Numbers $a, b$ and $c$ are positive integers and $\frac{1}{a}+\frac{1}{b}+\frac{ 1}{c}< 1.$ Show that \[\frac{1}{a}+\frac{1}{b}+\frac{ 1}{c}\leq \frac{41}{42}.\]

Find all functions $f:R\to R$ such that for any real $x, y$ , $$f(x+2^y)=f(2^x)+f(y)$$

A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied: (i) All the squares are congruent. (ii) If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares. (iii) Each square touches exactly three other squares. How many positive integers $n$ are there with $2018\leq n \leq 3018$, such that there exists a collection of $n$ squares that is tri-connected?

Let $\mathbb{R}_{\neq 0}$ be the set of nonzero real numbers. Find all functions $f:\mathbb{R}_{\neq 0}\rightarrow\mathbb{R}_{\neq 0}$ such that, for all $x,y\in\mathbb{R}_{\neq 0}$, \[(x-y)f(y^2)+f\left(xy\,f\left(\frac{x^2}{y}\right)\right)=f(y^2f(y)).\]

In a circle of diameter $ 1$, some chords are drawn. The sum of their lengths is greater than $ 19$. Prove that there is a diameter intersecting at least $ 7$ chords.

Let $P(X)$ be a polynomial with positive coefficients. Show that for every integer $n \geq 2$ and every $n$ positive numbers $x_1, x_2,..., x_n$ the following inequality is true: $$P\left(\frac{x_1}{x_2} \right)^2+P\left(\frac{x_2}{x_3} \right)^2+ ... +P\left(\frac{x_n}{x_1} \right)^2 \geq n \cdot P(1)^2.$$ When does the equality take place?

Given that $(1 + \tan 1^{\circ})(1 + \tan 2^{\circ}) \ldots (1 + \tan 45^{\circ}) = 2^n$, find $n$.

Determine the value of \[S=\sqrt{1+\dfrac{1}{1^2}+\dfrac{1}{2^2}}+\sqrt{1+\dfrac{1}{2^2}+\dfrac{1}{3^2}}+\cdots+\sqrt{1+\dfrac{1}{1999^2}+\dfrac{1}{2000^2}}\]

The largest divisor of $2,014,000,000$ is itself. What is its fifth largest divisor? $\textbf{(A) }125,875,000\qquad\textbf{(B) }201,400,000\qquad\textbf{(C) }251,750,000\qquad\textbf{(D) }402,800,000\qquad\textbf{(E) }503,500,000\qquad$

Let $a_1$, $a_2$ ,..., $a_{99}$ be a sequence of digits from the set ${0,...,9}$ such that if for some $n$ ∈ $N$, $a_n = 1$, then $a_{n+1} \ne 2$, and if $a_n = 3$ then $a_{n+1} \ne 4$. Prove that there exist indices $k,l$ ∈ ${1,...,98}$ such that $a_k = a_l$ and $a_{k+1} = a_{l+1}$.

$a_1,a_2,\ldots,a_n$ is a sequence of positive integers that has at least $\frac {2n}{3}+1$ distinct numbers and each positive integer has occurred at most three times in it. Prove that there exists a permutation  $b_1,b_2,\ldots,b_n$ of $a_i $'s such that all the $n$ sums $b_i+b_{i+1}$ are distinct ($1\le i\le n $ , $b_{n+1}\equiv b_1 $) [i]Proposed by Mohsen Jamali[/i]

In quadrilateral $ABCD$, $AB=2$, $AD=3$, $BC=CD=\sqrt7$, and $\angle DAB=60^\circ$. Semicircles $\gamma_1$ and $\gamma_2$ are erected on the exterior of the quadrilateral with diameters $\overline{AB}$ and $\overline{AD}$; points $E\neq B$ and $F\neq D$ are selected on $\gamma_1$ and $\gamma_2$ respectively such that $\triangle CEF$ is equilateral. What is the area of $\triangle CEF$?

We are given a rectangular table with a positive integer written in each of its cells. For each cell of the table, the number in it is equal to the total number of different values in the cells that are in the same row or column (including itself). Find all tables with this property.

Find all pairs of positive integers $x, y$ such that $x^3 - y^3 = 2005(x^2 - y^2)$.

An ant crawls clockwise along the contour of each face of a convex polyhedron. It is known that their speeds at any given time are not less than 1 mm/h. Prove that sooner or later two ants will collide. [i]Proposed by A. Klyachko[/i]

Consider two parabolas $y = x^2$ and $y = x^2 - 1$. Let $U$ be the set of points between the parabolas (including the points on the parabolas themselves). Does $U$ contain a line segment of length greater than $10^6$ ? Alexey Tolpygo

The altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ intersect at point $H$. A straight line passing through $H$ parallel to line $A_1C_1$ intersects the circumscribed circles of triangles $AHC_1$ and $CHA_1$ at points $X$ and $Y$, respectively. Prove that points $X$ and $Y$ are equidistant from the midpoint of segment $BH$.

