The function $f:\mathbb Z \to \mathbb Z$ has the property that for all integers $m$ and $n$ \[f(m)+f(n)+f(f(m^2+n^2))=1.\] We know that integers $a$ and $b$ exist such that $f(a)-f(b)=3$. Prove that integers $c$ and $d$ can be found such that $f(c)-f(d)=1$. [i]Proposed by Amirhossein Gorzi[/i]

On a large wooden block there are four twelve-hour analog clocks of varying accuracy. At 7PM on April 3, 2015, they all correctly displayed the time. The first clock is accurate, the second clock is two times as fast as the first clock, the third clock is three times as fast as the first clock, and the last clock doesn't move at all. How many hours must elapse (from 7PM) before the times displayed on the clocks coincide again? (The clocks do not distinguish between AM and PM.) [asy] import olympiad; import cse5; size(12cm); defaultpen(linewidth(0.9)+fontsize(11pt)); picture clock(real hh, real mm, string nn) { picture p; draw(p, unitcircle); for(int i=1;i<=12;i=i+1) { // draw(p, 0.9*dir(90-30*i)--dir(90-30*i)); label(p, "$"+(string) i+"$",0.84*dir(90-30*i), fontsize(9pt)); } dot(p, origin); pair hpoint = 0.5 * dir(90 - 30 * (hh + mm/60)); pair mpoint = 0.75 * dir(90 - 6 * mm); draw(p, origin--hpoint, EndArrow(HookHead, 3)); draw(p, origin--mpoint, EndArrow(HookHead, 5)); string tlabel; if (mm > 10) { tlabel = (string) hh + ":" + (string) mm; } else { tlabel = (string) hh + ":0" + (string) mm; } label(p, tlabel, dir(90)*1.2, dir(90)); label(p, tlabel, dir(90)*1.2, dir(90)); label(p, nn, dir(-90)*1.1, dir(-90)); return p; } // The block real h = 1; filldraw( (-1.2,-1)--(8.4,-1)--(8.4,-1-h)--(-1.2,-1-h)--cycle, 0.7*lightgrey, black); add(shift((0.0,0)) * clock(10,22, "I")); add(shift((2.4,0)) * clock( 1,44, "II")); add(shift((4.8,0)) * clock( 5,06, "III")); add(shift((7.2,0)) * clock( 7,00, "IV")); label("\emph{Omnes vulnerant, postuma necat}", (3.6, -1.8), origin); [/asy] [i]Proposed by Evan Chen[/i]

If you were to randomly select an answer to this question, what is the probability it would be correct? $\textbf{(A) }0\%\qquad\textbf{(B) }20\%\qquad\textbf{(C) }40\%\qquad\textbf{(D) }80\%\qquad\textbf{(E) }100\%$

$N$ cells chosen on a rectangular grid. Let $a_i$ is number of chosen cells in $i$-th row, $b_j$ is number of chosen cells in $j$-th column. Prove that $$ \prod_{i} a_i! \cdot \prod_{j} b_j! \leq N! $$

Let $ABC$ be a triangle with $AB<AC$. There are two points $X$ and $Y$ on the angle bisector of $B\widehat AC$ such that $X$ is between $A$ and $Y$ and $BX$ is parallel to $CY$. Let $Z$ be the reflection of $X$ with respect to $BC$. Line $YZ$ cuts line $BC$ at point $P$. If line $BY$ cuts line $CX$ at point $K$, prove that $KA=KP$.

In plane there are $n$ noncollinear points $A_1$, $A_2$,...,$A_n$. Prove that there exist a line which passes through exactly two of these points

$12$ friends play a tennis tournament, where each plays only one game with any of the other eleven. Winner gets one points. Loser getos zero points, and there is no draw. Final points of the participants are $B_1, B_2, ..., B_{12}$. Find the largest possible value of the sum $\Sigma_3=B_1^3+B_2^3+ ... + B_{12}^3$ .

If $n$ is a positive integer such that $1+2+3+\cdots+n=190$, then what is $n$?

If $x, y$, and $z$ are real numbers with $\frac{x - y}{z}+\frac{y - z}{x}+\frac{z - x}{y}= 36$, find $2012 +\frac{x - y}{z}\cdot \frac{y - z}{x}\cdot\frac{z - x}{y}$ .

