Let $a,b,$ be positive integers and $a>b>2$. Prove that $\frac{2^a+1}{2^b-1}$ is never an integer

Determine all pairs $(P, d)$ of a polynomial $P$ with integer coefficients and an integer $d$ such that the equation $P(x) - P(y) = d$ has infinitely many solutions in integers $x$ and $y$ with $x \neq y$.

Six altitudes are constructed from the three vertices of the base of a tetrahedron to the opposite sides of the three lateral faces. Prove that all three straight lines joining two base points of the altitudes in each lateral face are parallel to a certain plane. (IF Sharygin, Moscow)

Let $ABC$ be a triangle with incenter $I$ and let $\Gamma$ be its circumcircle. Let $M$ be the midpoint of $BC$, $K$ the midpoint of the arc $BC$ which does not contain $A$, $L$ the midpoint of the arc $BC$ which contains $A$ and $J$ the reflection of $I$ by the line $KL$. The line $LJ$ intersects $\Gamma$ again at the point $T\neq L$. The line $TM$ intersects $\Gamma$ again at the point $S\neq T$. Prove that $S, I, M, K$ lie on the same circle.

Define set $S_n=\{1,2,\cdots,n\}$. $X$ is a subset of $S_n$. We call sum of all numbers in $X$ [i]capacity[/i] ([i]capacity[/i] of empty set is $0$). If [i]capacity[/i] of $X$ is odd/even, then we call it [i]odd/even subset[/i]. [b](a)[/b] Prove that the number of [i]odd subsets[/i] and [i]even subsets[/i] of $S_n$ are the same. [b](b)[/b] Prove that the sum of [i]capacity[/i] of all [i]odd subsets[/i] and [i]even subsets[/i] are the same when $n\geq3$. [b](c)[/b] Calculate the sum of [i]capacity[/i] of all [i]odd subsets[/i] when $n\geq3$.

Prove that if $|x| < 1$, then \[ \frac{x}{(1-x)^2}+\frac{x^2}{(1+x^2)^2} + \frac{x^3}{(1-x^3)^2}+\cdots=\frac{x}{1-x}+\frac{2x^2}{1+x^2}+\frac{3x^3}{1-x^3}+\cdots\]

Let $ABC$ be a fixed triangle. Three similar (by point order) isosceles trapezoids are built on its sides: $ABXY, BCZW, CAUV$, such that the sides of the triangle are bases of the respective trapezoids. The circumcircles of triangles $XZU, YWV$ meet at two points $P, Q$. Prove that the line $PQ$ passes through a fixed point independent of the choice of trapezoids.

In the expression $ c\cdot a^b\minus{}d$, the values of $ a$, $ b$, $ c$, and $ d$ are $ 0$, $ 1$, $ 2$, and $ 3$, although not necessarily in that order. What is the maximum possible value of the result? $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ 10$

Given an acute triangle $ABC$ with $AB<AC$, let $D$ be the foot of altitude from $A$ to $BC$ and let $M\neq D$ be a point on segment $BC$.$\,J$ and $K$ lie on $AC$ and $AB$ respectively such that $K,J,M$ lies on a common line perpendicular to $BC$. Let the circumcircles of $\triangle ABJ$ and $\triangle ACK$ intersect at $O$. Prove that $J,O,M$ are collinear if and only if $M$ is the midpoint of $BC$. [i]Proposed by Wong Jer Ren[/i]

Call a subset $ S$ of $ \{1,2,\dots,n\}$ [i]mediocre[/i] if it has the following property: Whenever $ a$ and $ b$ are elements of $ S$ whose average is an integer, that average is also an element of $ S.$ Let $ A(n)$ be the number of mediocre subsets of $ \{1,2,\dots,n\}.$ [For instance, every subset of $ \{1,2,3\}$ except $ \{1,3\}$ is mediocre, so $ A(3)\equal{}7.$] Find all positive integers $ n$ such that $ A(n\plus{}2)\minus{}2A(n\plus{}1)\plus{}A(n)\equal{}1.$

Given that $x > 1$, compute $x$ such that $$\log_{16}(x) + \log_x(2)$$ is minimal.

Let $a_1, a_2, ... , a_{2012}$ be odd positive integers. Prove that the number $$A=\sqrt{a^2_1+ a^2_2+ ...+ a^2_{2012}-1}$$ is irrational.

Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $A'$, two circles through $B$ at $B'$ , two circles at $C$ at $C'$ and the two circles at $D$ at $D'$. Suppose the points $A',B',C'$ and $D'$ are distinct. Prove quadrilateral $A'B'C'D'$ is similar to $ABCD$.

