Let $a,b,c$ be positive real numbers such that \[ a+b+c\geq abc. \] Prove that at least two of the inequalities \[ \frac{2}{a}+\frac{3}{b}+\frac{6}{c}\geq6,\;\;\;\;\;\frac{2}{b}+\frac{3}{c}+\frac{6}{a}\geq6,\;\;\;\;\;\frac{2}{c}+\frac{3}{a}+\frac{6}{b}\geq6 \] are true.

Let $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for all $x$. Show that $f(x)=0$ for $-1\le x \le 1$.

A meeting is held at a round table. It is known that 7 women have a woman on their right side, and 12 women have a man on their right side. It is also known that 75% of the men have a woman on their right side. How many people are sitting at the round table?

We have that $(a+b)^3=216$, where $a$ and $b$ are positive integers such that $a>b$. What are the possible values of $a^2-b^2$?

Let $S$ be the set of positive integers in which the greatest and smallest elements are relatively prime. For natural $n$, let $S_n$ denote the set of natural numbers which can be represented as sum of at most $n$ elements (not necessarily different) from $S$. Let $a$ be greatest element from $S$. Prove that there are positive integer $k$ and integers $b$ such that $|S_n| = a \cdot n + b$ for all $ n > k $.

suppose that $\mathcal F\subseteq p(X)$ and $|X|=n$. we know that for every $A_i,A_j\in \mathcal F$ that $A_i\supseteq A_j$ we have $3\le |A_i|-|A_j|$. prove that: $|\mathcal F|\le \lfloor\frac{2^n}{3}+\frac{1}{2}\dbinom{n}{\lfloor\frac{n}{2}\rfloor}\rfloor$ (20 points)

What is the largest integer not exceeding $8x^3 +6x - 1$, where $x =\frac12 \left(\sqrt[3]{2+\sqrt5} + \sqrt[3]{2-\sqrt5}\right)$ ? (A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.

Let $ c\in\mathbb C$ and $ A_c \equal{} \{p\in \mathbb C[z]|p(z^2 \plus{} c) \equal{} p(z)^2 \plus{} c\}$. a) Prove that for each $ c\in C$, $ A_c$ is infinite. b) Prove that if $ p\in A_1$, and $ p(z_0) \equal{} 0$, then $ |z_0| < 1.7$. c) Prove that each element of $ A_c$ is odd or even. Let $ f_c \equal{} z^2 \plus{} c\in \mathbb C[z]$. We see easily that $ B_c: \equal{} \{z,f_c(z),f_c(f_c(z)),\dots\}$ is a subset of $ A_c$. Prove that in the following cases $ A_c \equal{} B_c$. d) $ |c| > 2$. e) $ c\in \mathbb Q\backslash\mathbb Z$. f) $ c$ is a non-algebraic number g) $ c$ is a real number and $ c\not\in [ \minus{} 2,\frac14]$.

Show that $ 3^{2008} \plus{} 4^{2009}$ can be written as product of two positive integers each of which is larger than $ 2009^{182}$.

For what natural numbers $n$ for any numbers $a, b , c$, which are values of the angles of an acute triangle, the following inequality is true: $$\sin na + \sin nb + \sin nc < 0?$$

Consider $10$ distinct positive integers that are all prime to each other (that is, there is no a prime factor common to all), but such that any two of them are not prime to each other. What is the smallest number of distinct prime factors that may appear in the product of $10$ numbers?

Given a non-isosceles triangle $ABC$ with incircle $k$ with center $S$. $k$ touches the side $BC,CA,AB$ at $P,Q,R$ respectively. The line $QR$ and line $BC$ intersect at $M$. A circle which passes through $B$ and $C$ touches $k$ at $N$. The circumcircle of triangle $MNP$ intersects $AP$ at $L$. Prove that $S,L,M$ are collinear.

Let $a, b, c, d, e, f, g, h, i$ be distinct integers from $1$ to $9$. The minimum possible positive value of $$\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}$$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

An isosceles triangle $ABC$ is inscribed in a circle with $\angle ACB = 90^o$ and $EF$ is a chord of the circle such that neither E nor $F$ coincide with $C$. Lines $CE$ and $CF$ meet $AB$ at $D$ and $G$ respectively. Prove that $|CE|\cdot |DG| = |EF| \cdot |CG|$.

