Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in the triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 1995$, $AC = 1994$, and $BC = 1993$, then $RS$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime integers. Find $m + n$

Find all $3$-digit numbers $\overline{abc}$ ($a,b \ne 0$) such that $\overline{bcd} \times  a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$

A trapezium is given with parallel bases having lengths $1$ and $4$. Split it into two trapeziums by a cut, parallel to the bases, of length $3$. We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in $m$ and $n$ trapeziums, respectively, so that all the $m + n$ trapezoids obtained have the same area. Determine the minimum possible value for $m + n$ and the lengths of the cuts to be made to achieve this minimum value.

Let $c$ be a complex number. Suppose there exist distinct complex numbers $r$, $s$, and $t$ such that for every complex number $z$, we have \[ (z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct). \] Compute the number of distinct possible values of $c$.

This is a very classical problem. Let the $ABCD$ be a cyclic quadrilateral with circumcircle $\omega_1$.Lines $AC$ and $BD$ intersect at point $E$,and lines $AD$,$BC$ intersect at point $F$.Circle $\omega_2$ is tangent to segments $EB,EC$ at points $M,N$ respectively,and intersects with circle $\omega_1$ at points $Q,R$.Lines $BC,AD$ intersect line $MN$ at $S,T$ respectively.Show that $Q,R,S,T$ are concyclic.

Consider the set $X$ = $\{ 1,2 \ldots 10 \}$ . Find two disjoint nonempty sunsets $A$ and $B$ of $X$ such that a) $A \cup B = X$; b) $\prod_{x\in A}x$ is divisible by $\prod_{x\in B}x$, where $\prod_{x\in C}x$ is the product of all numbers in $C$; c) $\frac{ \prod\limits_{x\in A}x}{ \prod\limits_{x\in B}x}$ is as small as possible.

$ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD$. The crease is $EF$, where $E$ is on $AB$ and $F$is on $CD$. The dimensions $AE=8$, $BE=17$, and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect(E,F)*B, Cp=reflect(E,F)*C; draw(F--D--A--E); draw(E--B--C--F, linetype("4 4")); filldraw(E--F--Cp--Bp--cycle, white, black); pair point=( 12.5, 35/3 ); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$B^\prime$", Bp, dir(point--Bp)); label("$C^\prime$", Cp, dir(point--Cp));[/asy]

Find all positive integers $x$ and $y$ such that $${x \choose y} = 1432$$

In a football tournament of one round (each team plays each other once, $2$ points for win , $1$ point for draw and $0$ points for loss), $28$ teams compete. During the tournament more than $75\%$ of the matches finished in a draw . Prove that there were two teams who finished with the same number of points. (M . Vora, gymnasium student , Hungary)

An integer $n>1$ is given . Find the smallest positive number $m$ satisfying the following conditions： for any set $\{a,b\}$ $\subset \{1,2,\cdots,2n-1\}$ ,there are non-negative integers $ x, y$ ( not all zero) such that $2n|ax+by$ and $x+y\leq m.$

In triangle $ABC, \angle A= 45^o, BH$ is the altitude, the point $K$ lies on the $AC$ side, and $BC = CK$. Prove that the center of the circumscribed circle of triangle $ABK$ coincides with the center of an excircle of triangle $BCH$.

Let $ABCD$ a cyclique quadrilateral. We consider the Following points: $A'$ the orthogonal projection of $A$ over $(BD)$, $B'$ the orthogonal projection of $B$ over $(AC)$, $C'$ the orthgonal projection of $C$ over $(BD)$ and $D'$ being the orthogonal projection of $D$ over $(AC)$ Prove that $A', B', C'$ and $D'$.

There are 7 pieces of paper in the hat. On the $ n $th piece of paper there is written the number $ 2^n-1 $ ($ n = 1, 2, \ldots, 7 $). We draw cards randomly until the sum exceeds 124. What is the most probable value of this sum?

$B$ is the center of unit circle. $A,C$ are points on the circle (the order of $A,B,C$ is clockwise), and $\angle ABC=2\alpha(0<\alpha<\frac{\pi}{3})$. Then we will rotate $\triangle ABC$ anticlockwise. In the first rotation, $A$ is the center of rotation, the result is that $B$ is on the circle. In the second rotation, $B$ is the center of rotation, the result is that $C$ is on the circle. In the third rotation, $C$ is the center of rotation, the result is that $A$ is on the circle. ... After we rotate for $100$ times, the distance $A$ travelled is $\text{(A)}22\pi(1+\sin\alpha)-66\alpha\qquad\text{(B)}\frac{67}{3}\pi\qquad\text{(C)}22\pi+\frac{68}{3}\pi\sin\alpha-66\alpha\qquad\text{(D)}33\pi-66\alpha$

$ABC$ is triangle with $AB=BC$. $X,Y$ are midpoints of $AC$ and $AB$. $Z$ is base of perpendicular from $B$ to $CY$. Prove, that circumcenter of $XYZ$ lies on $AC$

The edges of $K_{2017}$ are each labeled with $1,2,$ or $3$ such that any triangle has sum of labels at least $5.$ Determine the minimum possible average of all $\dbinom{2017}{2}$ labels. (Here $K_{2017}$ is defined as the complete graph on 2017 vertices, with an edge between every pair of vertices.) [i]Proposed by Michael Ma[/i]

Let $ a_0,a_1,\dots,a_{n \plus{} 1}$ be natural numbers such that $ a_0 \equal{} a_{n \plus{} 1} \equal{} 1$, $ a_i>1$ for all $ 1\leq i \leq n$, and for each $ 1\leq j\leq n$, $ a_i|a_{i \minus{} 1} \plus{} a_{i \plus{} 1}$. Prove that there exist one $ 2$ in the sequence.

Let $ABC$ be a right triangle with $\angle A= 90^{\circ}$. A circle $\omega$ centered on $BC$ is tangent to $AB$ at $D$ and $AC$ at $E$. Let $F$ and $G$ be the intersections of $\omega$ and $BC$ so that $F$ lies between $B$ and $G$. If lines $DG$ and $EF$ intersect at $X$, show that $AX=AD.$

If $a$ and $b$ are positive integers such that $a^2-b^4= 2009$, find $a+b$.

Chess king is standing in some square of chessboard. Every sunday it is moved to one square by diagonal, and every another day it is moved to one square by horisontal or vertical. What maximal numbers of moves can be made ?

Let $ABC$ be a right triangle in $A$ , whose leg measures $1$ cm. The bisector of the angle $BAC$ cuts the hypotenuse in $R$, the perpendicular to $AR$ on $R$ , cuts the side $AB$ at its midpoint. Find the measurement of the side $AB$ .

Let $k$ be a real number. Define on the set of reals the operation $x*y$= $\frac{xy}{x+y+k}$ whenever $x+y$ does not equal $-k$. Let $x_1<x_2<x_3<x_4$ be the roots of $t^4=27(t^2+t+1)$.suppose that $[(x_1*x_2)*x_3]*x_4=1$. Find all possible values of $k$

A particle performing a random walk on the integer points of the semi-axis $x \ge 0$ moves a distance $1$ to the right with probability $a$, and to the left with probability $b$, and stands still in the remaining cases (if $x = 0$, it stands still instead of moving to the left). Determine the steady-state probability distribution, and also the expectation of $x$ and $x^2$ over a long time, if the particle starts at the point $0$.

Five distinct $2$-digit numbers are in a geometric progression. Find the middle term.

Given a cyclic quadrilateral $ABCD$ so that $AB = AD$ and $AB + BC <CD$. Prove that the angle $ABC$ is more than $120$ degrees.