Given an acute-angled triangle $ABC$ with orthocenter $H$. Reflection of nine-point circle about $AH$ intersects circumcircle at points $X$ and $Y$. Prove that $AH$ is the external bisector of $\angle XHY$. [i]Proposed by Mohammad Javad Shabani[/i]

Consider the grid of points $X = \{(m,n) | 0 \leq m,n \leq 4 \}$. We say a pair of points $\{(a,b),(c,d)\}$ in $X$ is a knight-move pair if $( c = a \pm 2$ and $d = b \pm 1)$ or $( c = a \pm 1$ and $d = b \pm 2)$. The number of knight-move pairs in $X$ is:

If $f(x) = x^2 + x$, prove that the equation $4f(a) = f(b)$ has no solutions in positive integers $a$ and $b$.

Assume that, for a certain school, it is true that [list]I: Some students are not honest II: All fraternity members are honest[/list] A necessary conclusion is: $\textbf{(A)}\ \text{Some students are fraternity members} \qquad\\ \textbf{(B)}\ \text{Some fraternity members are not students} \qquad\\ \textbf{(C)}\ \text{Some students are not fraternity members} \qquad\\ \textbf{(D)}\ \text{No fraternity member is a student} \qquad\\ \textbf{(E)}\ \text{No student is a fraternity member} $

$ABC$ is a triangle, with inradius $r$ and circumradius $R.$ Show that: \[ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{B}{2} \right) + \sin \left( \frac{B}{2} \right) \cdot \sin \left( \frac{C}{2} \right) + \sin \left( \frac{C}{2} \right) \cdot \sin \left( \frac{A}{2} \right) \leq \frac{5}{8} + \frac{r}{4 \cdot R}. \]

$ABCD$ is a trapezoid with $AB || CD$. There are two circles $\omega_1$ and $\omega_2$ is the trapezoid such that $\omega_1$ is tangent to $DA$, $AB$, $BC$ and $\omega_2$ is tangent to $BC$, $CD$, $DA$. Let $l_1$ be a line passing through $A$ and tangent to $\omega_2$(other than $AD$), Let $l_2$ be a line passing through $C$ and tangent to $\omega_1$ (other than $CB$). Prove that $l_1 || l_2$.

Given a triangle $ABC$ , a circle centered at some point $O$ meets the segments $BC$ , $CA$ , $AB$ in the pairs of points $X$ and $X^{'}$ , $Y$ and $Y^{'}$ , $Z$ and $Z^{'}$ , respectively ,labelled in circular order : $X,X^{'},Y,Y^{'},Z,Z^{'}$. Let $M$ be the Miquel point of the triangle $XYZ$ and let $M^{'}$ be that of the triangle $X^{'}Y^{'}Z^{'}$ . Prove that the segments $OM$ and $OM^{'}$ have equal lehgths.

In a board with $3$ rows and $555$ columns, $3$ squares are colored red, one in each of the $3$ rows. If the numbers from $1$ to $1665$ are written in the boxes, in row order, from left to right (in the first row from $1$ to $555$, in the second from $556$ to $1110$ and in the third from $1111$ to $1665$) there are $3$ numbers that are written in red squares. If they are written in the boxes, ordered by columns, from top to bottom, the numbers from $1$ to $1665$ (in the first column from $1$ to $3$, in the second from $4$ to $6$, in the third from $7$ to $9$,... ., and in the last one from $1663$ to $1665$) there are $3$ numbers that are written in red boxes. We call [i]red[/i] numbers those that in one of the two distributions are written in red boxes. Indicate which are the $3$ squares that must be colored red so that there are only $3$ red numbers. Show all the possibilities.

[b]a)[/b] Let be two nonegative integers $ n\ge 1,k, $ and $ n $ real numbers $ a,b,\ldots ,c. $ Prove that $$ (1/a+1/b+\cdots 1/c)\left( a^{1+k} +b^{1+k}+\cdots c^{1+k} \right)\ge n\left(a^k+b^k+\cdots +c^k\right) . $$ [b]b)[/b] If $ 1\le d\le e\le f\le g\le h\le i\le 1000 $ are six real numbers, determine the minimum value the expression $$ d/e+f/g+h/i $$ can take.

Find the smallest positive integer $n$ for which the expansion of $(xy - 3x +7y - 21)^n,$ after like terms have been collected, has at least 1996 terms.

find the number of ordered triples $(x,y,z)$ of real numbers that satisfy the system of equations: $x+y+z=7; x^2+y^2+z^2=27; xyz=5$.

Find all natural numbers $n$ such that $n$ equals the cube of the sum of its digits.

Each of the $9$ referees on the figure skating championship estimates the program of $20$ sportsmen by assigning him a place (from $1$ to $20$). The winner is determined by adding those numbers. (The less is the sum - the higher is the final place). It was found, that for the each sportsman, the difference of the places, received from the different referees was not greater than $3$. What can be the maximal sum for the winner?

Two circles $C_{1}$ and $C_{2}$ are (externally or internally) tangent at a point $P$. The straight line $D$ is tangent at $A$ to one of the circles and cuts the other circle at the points $B$ and $C$. Prove that the straight line $PA$ is an interior or exterior bisector of the angle $\angle BPC$.

Given real numbers $a, b$, find all functions $f: R \to R$ satisfying $f(x,y) = af (x,z) + bf(y,z)$ for all $x,y,z \in R$.

If the degree measures of the angles of a triangle are in the ratio $3:3:4$, what is the degree measure of the largest angle of the triangle? $\textbf{(A) }18\qquad\textbf{(B) }36\qquad\textbf{(C) }60\qquad\textbf{(D) }72\qquad\textbf{(E) }90$

Show that $\frac{1}{x_1^n} + \frac{1}{x_2^n} +...+ \frac{1}{x_k^n} \ge k^{n+1}$ for positive real numbers $x_i $ with sum $1$.

Given a table with $2^n * n$ 1*1 squares ( $2^n$ rows and n column). In any square we put a number in {1, -1} such that no two rows are the same. Then we change numbers in some squares by 0. Prove that in new table we can choose some rows such that sum of all numbers in these rows equal to 0.

Let $n$ be the number of real values of $p$ for which the roots of \[ x^2-px+p=0 \] are equal. Then $n$ equals: ${{ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \text{a finite number greater than 2} }\qquad\textbf{(E)}\ \text{an infinitely large number} } $

Find $ \inf_{z\in\mathbb{C}} \left( |z^2+z+1|+|z^2-z+1| \right) . $ [i]Gheorghe Andrei[/i] and [i]Doru Constantin Caragea[/i]

Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfy the following property for all real numbers $x$ and all polynomials $P$ with real coefficients: If $P(f(x)) = 0$, then $f(P(x)) = 0$.

Consider an $n\times n$ unit-square board. The main diagonal of the board is the $n$ unit squares along the diagonal from the top left to the bottom right. We have an unlimited supply of tiles of this form: [asy] size(1.5cm); draw((0,1)--(1,1)--(1,2)--(0,2)--(0,1)--(0,0)--(1,0)--(2,0)--(2,1)--(1,1)--(1,0)); [/asy] The tiles may be rotated. We wish to place tiles on the board such that each tile covers exactly three unit squares, the tiles do not overlap, no unit square on the main diagonal is covered, and all other unit squares are covered exactly once. For which $n\geq 2$ is this possible? [i]Proposed by Daniel Kohen[/i]

Jones covered a distance of $ 50$ miles on his first trip. On a later trip he traveled $ 300$ miles while going three times as fast. His new time compared with the old time was: $ \textbf{(A)}\ \text{three times as much} \qquad\textbf{(B)}\ \text{twice as much} \qquad\textbf{(C)}\ \text{the same}$ $ \textbf{(D)}\ \text{half as much} \qquad\textbf{(E)}\ \text{a third as much}$

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[ f(2x + y + f(x + y)) + f(xy) = y f(x) \] for all real numbers $x$ and $y$.

To obtain a square $P$ of side length $2$ cm divided into $4$ unit squares it is sufficient to draw $3$ squares: $P$ and another $2$ unit squares with a common vertex, as shown below: [img]https://cdn.artofproblemsolving.com/attachments/1/d/827516518871ec8ff00a66424f06fda9812193.png[/img] Find the minimum number of squares sufficient to obtain a square.of side length $n$ cm divided into $n^2$ unit squares ($n \ge 3$ is an integer).