Let $f:\mathbb{Z}_{+}\rightarrow\mathbb{Z}_{+}$ is one to one and bijective function. Prove that $f(mn)=f (m)f (n)$ if and only if $lcm (f (m),f (n))=f(lcm(m,n)) $

For odd numbers $a_1 , ..., a_k$ and even numbers $b_1 , ..., b_k$ , we know that $\sum_ {j = 1}^k a_j^n = \sum_{j = 1}^k b_j^n$ is satisfied for n = 1,2, ..., N. Prove that $k\geq 2^N$ and that for $k = 2^N$ there exists a solution $(a_1,...,b_1,...)$ with the above properties.

Let $ABC$ be a triangle such that $AB < BC$. Plot the height $BH$ with $H$ in $AC$. Let I be the incenter of triangle $ABC$ and $M$ the midpoint of $AC$. If line $MI$ intersects $BH$ at point $N$, prove that $BN < IM$.

Find the minimu value of $\frac{1}{\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \{x\cos t+(1-x)\sin t\}^2dt.$ [i]2010 Ibaraki University entrance exam/Science[/i]

Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.

Let be given an arbitrary tetrahedron $ABCD$ with volume $V$. Consider all lines which pass through the barycenter $S$ of the tetrahedron and intersect the edges $AD,BD,CD$ at points $A',B',C$ respectively. It is known that among the obtained tetrahedra there exists one with the minimal volume. Express this minimal volume in terms of $V$

Let $AB$ be a line segment with length 2, and $S$ be the set of points $P$ on the plane such that there exists point $X$ on segment $AB$ with $AX = 2PX$. Find the area of $S$.

$ABC$ is a triangle. $U$ is a point on the line $BC$. $I$ is the midpoint of $BC$. The line through $C$ parallel to $AI$ meets the line $AU$ at $E$. The line through $E$ parallel to $BC$ meets the line $AB$ at $F$. The line through $E$ parallel to $AB$ meets the line $BC$ at $H$. The line through $H$ parallel to $AU$ meets the line $AB$ at $K$. The lines $HK$ and $FG$ meet at $T. V$ is the point on the line $AU$ such that $A$ is the midpoint of $UV$. Show that $V, T$ and $I$ are collinear.

Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the probability that he wins the sixth game is $m/n$, where $m$ and $n$ are relatively prime positive integers. What are the last three digits of $m + n$?

Let $ABCD$ be a parallelogram. Extend $\overline{DA}$ through $A$ to a point $P,$ and let $\overline{PC}$ meet $\overline{AB}$ at $Q$ and $\overline{DB}$ at $R.$ Given that $PQ=735$ and $QR=112,$ find $RC.$

The following sequence lists all the positive rational numbers that do not exceed $\frac12$ by first listing the fraction with denominator 2, followed by the one with denominator 3, followed by the two fractions with denominator 4 in increasing order, and so forth so that the sequence is \[ \frac12,\frac13,\frac14,\frac24,\frac15,\frac25,\frac16,\frac26,\frac36,\frac17,\frac27,\frac37,\cdots. \] Let $m$ and $n$ be relatively prime positive integers so that the $2012^{\text{th}}$ fraction in the list is equal to $\frac{m}{n}$. Find $m+n$.

Given a triangle $ABC$ with side $AB$ greater than $BC$, let $M$ be the midpoint of $AC$ and $L$ be the point at which the bisector of angle $\angle B$ intersects side $AC$. The line parallel to $AB$, which intersects the bisector $BL$ at $D$, is drawn by $M$, and the line parallel to the side $BC$ that intersects the median $BM$ at $E$ is drawn by $L$. Show that $ED$ is perpendicular to $BL$.

$ \frac{3 \times 5}{9 \times 11} \times \frac{7 \times 9 \times 11}{3 \times 5 \times 7}\equal{}$ \[ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 49 \qquad \textbf{(D)}\ \frac{1}{49} \qquad \textbf{(E)}\ 50 \]

Find all positive integers $a, b, c$, such that $(8a-5b)^2 + (3b-2c)^2 + (3c-7a)^2 = 2$.

