Let $a, b \ge 2$ be positive integers with $gcd (a, b) = 1$. Let $r$ be the smallest positive value that $\frac{a}{b}- \frac{c}{d}$ can take, where $c$ and $d$ are positive integers satisfying $c \le a$ and $d \le b$. Prove that $\frac{1}{r}$ is an integer.

A quadratic polynomial $p(x)$ with real coefficients and leading coefficient $1$ is called disrespectful if the equation $p(p(x)) = 0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)?$ $\textbf{(A) }\dfrac5{16} \qquad \textbf{(B) }\dfrac12 \qquad \textbf{(C) }\dfrac58 \qquad \textbf{(D) }1 \qquad \textbf{(E) }\dfrac98$

Two students $ A$ and $ B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $ A:$ “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student $ B.$ If $ B$ answers “no,” the referee puts the question back to $ A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”

Dima wrote several natural numbers on the blackboard and underlined some of them. Misha wants to erase several numbers (but not all) such that a multiple of three underlined numbers remain and the total amount of the remaining numbers would be divisible by $2013$; but after trying for a while he realizes that it's impossible to do this. What is the largest number of the numbers on the board?

Every nonnegative periodic decimal fraction represents a rational number, also in the form $\frac{p}{q}$ can be represented ($p$ and $q$ are natural numbers and coprime, $p\ge 0$, $q > 0)$. Now let $a_1$, $a_2$, $a_3$ and $a_4$ be digits to represent numbers in the decadic system. Let $a_1 \ne a_3$ or $a_2 \ne a_4$.Prove that it for the numbers: $z_1 = 0, \overline{a_1a_2a_3a_4} = 0,a_1a_2a_3a_4a_1a_2a_3a_4...$ $z_2 = 0, \overline{a_4a_1a_2a_3}$ $z_3 = 0, \overline{a_3a_4a_1a_2}$ $z_4 = 0, \overline{a_2a_3a_4a_1}$ In the above representation $p/q$ always have the same denominator. [hide=original wording]Jeder nichtnegative periodische Dezimalbruch repr¨asentiert eine rationale Zahl, die auch in der Form p/q dargestellt werden kann (p und q nat¨urliche Zahlen und teilerfremd, p >= 0, q > 0). Nun seien a1, a2, a3 und a4 Ziffern zur Darstellung von Zahlen im dekadischen System. Dabei sei a1 $\ne$ a3 oder a2 $\ne$ a4. Beweisen Sie! Die Zahlen z1 = 0, a1a2a3a4 = 0,a1a2a3a4a1a2a3a4... z2 = 0, a4a1a2a3 z3 = 0, a3a4a1a2 z4 = 0, a2a3a4a1 haben in der obigen Darstellung p/q stets gleiche Nenner.[/hide]

A diabolical combination lock has $n$ dials (each with $c$ possible states), where $n,c>1$. The dials are initially set to states $d_1, d_2, \ldots, d_n$, where $0\le d_i\le c-1$ for each $1\le i\le n$. Unfortunately, the actual states of the dials (the $d_i$'s) are concealed, and the initial settings of the dials are also unknown. On a given turn, one may advance each dial by an integer amount $c_i$ ($0\le c_i\le c-1$), so that every dial is now in a state $d_i '\equiv d_i+c_i \pmod{c}$ with $0\le d_i ' \le c-1$. After each turn, the lock opens if and only if all of the dials are set to the zero state; otherwise, the lock selects a random integer $k$ and cyclically shifts the $d_i$'s by $k$ (so that for every $i$, $d_i$ is replaced by $d_{i-k}$, where indices are taken modulo $n$). Show that the lock can always be opened, regardless of the choices of the initial configuration and the choices of $k$ (which may vary from turn to turn), if and only if $n$ and $c$ are powers of the same prime. [i]Bobby Shen.[/i]

Is there a polyhedron whose area ratio of any two faces is at least $2$ ?

For an integer $n\ge 2$ find all $a_1,a_2,\cdots ,a_n, b_1,b_2,\cdots , b_n$ so that (a) $0\le a_1\le a_2\le \cdots \le a_n\le 1\le b_1\le b_2\le \cdots \le b_n;$ (b) $\sum_{k=1}^n (a_k+b_k)=2n;$ (c) $\sum_{k=1}^n (a_k^2+b_k^2)=n^2+3n.$

Prove that for all real numbers $\alpha_1,\alpha_2,\ldots,\alpha_n$ we have \[ \sum_{i=1}^n \sum_{j=1}^n ij \cos {(\alpha_i - \alpha_j )} \geq 0. \] [i]Octavian Stanasila[/i]

Let $M$ and $N$ be the midpoints of the respective sides $AB$ and $AC$ of an acute-angled triangle $ABC$. Let $P$ be the foot of the perpendicular from $N$ onto $BC$ and let $A_1$ be the midpoint of $MP$. Points $B_1$ and $C_1$ are obtained similarly. If $AA_1$, $BB_1$ and $CC_1$ are concurrent, show that the triangle $ABC$ is isosceles. [i]Mircea Becheanu[/i]

Prove that for any integer $n$ the number $n (n^2 + 5)$ is divisible by $6$.

Kostya marked the points $A(0, 1), B(1, 0), C(0, 0)$ in the coordinate plane. On the legs of the triangle ABC he marked the points with coordinates $(\frac{1}{2},0), (\frac{1}{3},0), \cdots, (\frac{1}{n+1},0)$ and $(0,\frac{1}{2}), (0,\frac{1}{3}), \cdots, (0,\frac{1}{n+1}).$ Then Kostya joined each pair of marked points with a segment. Sasha drew a $1 \times n$ rectangle and joined with a segment each pair of integer points on its border. As a result both the triangle and the rectangle are divided into polygons by the segments drawn. Who has the greater number of polygons: Sasha or Kostya? [i](M. Alekseyev )[/i]

Let $m_1, m_2, \cdots ,m_k$ be positive numbers with the sum $S$. Prove that \[\displaystyle\sum_{i=1}^k\left(m_i +\frac{1}{m_i}\right)^2 \ge k\left(\frac{k}{S}+\frac{S}{k}\right)^2\]

There are $n>3$ different natural numbers, less than $(n-1)!$ For every pair of numbers Ivan divides bigest on lowest and write integer quotient (for example, $100$ divides $7$ $= 14$) and write result on the paper. Prove, that not all numbers on paper are different.

a)Prove that $\frac{1}{2}+\frac{1}{3}+...+\frac{1}{{{2}^{m}}}<m$, for any $m\in {{\mathbb{N}}^{*}}$. b)Let ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ be the prime numbers less than ${{2}^{100}}$. Prove that $\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+...+\frac{1}{{{p}_{n}}}<10$

Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.