Suppose that the number $ a$ satisfies the equation $ 4 \equal{} a \plus{} a^{ \minus{} 1}$. What is the value of $ a^{4} \plus{} a^{ \minus{} 4}$? $ \textbf{(A)}\ 164 \qquad \textbf{(B)}\ 172 \qquad \textbf{(C)}\ 192 \qquad \textbf{(D)}\ 194 \qquad \textbf{(E)}\ 212$

Let $ABCD$ be a square with $2cm$ side length and with center $T$. A rhombus $ARTE$ is drawn where point $E$ lies on line $DC$. What is the area of $ARTE$?

positive integers $a_1, a_2, . . . , a_9$ satisfying $a_1+a_2+ . . . +a_9 =90$ find maximum of $$\frac{1^{a_1} \cdot 2^{a_2} \cdot . . . \cdot 9^{a_9}}{a_1! \cdot a_2! \cdot . . . \cdot a_9!}$$ [hide=mention] I was really shocked because there are no inequality problems at KJMO and the test difficulty even more lower...[/hide]

In a certain country there are 9 cities and two airline companies: AeroSol and AeroLuna. Between each pair of cities there are flights from one and only one of them. the two companies. Furthermore, for any triple of cities $X$, $Y$,$ Z$ σt least one of the flights between them is served by AeroLuna. It is possible to find $4$ cities such that all flights between them be served by AeroLuna?

Let $\operatorname{cif}(x)$ denote the sum of the digits of the number $x$ in the decimal system. Put $a_1=1997^{1996^{1997}}$, and $a_{n+1}=\operatorname{cif}(a_n)$ for every $n>0$. Find $\lim_{n\to\infty}a_n$.

Let $ ABC$ be a triangle with integer side lengths, and let its incircle touch $ BC$ at $ D$ and $ AC$ at $ E$. If $ |AD^2\minus{}BE^2| \le 2$, show that $ AC\equal{}BC$.

Ten girls, numbered from 1 to 10, sit at a round table, in a random order. Each girl then receives a new number, namely the sum of her own number and those of her two neighbours. Prove that some girl receives a new number greater than 17.

Observe that the fraction $ \frac{1}{7}\equal{}0,142857$ is a pure periodical decimal with period $ 6\equal{}7\minus{}1$,and in one period one has $ 142\plus{}857\equal{}999$.For $ n\equal{}1,2,\dots$ find a sufficient and necessary condition that the fraction $ \frac{1}{2n\plus{}1}$ has the same properties as above and find two such fractions other than $ \frac{1}{7}$.

Let $a,b,c$ be pairwise distinct positive real, Prove that$$\dfrac{ab+bc+ca}{(a+b)(b+c)(c+a)}<\dfrac17(\dfrac{1}{|a-b|}+\dfrac{1}{|b-c|}+\dfrac{1}{|c-a|}).$$

In Geneva there are $16$ secret agents, each of whom is watching one or more other agents. It is known that if agent $A$ is watching agent $B$, then $B$ is not watching $A$. Moreover, any $10$ agents can be ordered so that the first is watching the second, the second is watching the third, etc, the last is watching the first. Show that any $11$ agents can also be so ordered.

Four segments divide a convex quadrilateral into nine quadrilaterals. The points of intersections of these segments lie on the diagonals of the quadrilateral (see figure). It is known that the quadrilaterals 1, 2, 3, 4 admit inscribed circles. Prove that the quadrilateral 5 also has an inscribed circle. [asy] pair A,B,C,D,E,F,G,H,I,J,K,L; A=(-4.0,4.0);B=(-1.06,4.34);C=(-0.02,4.46);D=(4.14,4.93);E=(3.81,0.85);F=(3.7,-0.42); G=(3.49,-3.05);H=(1.37,-2.88);I=(-1.46,-2.65);J=(-2.91,-2.52);K=(-3.14,-1.03);L=(-3.61,1.64); draw(A--D);draw(D--G);draw(G--J);draw(J--A); draw(A--G);draw(D--J); draw(B--I);draw(C--H);draw(E--L);draw(F--K); pair R,S,T,U,V; R=(-2.52,2.56);S=(1.91,2.58);T=(-0.63,-0.11);U=(-2.37,-1.94);V=(2.38,-2.06); label("1",R,N);label("2",S,N);label("3",T,N);label("4",U,N);label("5",V,N); [/asy] [i]Proposed by Nairi M. Sedrakyan, Armenia[/i]

Suppose that $4$ real numbers $a, b,c,d$ satisfy the conditions $\begin{cases} a^2 + b^2 = 4\\ c^2 + d^2 = 4 \\ ac + bd = 2 \end{cases}$ Find the set of all possible values the number $M = ab + cd$ can take.