x)$ for... | 9 |
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