Prove the inequality: $$\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{1974}-\frac{1}{1975}<\frac{2}{5}$$

Let $Q=\{0,1\}^n$, and let $A$ be a subset of $Q$ with $2^{n-1}$ elements. Prove that there are at least $2^{n-1}$ pairs $(a,b)\in A\times (Q\setminus A)$ for which sequences $a$ and $b$ differ in only one term.

Solve the equation $a + b + 4 = 4\sqrt{a\sqrt{b}}$ in real numbers

Show that the equation $x^2+y^5=z^3$ has infinitely many solutions in integers $x, y,z$ for which $xyz \neq 0$.

The following analog clock has two hands that can move independently of each other. [asy] unitsize(2cm); draw(unitcircle,black+linewidth(2)); for (int i = 0; i < 12; ++i) { draw(0.9*dir(30*i)--dir(30*i)); } for (int i = 0; i < 4; ++i) { draw(0.85*dir(90*i)--dir(90*i),black+linewidth(2)); } for (int i = 0; i < 12; ++i) { label("\small" + (string) i, dir(90 - i * 30) * 0.75); } draw((0,0)--0.6*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); draw((0,0)--0.4*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); [/asy] Initially, both hands point to the number 12. The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock while the other hand does not move. Let $N$ be the number of sequences of 144 hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the end of the 144 movements, the hands have returned to their initial position. Find the remainder when $N$ is divided by 1000.

Let $ K$ be the product of all factors $ (b\minus{}a)$ (not necessarily distinct) where $ a$ and $ b$ are integers satisfying $ 1\le a < b \le 20$. Find the greatest positive integer $ n$ such that $ 2^n$ divides $ K$.

We say that a positive integer $n$ is [i]good[/i] if the result of multiplying the first $n$ positive odd integers consists only of the digits $1$, $3$, $5$ and $9$. For example, $n = 3$ is good because $1 \times 3 \times 5 = 15$, but $n = 4$ is not good because $1 \times 3 \times 5 \times 7 = 105$. Determine all the good numbers.

Circles $C_{1}$ and $C_{2}$ are tangent to each other at $K$ and are tangent to circle $C$ at $M$ and $N$. External tangent of $C_{1}$ and $C_{2}$ intersect $C$ at $A$ and $B$. $AK$ and $BK$ intersect with circle $C$ at $E$ and $F$ respectively. If AB is diameter of $C$, prove that $EF$ and $MN$ and $OK$ are concurrent. ($O$ is center of circle $C$.)

How many of the following statements are false? a. $2005$ distinct positive integers exist such that the sum of their squares is a cube and the sum of their cubes is a square. b. There are $2$ integral solutions to $x^2 + y^2 + z^2 = x^2y^2$. c. If the vertices of a triangle are lattice points in a plane, the diameter of the triangle’s circumcircle will never exceed the product of the triangle’s side lengths.

A scalene triangle $ABC$ is inscribed in a circle $\Gamma$. The bisector of angle $A$ meets $BC$ at $E$. Let $M$ be the midpoint of the arc $BAC$. The line $ME$ intersects $\Gamma$ again at $D$. Show that the circumcentre of triangle $AED$ coincides with the intersection point of the tangent to $\Gamma$ at $D$ and the line $BC$.

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

It is known about the numbers $a, b$ and $c$ that $\frac{a}{b+c-a}=\frac{b}{a ​​+ c-b}= \frac{c}{a ​​+ b-c}$. What values ​​can an expression take $\frac{(a + b) (b + c) (a + c)}{abc}$ ?

Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy \[(ab + 1)(bc + 1)(ca + 1) = 84.\]

Consider all the real sequences $x_0,x_1,\cdots,x_{100}$ satisfying the following two requirements: (1)$x_0=0$; (2)For any integer $i,1\leq i\leq 100$,we have $1\leq x_i-x_{i-1}\leq 2$. Find the greatest positive integer $k\leq 100$,so that for any sequence $x_0,x_1,\cdots,x_{100}$ like this,we have \[x_k+x_{k+1}+\cdots+x_{100}\geq x_0+x_1+\cdots+x_{k-1}.\]

Given triangle ABC with incenter I. Let P,Q be point on circumcircle such that $\angle API=\angle CPI$ and $\angle BQI=\angle CQI$.Prove that $BP,AQ$ and $OI$ are concurrent.

We draw two lines $(\ell_1) , (\ell_2)$ through the orthocenter $H$ of the triangle $ABC$ such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle.