Let $a$, $b$ be the positive integers greater than $1$. Prove that if $$ \frac{a}{b},\; \frac{a-1}{b-1} $$ differ by 1, then both are integers.

In the triangle $ABC, \angle BAC$ is acute, the angle bisector of $\angle BAC$ meets $BC$ at $D, K$ is the foot of the perpendicular from $B$ to $AC$, and $\angle ADB = 45^o$. Point $P$ lies between $K$ and $C$ such that $\angle KDP = 30^o$. Point $Q$ lies on the ray $DP$ such that $DQ = DK$. The perpendicular at $P$ to $AC$ meets $KD$ at $L$. Prove that $PL^2 = DQ \cdot PQ$.

Let $m$ and $n$ be positive integers such that $n \leq m$. Prove that \[ 2^n n! \leq \frac{(m+n)!}{(m-n)!} \leq (m^2 + m)^n \]

For $[0,1]\subset E\subset [0,+\infty)$ where $E$ is composed of a finite number of closed interval,we start a two dimensional Brownian motion from the point $x<0$ terminating when we first hit $E$.Let $p(x)$ be the probability of the finishing point being in $[0,1]$.Prove that $p(x)$ is increasing on $[-1,0)$.

Given a $ \vartriangle ABC $ with $ AB> AC $ and $ \angle BAC = 60^o$. Denote the circumcenter and orthocenter as $ O $ and $ H $ respectively. We also have that $ OH $ intersects $ AB $ in $ P $ and $ AC $ in $ Q $. Prove that $ PO = HQ $.

Each point of the sphere of radius $r\ge 1$ is colored in one of $n$ colors ($n \ge 2$), and for each color there is a point on the sphere colored in this color. Prove that there are points $A_i$, $B_i$, $i= 1, ..., n$ on the sphere such that the colors of the points $A_1, ..., A_n$ are pairwise different and the color of the point $B_i$ at a distance of $1$ from $A_i$ is different from the color of the point $A_1, i= 1, ..., n$

Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$? $\textbf{(A) } 15 \qquad\textbf{(B) } 15.5 \qquad\textbf{(C) } 16 \qquad\textbf{(D) } 16.5 \qquad\textbf{(E) } 17 $

Let $k$ be a positive integer and $a_1, a_2, ...$ be a sequence of terms from set $\{ 0, 1, ..., k \}$. Let $b_n = \sqrt[n] {a_1^n + a_2^n + ... + a_n^n}$ for all positive integers $n$. Prove, that if in sequence $b_1, b_2, b_3, ...$ are infinitely many integers, then all terms of this series are integers.

Prove that you can't split a square into finitely many hexagons, whose inner angles are all less than $180^o$.

Let $n \geq 4$, and consider a (not necessarily convex) polygon $P_1P_2\hdots P_n$ in the plane. Suppose that, for each $P_k$, there is a unique vertex $Q_k\ne P_k$ among $P_1,\hdots, P_n$ that lies closest to it. The polygon is then said to be [i]hostile[/i] if $Q_k\ne P_{k\pm 1}$ for all $k$ (where $P_0 = P_n$, $P_{n+1} = P_1$). (a) Prove that no hostile polygon is convex. (b) Find all $n \geq 4$ for which there exists a hostile $n$-gon.

How many of the numbers $1\cdot 2\cdot 3$, $2\cdot 3\cdot 4$,..., $2020 \cdot 2021 \cdot 2022$ are divisible by $2020$?

Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that the points $A, X$, and $Y$ are collinear.

Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$ has a solution in integers such that $p\nmid x_1x_2x_3x_4$.

Find all pairs $(x, y)$ of positive integers such that $x^2 + y^2 + 33^2 =2010\sqrt{x-y}$.

Show the the sum of any three consecutive positive integers is a divisor of the sum of their cubes.

Let $ a$, $ a'$, $ b$, and $ b'$ be real numbers with $ a$ and $ a'$ nonzero. The solution to $ ax \plus{} b \equal{} 0$ is less than the solution to $ a'x \plus{} b' \equal{} 0$ if and only if $ \textbf{(A)}\ a'b < ab' \qquad \textbf{(B)}\ ab' < a'b \qquad \textbf{(C)}\ ab < a'b' \qquad \textbf{(D)}\ \frac {b}{a} < \frac {b'}{a'}$ $ \textbf{(E)}\ \frac {b'}{a'} < \frac {b}{a}$

Determine all $ f:R\rightarrow R $ such that $$ f(xf(y)+y^3)=yf(x)+f(y)^3 $$

Let $a_0,a_1,\ldots$ be an infinite sequence of positive numbers. Prove that the inequality $1+a_n>\sqrt[n]2a_{n-1}$ holds for infinitely many positive integers $n$.

Let side $BC$ of $\bigtriangleup ABC$ be the diameter of a semicircle which cuts $AB$ and $AC$ at $D$ and $E$ respectively. $F$ and $G$ are the feet of the perpendiculars from $D$ and $E$ to $BC$ respectively. $DG$ and $EF$ intersect at $M$. Prove that $AM \perp BC$.

$\frac{2}{25}=$ $\text{(A)}\ .008 \qquad \text{(B)}\ .08 \qquad \text{(C)}\ .8 \qquad \text{(D)} 1.25 \qquad \text{(E)}\ 12.5$

A triangle is given whose altitudes' lengths are $\frac{1}{5},\frac{1}{5},\frac{1}{8}$. Evaluate the triangle's area.

A straight line is drawn through an arbitrary internal point $K$ of the trapezoid $ABCD$, intersecting the bases of $BC$ and $AD$ at points $P$ and $Q$, respectively. The circles circumscribed around the triangles $BPK$ and $DQK$ intersect, besides the point $K$, also at the point $L$. Prove that the point $L$ lies on the diagonal $BD$.

For a positive integer $n$, consider all nonincreasing functions $f : \{1,\hdots,n\}\to\{1,\hdots,n\}$. Some of them have a fixed point (i.e. a $c$ such that $f(c) = c$), some do not. Determine the difference between the sizes of the two sets of functions. [i]Remark.[/i] A function $f$ is [i]nonincreasing[/i] if $f(x) \geq f(y)$ holds for all $x \leq y$

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ with the property that $$f(x)f(y) = (xy - 1)^2f\left(\frac{x + y - 1}{xy - 1}\right)$$ for all real numbers $x, y$ with $xy \neq 1$.

Three wise men live in an old region. As they do not always agree on their advice to the king, he decided to stay with the wisest of the three, killing the others. To decide which of them was saved, performed the following test: He put each sage a hat, without him seeing its color, then locked them in a common room and told them: $\bullet$ Only the first to guess the color of his own hat will save his life. $\bullet$ In total there are five hats, three are white and two are black. $\bullet$ You cannot communicate with each other, but you can look at each other. After a long time, one of the wise men says: "I know the color of my hat." What color did they have? How did you figure it out? What color were the other hats used?