Let $X$ be a finite set of real numbers. For any $x,x' \in X$ with $x<x'$, define a function $f(x,x')$, then $f$ is called an ordered pair function on $X$. For any given ordered pair function $f$ on $X$, if there exist elements $x_1 <x_2 <\cdots<x_k$ in $X$ such that $f(x_1 ,x_2 ) \le f(x_2 ,x_3 ) \le \cdots \le f(x_{k-1} ,x_k )$, then $x_1 ,x_2 ,\cdots,x_k$ is called an $f$-ascending sequence of length $k$ in $X$. Similarly, define an $f$-descending sequence of length $l$ in $X$. For integers $k,l \ge 3$, let $h(k,l)$ denote the smallest positive integer such that for any set $X$ of $s$ real numbers and any ordered pair function $f$ on $X$, there either exists an $f$-ascending sequence of length $k$ in $X$ or an $f$-descending sequence of length $l$ in $X$ if $s \ge h(k,l)$. Prove: 1.For $k,l>3,h(k,l) \le h(k-1,l)+h(k,l-1)-1$; 2.$h(k,l) \le \binom{l-2}{k+l-4} +1$.

Two straight pipes (circular cylinders), with radii $1$ and $\frac{1}{4}$, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both? [asy] size(6cm); draw(circle((0,1),1), linewidth(1.2)); draw((-1,0)--(1.25,0), linewidth(1.2)); draw(circle((1,1/4),1/4), linewidth(1.2)); [/asy] $\textbf{(A)}~\displaystyle\frac{1}{9} \qquad\textbf{(B)}~1 \qquad\textbf{(C)}~\displaystyle\frac{10}{9} \qquad\textbf{(D)}~\displaystyle\frac{11}{9} \qquad\textbf{(E)}~\displaystyle\frac{19}{9}$

Let $K$ be a convex quadrangle and let $l$ be a line through the point of intersection of the diagonals of $K$. Show that the length of the segment of intersection $l\cap K$ does not exceed the length of (at least) one of the diagonals of $K$.

Take an arbitrary point $D$ on side $BC$ of triangle $ABC$ and draw a circle through point $D$ and the centers of the circles inscribed in triangles $ABD$ and $ACD$. Prove that all circles obtained for different points $D$ of side $BC$ have a common point.

There are $14$ players participating at a chess tournament, each playing one game with every other player. After the end of the tournament, the players were ranked in descending order based on their points. The sum of the points of the first three players is equal with the sum of the points of the last nine players. What is the highest possible number of draws in the tournament.(For a victory the player gets $1$ point, for a loss $0$ points, in a draw both players get $0,5$ points.)

We mean a traingle in $\mathbb Q^{n}$, 3 points that are not collinear in $\mathbb Q^{n}$ a) Suppose that $ABC$ is triangle in $\mathbb Q^{n}$. Prove that there is a triangle $A'B'C'$ in $\mathbb Q^{5}$ that $\angle B'A'C'=\angle BAC$. b) Find a natural $m$ that for each traingle that can be embedded in $\mathbb Q^{n}$ it can be embedded in $\mathbb Q^{m}$. c) Find a triangle that can be embedded in $\mathbb Q^{n}$ and no triangle similar to it can be embedded in $\mathbb Q^{3}$. d) Find a natural $m'$ that for each traingle that can be embedded in $\mathbb Q^{n}$ then there is a triangle similar to it, that can be embedded in $\mathbb Q^{m}$. You must prove the problem for $m=9$ and $m'=6$ to get complete mark. (Better results leads to additional mark.)

For a positive real number $ a > 1$, prove the following inequality. $ \frac {1}{a \minus{} 1}\left(1 \minus{} \frac {\ln a}{a\minus{}1}\right) < \int_0^1 \frac {x}{a^x}\ dx < \frac {1}{\ln a}\left\{1 \minus{} \frac {\ln (\ln a \plus{} 1)}{\ln a}\right\}$

Positive numbers $x, y, z$ satisfy $x^2+y^2+z^2+xy+yz+zy \le 1$. Prove that $\big( \frac{1}{x}-1\big) \big( \frac{1}{y}-1\big)\big( \frac{1}{z}-1\big) \ge 9 \sqrt6 -19$.

Let $x$ and $y$ be real numbers. Show that\[x^2+y^2+1>x\sqrt{y^2+1}+y\sqrt{x^2+1}.\]

For $S_n=\{1, 2, ..., n\}$, find the maximum value of $m$ that makes the following proposition true. [b]Proposition[/b] There exists $m$ different subsets of $S$, say $A_1, A_2, ..., A_m$, such that for every $i, j=1, 2, ..., m$, the set $A_i \cup A_j$ is not $S$.

On sides $AB$ and $BC$ of a non-isosceles triangle $ABC$ are selected points $C_1$ and $A_1$ such that the quadrilateral $AC_1A_1C$ is cyclic. Lines $CC_1$ and $AA_1$ intersect at point $P$. Line $BP$ intersects the circumscribed circle of triangle $ABC$ at the point $Q$. Prove that the lines $QC_1$ and $CM$, where $M$ is the midpoint of $A_1C_1$, intersect at the circumscribed circles of triangle $ABC$.