A frog jumps on the coordinate lattice, starting from the point $(1,1)$, according to the following rules: (i) From point $(a,b)$ the frog can jump to either $(2a,b)$ or $(a,2b)$; (ii) If $a>b$, the frog can also jump from $(a,b)$ to $(a-b,b)$, while for $a<b$ it can jump from $(a,b)$ to $(a,b-a)$. Can the frog get to the point: (a) $(24,40)$; (b) $(40,60)$; (c) $(24,60)$; (d) $(200,4)$?

Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] $(i)$ $f(n) \neq 0$ for at least one $n$; [*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; [*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$. [/list]

Define the sequence $oa_n$ as follows: $oa_0=1, oa_n= oa_{n-1} \cdot cos\left( \dfrac{\pi}{2^{n+1}} \right)$. Find $\lim\limits_{n\rightarrow+\infty} oa_n$.

Let $f\in C^2[0,N]$ and $|f'(x)|<1$, $f''(x)>0$ for every $x\in [0, N]$. Let $0\leq m_0\ <m_1 < \cdots < m_k\leq N$ be integers such that $n_i=f(m_i)$ are also integers for $i=0,1,\ldots, k$. Denote $b_i=n_i-n_{i-1}$ and $a_i=m_i-m_{i-1}$ for $i=1,2,\ldots, k$. a) Prove that $$-1<\frac{b_1}{a_1}<\frac{b_2}{a_2}<\cdots < \frac{b_k}{a_k}<1$$ b) Prove that for every choice of $A>1$ there are no more than $N / A$ indices $j$ such that $a_j>A$. c) Prove that $k\leq 3N^{2/3}$ (i.e. there are no more than $3N^{2/3}$ integer points on the curve $y=f(x)$, $x\in [0,N]$).

2011 numbers are written on a board. For any three numbers, their sum is also among numbers written on the board. What is the smallest number of zeros among all 2011 numbers? (Author: I. Bogdanov)

We are given an infinite set of points in the plane such that any two of them have a distance of at most one. Prove that all the axes of symmetry of this set are concurrent, provided that there are at least two of them. [i]Proposed by David Anghel[/i]

The sides $a,b,c$ of triangle $ABC$ satisfy $a\le b\le c$. The circumradius and inradius of triangle $ABC$ are $R$ and $r$ respectively. Let $f=a+b-2R-2r$. Determine the sign of $f$ by the measure of angle $C$.

If an acute-angled triangle $ABC$ is given, construct an equilateral triangle $A'B'C'$ in space such that lines $AA',BB', CC'$ pass through a given point.

Each student chooses $1$ math problem and $1$ physics problem among $20$ math problems and $11$ physics problems. No same pair of problem is selected by two students. And at least one of the problems selected by any student is selected by at most one other student. At most how many students are there?

Find all integer values of $a$ such that the quadratic expression $(x+a)(x+1991) +1$ can be factored as a product $(x+b)(x+c)$ where $b,c$ are integers.

Let $n>1$ be a positive integer. Claire writes $n$ distinct positive real numbers $x_1, x_2, \dots, x_n$ in a row on a blackboard. In a $\textit{move},$ William can erase a number $x$ and replace it with either $\tfrac{1}{x}$ or $x+1$ at the same location. His goal is to perform a sequence of moves such that after he is done, the number are strictly increasing from left to right. [list] [*]Prove that there exists a positive constant $A,$ independent of $n,$ such that William can always reach his goal in at most $An \log n$ moves. [*]Prove that there exists a positive constant $B,$ independent of $n,$ such that Claire can choose the initial numbers such that William cannot attain his goal in less than $Bn \log n$ moves. [/list]

Compute the number of positive integers $n \leq 50$ such that there exist distinct positive integers $a,b$ satisfying \[ \frac{a}{b} +\frac{b}{a} = n \left(\frac{1}{a} + \frac{1}{b}\right). \]

Find the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive integers such that \[ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.\] [i]Proposed by Daniel Brown, Canada[/i]

For two given triangles $A_1A_2A_3$ and $B_1B_2B_3$ with areas $\Delta_A$ and $\Delta_B$, respectively, $A_iA_k \ge B_iB_k, i, k = 1, 2, 3$. Prove that $\Delta_A \ge \Delta_B$ if the triangle $A_1A_2A_3$ is not obtuse-angled.