Show that for every integer $n \geq 2$ the number $$a=n^{5n-1}+n^{5n-2}+n^{5n-3}+n+1$$ is a composite number.

Let $n$ be a natural number and let $P (t) = 1 + t + t^2 + ... + t^{2n}$. If $x \in R$ such that $P (x)$ and $P (x^2)$ are rational numbers, prove that $x$ is rational number.

A large cube is formed by stacking $27$ unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is [asy] size(120); defaultpen(linewidth(0.7)); pair slant = (2,1); for(int i = 0; i < 4; ++i) draw((0,i)--(3,i)^^(i,0)--(i,3)^^(3,i)--(3,i)+slant^^(i,3)--(i,3)+slant); for(int i = 1; i < 4; ++i) draw((0,3)+slant*i/3--(3,3)+slant*i/3^^(3,0)+slant*i/3--(3,3)+slant*i/3);[/asy] $\textbf{(A)}\ 16\qquad \textbf{(B)}\ 17 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 19 \qquad \textbf{(E)}\ 20$

Some positive integers are initially written on a board, where each $2$ of them are different. Each time we can do the following moves: (1) If there are 2 numbers (written in the board) in the form $n, n+1$ we can erase them and write down $n-2$ (2) If there are 2 numbers (written in the board) in the form $n, n+4$ we can erase them and write down $n-1$ After some moves, there might appear negative numbers. Find the maximum value of the integer $c$ such that: Independetly of the starting numbers, each number which appears in any move is greater or equal to $c$

A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary. [i]Author: Kei Irie, Japan[/i]

Two circles $ \Omega_{1}$ and $ \Omega_{2}$ are externally tangent to each other at a point $ I$, and both of these circles are tangent to a third circle $ \Omega$ which encloses the two circles $ \Omega_{1}$ and $ \Omega_{2}$. The common tangent to the two circles $ \Omega_{1}$ and $ \Omega_{2}$ at the point $ I$ meets the circle $ \Omega$ at a point $ A$. One common tangent to the circles $ \Omega_{1}$ and $ \Omega_{2}$ which doesn't pass through $ I$ meets the circle $ \Omega$ at the points $ B$ and $ C$ such that the points $ A$ and $ I$ lie on the same side of the line $ BC$. Prove that the point $ I$ is the incenter of triangle $ ABC$. [i]Alternative formulation.[/i] Two circles touch externally at a point $ I$. The two circles lie inside a large circle and both touch it. The chord $ BC$ of the large circle touches both smaller circles (not at $ I$). The common tangent to the two smaller circles at the point $ I$ meets the large circle at a point $ A$, where the points $ A$ and $ I$ are on the same side of the chord $ BC$. Show that the point $ I$ is the incenter of triangle $ ABC$.

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

A car has a 4-digit integer price, which is written digitally. (so in digital numbers, like on your watch probably) While the salesmen isn't watching, the buyer turns the price upside down and gets the car for 1626 less. How much did the car initially cost?

$ABC$ is an equilateral triangle. $A_{1}, B_{1}, C_{1}$ are the midpoints of $BC, CA, AB$ respectively. $p$ is an arbitrary line through $A_{1}$. $q$ and $r$ are lines parallel to $p$ through $B_{1}$ and $C_{1}$ respectively. $p$ meets the line $B_{1}C_{1}$ at $A_{2}$. Similarly, $q$ meets $C_{1}A_{1}$ at $B_{2}$, and $r$ meets $A_{1}B_{1}$ at $C_{2}$. Show that the lines $AA_{2}, BB_{2}, CC_{2}$ meet at some point $X$, and that $X$ lies on the circumcircle of $ABC$.

Find all function $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that \[ f(x \plus{} y)(f(x) \minus{} y) \equal{} xf(x) \minus{} yf(y) \] for all $ x,y \in \mathbb{R}$.

Let $r$ be a nonnegative continuous function on the real line. Show that there exists a function $f\in C^1(\mathbb{R})$, not identically zero, such that $f'(x)=f(x-r(f(x)))$, $x\in\mathbb{R}$. (translated by L. Erdős)