What is the smallest number ? (A) $3$ (B) $2^{\sqrt2}$ (C) $2^{1+\frac{1}{\sqrt2}}$ (D) $2^{\frac12} + 2^{\frac23}$ (E) $2^{\frac53}$

Circles $A, B$, and $C$ each have radius $r$, and their centers are the vertices of an equilateral triangle of side length $6r$. Two lines are drawn, one tangent to $A$ and $C$ and one tangent to $B$ and $C$, such that $A$ is on the opposite side of each line from $B$ and $C$. Find the sine of the angle between the two lines. [img]http://4.bp.blogspot.com/-IZv8q-3NYZg/XXmrroy2PnI/AAAAAAAAKxg/jSOcOOQ8Kyw0EwHUifXJ1jOd2ENAo1FfACK4BGAYYCw/s200/2008%2Bpumac%2Bb6.png[/img]

A minus sign is placed on one square of a $5 \times 5$ board and plus signs are placed on the remaining squares. A move is to select a $2 \times 2, 3 \times 3, 4 \times 4$ or $5 \times 5$ square and change all the signs in it. Which initial positions allow a series of moves to change all the signs to plus?

Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$

Real numbers $a, b, c, x, y, z$ satisfy $$\begin{aligned} \begin{cases} a^2+2bc=x^2+2yz,\\ b^2+2ca=y^2+2zx,\\ c^2+2ab=z^2+2xy.\\ \end{cases} \end{aligned}$$ Prove that $a^2+b^2+c^2=x^2+y^2+z^2$.

Determine all real numbers $\alpha$ with the property that all numbers in the sequence $\cos\alpha,\cos2\alpha,\cos2^2\alpha,\ldots,\cos2^n\alpha,\ldots$ are negative.

What maximal area can have a triangle if its sides $a,b,c$ satisfy inequality $0\le a\le 1\le b\le 2\le c\le 3$ ?

Determine all $k$ for which there exists a natural number n such that $1^n + 2^n + 3^n + 4^n$ with exactly $k$ zeros at the end.

A pair of positive integers $(m,n)$ is called [i]compatible[/i] if $m \ge \tfrac{1}{2} n + 7$ and $n \ge \tfrac{1}{2} m + 7$. A positive integer $k \ge 1$ is called [i]lonely[/i] if $(k,\ell)$ is not compatible for any integer $\ell \ge 1$. Find the sum of all lonely integers. [i]Proposed by Evan Chen[/i]

$a_1,a_2,...,a_{2014}$ is a permutation of $1,2,3,...,2014$. What is the greatest number of perfect squares can have a set ${ a_1^2+a_2,a_2^2+a_3,a_3^2+a_4,...,a_{2013}^2+a_{2014},a_{2014}^2+a_1 }?$

Let $r, s \geq 1$ be integers and $a_{0}, a_{1}, . . . , a_{r-1}, b_{0}, b_{1}, . . . , b_{s-1} $ be real non-negative numbers such that $(a_0+a_1x+a_2x^2+. . .+a_{r-1}x^{r-1}+x^r)(b_0+b_1x+b_2x^2+. . .+b_{s-1}x^{s-1}+x^s) =1 +x+x^2+. . .+x^{r+s-1}+x^{r+s}$. Prove that each $a_i$ and each $b_j$ equals either $0$ or $1$.

Determine all values $\alpha\in\mathbb R$ with the following property: if positive numbers $(x,y,z)$ satisfy the inequality \[x^2+y^2+z^2\le\alpha(xy+yz+zx),\] then $x,y,z$ are sides of a triangle.

An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A [i]beam[/i] is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions: [list=] [*]The two $1 \times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \cdot {2020}^2$ possible positions for a beam.) [*]No two beams have intersecting interiors. [*]The interiors of each of the four $1 \times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam. [/list] What is the smallest positive number of beams that can be placed to satisfy these conditions? [i]Proposed by Alex Zhai[/i]

Consider an infinite strip of unit squares. The squares are numbered "1", "2", "3", ... A pawn starts on one of the squares and it can move according to the following rules: (1) from the square numbered "$n$" to the square numbered "$2n$", and vice versa; (2) from the square numbered "$n$" to the square numbered "$3n + 1$", and vice versa. Show that the pawn can reach the square numbered "$1$" in a finite number of moves.

$2024$ people attended a party. Eunson, the host of the party, wanted to make the participant shake hands in pairs. As a professional daydreamer, Eunsun wondered which would be greater: the number of ways each person could shake hands with $4$ others or the number of ways each person could shake hands with $3$ others. Solve Eunsun's peculiar question.

Given $ a, b, c > 0$. Prove that: $ (1\plus{}a\plus{}b\plus{}c)(1\plus{}ab\plus{}bc\plus{}ca) \ge 4\sqrt{2}\sqrt{(a\plus{}bc)(b\plus{}ca)(c\plus{}ab)}$ :)

Consider a sequence $T_0, T_1, \dots$ of polynomials defined recursively by $T_0(x) = 2$, $T_1(x)=x$, and $T_{n+2}(x) = xT_{n+1}(x) - T_n(x)$ for each nonnegative integer $n$. Let $L_n$ be the sequence of Lucas Numbers, defined by $L_0 = 2$, $L_1 = 1$, and $L_{n+2} = L_n+L_{n+1}$ for every nonnegative integer $n$. Find the remainder when $ T_0\left( L_0 \right) + T_1 \left( L_2 \right) + T_2 \left( L_4 \right) + \dots + T_{359} \left( L_{718} \right)$ is divided by $359$. [i]Proposed by Yang Liu[/i]