Let $ (x_n)$ be the sequence of natural number such that: $ x_1\equal{}1$ and $ x_n<x_{n\plus{}1}\leq 2n$ for $ 1\leq n$. Prove that for every natural number $ k$, there exist the subscripts $ r$ and $ s$, such that $ x_r\minus{}x_s\equal{}k$.

Let $n$ be a positive integer with $d^2$ positive divisors. We fill a $d\times d$ board with these divisors. At a move, we can choose a row, and shift the divisor from the $i^{\text{th}}$ column to the $(i+1)^{\text{th}}$ column, for all $i=1,2,\ldots, d$ (indices reduced modulo $d$). A configuration of the $d\times d$ board is called [i]feasible[/i] if there exists a column with elements $a_1,a_2,\ldots,a_d,$ in this order, such that $a_1\mid a_2\mid\ldots\mid a_d$ or $a_d\mid a_{d-1}\mid\ldots\mid a_1.$ Determine all values of $n$ for which ragardless of how we initially fill the board, we can reach a feasible configuration after a finite number of moves.

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $a, b, c, d$ be positive integers and $p$ be prime number such that $a^2+b^2=p$ and $c^2+d^2$ is divisible by $p.$ Prove that there exist positive integers $e$ and $f$ such that $e^2+f^2=\frac{c^2+d^2}{p}.$

Suppose that $ M$ is an arbitrary point on side $ BC$ of triangle $ ABC$. $ B_1,C_1$ are points on $ AB,AC$ such that $ MB = MB_1$ and $ MC = MC_1$. Suppose that $ H,I$ are orthocenter of triangle $ ABC$ and incenter of triangle $ MB_1C_1$. Prove that $ A,B_1,H,I,C_1$ lie on a circle.

Let $R$ be a square region and $n\ge4$ an integer. A point $X$ in the interior of $R$ is called $n\text{-}ray$ partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional? $\textbf{(A)}\,1500 \qquad\textbf{(B)}\,1560 \qquad\textbf{(C)}\,2320 \qquad\textbf{(D)}\,2480 \qquad\textbf{(E)}\,2500$

(a) If $m, n$ are positive integers such that $2^n-1$ divides $m^2 + 9$, prove that $n$ is a power of $2$; (b) If $n$ is a power of $2$, prove that there exists a positive integer $m$ such that $2^n-1$ divides $m^2 + 9$.

Find the unique 3 digit number $N=\underline{A}$ $\underline{B}$ $\underline{C},$ whose digits $(A, B, C)$ are all nonzero, with the property that the product $P=\underline{A}$ $\underline{B}$ $\underline{C}$ $\times$ $\underline{A}$ $\underline{B}$ $\times$ $\underline{A}$ is divisible by $1000$. [i]Proposed by Kyle Lee[/i]

Define the sequence $(a_n)_{n\geq 1}$ by $a_1=1$, $a_2=p$ and \[a_{n+1}=pa_n-a_{n-1} \textrm { for all } n>1.\] Prove that for $n>1$ the polynomial $x^n-a_nx+a_{n-1}$ is divisible by $x^2-px+1$. Using this result, solve the equation \[x^4-56x+15=0.\]

Juan did not like the criticism of his classmates published in his school newspaper. He found nothing better than to start ripping up the diary. First he tore it into $4$ parts and then he continued to break it in a very methodical way: namely, each piece of newspaper he found he would tear it back into $4$ or $10$ pieces randomly. Breaking this way, was he able to get exactly $2003$ pieces of the diary?

Given is a grid 8x8. Every square is colored in black or white, so that in every 3x3, the number of white squares is even. What is the minimum number of black squares

Prove the inequality: $ctg \frac{a}{2}> 1 + ctg a$ for $0 <a <\frac{\pi}{2}$