For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.

Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat? $\textbf{(A)}\hspace{.05in} \dfrac1{24}\qquad \textbf{(B)}\hspace{.05in}\dfrac1{12} \qquad \textbf{(C)}\hspace{.05in}\dfrac18 \qquad \textbf{(D)}\hspace{.05in}\dfrac16 \qquad \textbf{(E)}\hspace{.05in}\dfrac14 $

The square $BCDE$ is inscribed in circle $\omega$ with center $O$. Point $A$ is the reflection of $O$ over $B$. A "hook" is drawn consisting of segment $AB$ and the major arc $\widehat{BE}$ of $\omega$ (passing through $C$ and $D$). Assume $BCDE$ has area $200$. To the nearest integer, what is the length of the hook? [i]Proposed by Evan Chen[/i]

Let $p \geq 3$ be a prime and $a_1,a_2,\cdots , a_{p-2}$ be a sequence of positive integers such that for every $k \in \{1,2,\cdots,p-2\}$ neither $a_k$ nor $a_k^k-1$ is divisible by $p$. Prove that product of some of members of this sequence is equivalent to $2$ modulo $p$.

Two persons play a game on a board divided into $3\times 100$ squares. They move in turn: the first places tiles of size $1\times2$ lengthwise (along the long axis of the board), the second, in the perpendicular direction. The loser is the one who cannot make a move. Which of the players can always win (no matter how his opponent plays), and what is the winning strategy?

Given a positive integer $n$, determine the maximum number of lattice points in the plane a square of side length $n +\frac{1}{2n+1}$ may cover.

Let $\mathcal{C}$ be the family of circumferences in $\mathbb{R}^2$ that satisfy the following properties: (i) if $C_n$ is the circumference with center $(n,1/2)$ and radius $1/2$, then $C_n\in \mathcal{C}$, for all $n\in \mathbb{Z}$. (ii) if $C$ and $C'$, both in $\mathcal{C}$, are externally tangent, then the circunference externally tangent to $C$ and $C'$ and tanget to $x$-axis also belongs to $\mathcal{C}$. (iii) $\mathcal{C}$ is the least family which these properties. Determine the set of the real numbers which are obtained as the first coordinate of the points of intersection between the elements of $\mathcal{C}$ and the $x$-axis.

Evaluate $ \int_{\frac{\pi}{4}}^{\frac {\pi}{2}} \frac {1}{(\sin x \plus{} \cos x \plus{} 2\sqrt {\sin x\cos x})\sqrt {\sin x\cos x}}dx$.

Consider a triangle $XYZ$ and a point $O$ in its interior. Three lines through $O$ are drawn, parallel to the respective sides of the triangle. The intersections with the sides of the triangle determine six line segments from $O$ to the sides of the triangle. The lengths of these segments are integer numbers $a, b, c, d, e$ and $f$ (see figure). Prove that the product $a \cdot b \cdot c\cdot d \cdot e \cdot f$ is a perfect square. [asy] unitsize(1 cm); pair A, B, C, D, E, F, O, X, Y, Z; X = (1,4); Y = (0,0); Z = (5,1.5); O = (1.8,2.2); A = extension(O, O + Z - X, X, Y); B = extension(O, O + Y - Z, X, Y); C = extension(O, O + X - Y, Y, Z); D = extension(O, O + Z - X, Y, Z); E = extension(O, O + Y - Z, Z, X); F = extension(O, O + X - Y, Z, X); draw(X--Y--Z--cycle); draw(A--D); draw(B--E); draw(C--F); dot("$A$", A, NW); dot("$B$", B, NW); dot("$C$", C, SE); dot("$D$", D, SE); dot("$E$", E, NE); dot("$F$", F, NE); dot("$O$", O, S); dot("$X$", X, N); dot("$Y$", Y, SW); dot("$Z$", Z, dir(0)); label("$a$", (A + O)/2, SW); label("$b$", (B + O)/2, SE); label("$c$", (C + O)/2, SE); label("$d$", (D + O)/2, SW); label("$e$", (E + O)/2, SE); label("$f$", (F + O)/2, NW); [/asy]

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

Find the greatest positive real number $ k$ such that the inequality below holds for any positive real numbers $ a,b,c$: \[ \frac ab \plus{} \frac bc \plus{} \frac ca \minus{} 3 \geq k \left( \frac a{b \plus{} c} \plus{} \frac b{c \plus{} a} \plus{} \frac c{a \plus{} b} \minus{} \frac 32 \right). \]

Let $a_1,a_2,\cdots ,a_{100}$ be non-negative integers such that $(1)$ There are positive integers$ k\leq 100$ such that $a_1\leq a_2\leq \cdots\leq a_{k}$ and $a_i=0$ $(i>k);$ $(2)$ $ a_1+a_2+a_3+\cdots +a_{100}=100;$ $(3)$ $ a_1+2a_2+3a_3+\cdots +100a_{100}=2022.$ Find the minimum of $ a_1+2^2a_2+3^2a_3+\cdots +100^2a_{100}.$

Calculate the following $$P=(4\sin^2{0} -3)(4\sin^2\frac{\pi}{2^{2005}} -3)(4\sin^2\frac{2\pi}{2^{2005}} -3)(4\sin^2\frac{3\pi}{2^{2005}} -3)...$$ $$...\,\,\,\,(4\sin^2\frac{(2^{2004}-1)\pi}{2^{2005}} -3)(4\sin^2\frac{\pi}{2} -3)$$

Let $p$ be a (positive) prime number. Suppose that real numbers $a_1, a_2, . . ., a_{p+1}$ have the property that, whenever one of the numbers is deleted, the remaining numbers can be partitioned into two classes with the same arithmetic mean. Show that these numbers must be equal.

In triangle $ABC$, the incircle touches sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. Let $X$ be the feet of the altitude of the vertex $D$ on side $EF$ of triangle $DEF$. Prove that $AX,BY$ and $CZ$ are concurrent on the Euler line of the triangle $DEF$.

Edges of a complete graph with $2m$ vertices are properly colored with $2m-1$ colors. It turned out that for any two colors all the edges colored in one of these two colors can be described as union of several $4$-cycles. Prove that $m$ is a power of $2$.