Let $n \geq 2$. Let $a_1 , a_2 , a_3 , \ldots a_n$ be $n$ real numbers all less than $1$ and such that $|a_k - a_{k+1} | < 1$ for $1 \leq k \leq n-1$. Show that \[ \dfrac{a_1}{a_2} + \dfrac{a_2}{a_3} + \dfrac{a_3}{a_4} + \ldots + \dfrac{a_{n-1}}{a_n} + \dfrac{a_n}{a_1} < 2 n - 1 . \]

In a regular polygon with 100 vertices, 10 vertices are painted blue, and 10 other vertices are painted red. 1. Prove that there exist two distinct blue vertices \( A_1 \) and \( A_2 \), and two distinct red vertices \( R_1 \) and \( R_2 \), such that the distance between \( A_1 \) and \( R_1 \) is equal to the distance between \( A_2 \) and \( R_2 \). 2. Prove that there exist two distinct blue vertices \( A_1 \) and \( A_2 \), and two distinct red vertices \( R_1 \) and \( R_2 \), such that the distance between \( A_1 \) and \( A_2 \) is equal to the distance between \( R_1 \) and \( R_2 \).

Compute the number of ways to arrange $3$ copies of each of the $26$ lowercase letters of the English alphabet such that for any two distinct letters $x_1$ and $x_2,$ the number of $x_2$’s between the first and second occurrences of $x_1$ equals the number of $x_2$’s between the second and third occurrences of $x_1.$

Kei draws a $6\times 6$ grid. He colors $13$ of the unit squares silver and the remaining squares gold. Kei then folds the grid in half vertically, forming pairs of overlapping unit squares. Let $m$ and $M$ the least and greatest possible number of gold-on-gold pairs, respectively. What is $m + M?$ $\textbf{(A) } 12 \qquad\textbf{(B) }14 \qquad\textbf{(C) }16\qquad\textbf{(D) }18 \qquad\textbf{(E) }20$\\

A function $ f$ is defined by $ f(z) \equal{} i\bar z$, where $ i \equal{}\sqrt{\minus{}\!1}$ and $ \bar z$ is the complex conjugate of $ z$. How many values of $ z$ satisfy both $ |z| \equal{} 5$ and $ f (z) \equal{} z$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

A $8\times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maximum value of $k$.

Let $A_1A_2A_3\dots A_{20}$ be a $20$-sided polygon $P$ in the plane, where all of the side lengths of $P$ are equal, the interior angle at $A_i$ measures $108$ degrees for all odd $i$, and the interior angle $A_i$ measures $216$ degrees for all even $i$. Prove that the lines $A_2A_8$, $A_4A_{10}$, $A_5A_{13}$, $A_6A_{16}$, and $A_7A_{19}$ all intersect at the same point. [asy] import graph; size(10cm); pair temp= (-1,0); pair A01 = (0,0); pair A02 = rotate(306,A01)*temp; pair A03 = rotate(144,A02)*A01; pair A04 = rotate(252,A03)*A02; pair A05 = rotate(144,A04)*A03; pair A06 = rotate(252,A05)*A04; pair A07 = rotate(144,A06)*A05; pair A08 = rotate(252,A07)*A06; pair A09 = rotate(144,A08)*A07; pair A10 = rotate(252,A09)*A08; pair A11 = rotate(144,A10)*A09; pair A12 = rotate(252,A11)*A10; pair A13 = rotate(144,A12)*A11; pair A14 = rotate(252,A13)*A12; pair A15 = rotate(144,A14)*A13; pair A16 = rotate(252,A15)*A14; pair A17 = rotate(144,A16)*A15; pair A18 = rotate(252,A17)*A16; pair A19 = rotate(144,A18)*A17; pair A20 = rotate(252,A19)*A18; dot(A01); dot(A02); dot(A03); dot(A04); dot(A05); dot(A06); dot(A07); dot(A08); dot(A09); dot(A10); dot(A11); dot(A12); dot(A13); dot(A14); dot(A15); dot(A16); dot(A17); dot(A18); dot(A19); dot(A20); draw(A01--A02--A03--A04--A05--A06--A07--A08--A09--A10--A11--A12--A13--A14--A15--A16--A17--A18--A19--A20--cycle); label("$A_{1}$",A01,E); label("$A_{2}$",A02,W); label("$A_{3}$",A03,NE); label("$A_{4}$",A04,SW); label("$A_{5}$",A05,N); label("$A_{6}$",A06,S); label("$A_{7}$",A07,N); label("$A_{8}$",A08,SE); label("$A_{9}$",A09,NW); label("$A_{10}$",A10,E); label("$A_{11}$",A11,W); label("$A_{12}$",A12,E); label("$A_{13}$",A13,SW); label("$A_{14}$",A14,NE); label("$A_{15}$",A15,S); label("$A_{16}$",A16,N); label("$A_{17}$",A17,S); label("$A_{18}$",A18,NW); label("$A_{19}$",A19,SE); label("$A_{20}$",A20,W);[/asy]

Let $A = (2, 0)$, $B = (0, 2)$, $C = (-2, 0)$, and $D = (0, -2)$. Compute the greatest possible value of the product $PA \cdot PB \cdot PC \cdot PD$, where $P$ is a point on the circle $x^2 + y^2 = 9$.