If $ x,y$ are positive real numbers with sum $ 2a$, prove that : $ x^3y^3(x^2\plus{}y^2)^2 \leq 4a^{10}$ When does equality hold ? Babis

Let $x_1,x_2,\ldots,x_n $ be real numbers, where $n\ge 2$ is a given integer, and let $\lfloor{x_1}\rfloor,\lfloor{x_2}\rfloor,\ldots,\lfloor{x_n}\rfloor $ be a permutation of $1,2,\ldots,n$. Find the maximum and minimum of $\sum\limits_{i=1}^{n-1}\lfloor{x_{i+1}-x_i}\rfloor$ (here $\lfloor x\rfloor $ is the largest integer not greater than $x$).

Ash and Gary independently come up with their own lineups of $15$ fire, grass, and water monsters. Then, the first monster of both lineups will fight, with fire beating grass, grass beating water, and water beating fire. The defeated monster is then substituted with the next one from their team’s lineup; if there is a draw, both monsters get defeated. Gary completes his lineup randomly, with each monster being equally likely to be any of the three types. Without seeing Gary’s lineup, Ash chooses a lineup that maximizes the probability p that his monsters are the last ones standing. Compute $p.$

Which of the following represents the force corresponding to the given potential? [asy] // Code by riben size(400); picture pic; // Rectangle draw(pic,(0,0)--(22,0)--(22,12)--(0,12)--cycle); label(pic,"-15",(2,0),S); label(pic,"-10",(5,0),S); label(pic,"-5",(8,0),S); label(pic,"0",(11,0),S); label(pic,"5",(14,0),S); label(pic,"10",(17,0),S); label(pic,"15",(20,0),S); label(pic,"-2",(0,2),W); label(pic,"-1",(0,4),W); label(pic,"0",(0,6),W); label(pic,"1",(0,8),W); label(pic,"2",(0,10),W); label(pic,rotate(90)*"F (N)",(-2,6),W); label(pic,"x (m)",(11,-2),S); // Tick Marks draw(pic,(2,0)--(2,0.3)); draw(pic,(5,0)--(5,0.3)); draw(pic,(8,0)--(8,0.3)); draw(pic,(11,0)--(11,0.3)); draw(pic,(14,0)--(14,0.3)); draw(pic,(17,0)--(17,0.3)); draw(pic,(20,0)--(20,0.3)); draw(pic,(0,2)--(0.3,2)); draw(pic,(0,4)--(0.3,4)); draw(pic,(0,6)--(0.3,6)); draw(pic,(0,8)--(0.3,8)); draw(pic,(0,10)--(0.3,10)); draw(pic,(2,12)--(2,11.7)); draw(pic,(5,12)--(5,11.7)); draw(pic,(8,12)--(8,11.7)); draw(pic,(11,12)--(11,11.7)); draw(pic,(14,12)--(14,11.7)); draw(pic,(17,12)--(17,11.7)); draw(pic,(20,12)--(20,11.7)); draw(pic,(22,2)--(21.7,2)); draw(pic,(22,4)--(21.7,4)); draw(pic,(22,6)--(21.7,6)); draw(pic,(22,8)--(21.7,8)); draw(pic,(22,10)--(21.7,10)); // Paths path A=(0,6)--(5,6)--(5,4)--(11,4)--(11,8)--(17,8)--(17,6)--(22,6); path B=(0,6)--(5,6)--(5,2)--(11,2)--(11,10)--(17,10)--(17,6)--(22,6); path C=(0,6)--(5,6)--(5,5)--(11,5)--(11,7)--(17,7)--(17,6)--(22,6); path D=(0,6)--(5,6)--(5,7)--(11,7)--(11,5)--(17,5)--(17,6)--(22,6); path E=(0,6)--(5,6)--(5,8)--(11,8)--(11,4)--(17,4)--(17,6)--(22,6); draw(A); label("(A)",(9.5,-3),4*S); draw(shift(35*right)*B); label("(B)",(45.5,-3),4*S); draw(shift(20*down)*C); label("(C)",(9.5,-23),4*S); draw(shift(35*right)*shift(20*down)*D); label("(D)",(45.5,-23),4*S); draw(shift(40*down)*E); label("(E)",(9.5,-43),4*S); add(pic); picture pic2=shift(35*right)*pic; picture pic3=shift(20*down)*pic; picture pic4=shift(35*right)*shift(20*down)*pic; picture pic5=shift(40*down)*pic; add(pic2); add(pic3); add(pic4); add(pic5); [/asy]

Let $f$ be a continuously differentiable function on $[0, 1]$ and $m \in \mathbb{N}$. Let $A = f(1)$ and let $B=\int \limits_{0}^1 x^{-\frac{1}{m}}f(x)dx$. Calculate $$\lim \limits_{n \to \infty} n\left(\int \limits_{0}^1 f(x)dx-\sum \limits_{k=1}^n \left(\frac{k^m}{n^m}-\frac{(k-1)^m}{n^m}\right)f\left(\frac{(k-1)^m}{n^m}\right)\right)$$in terms of $A$ and $B$.

Two circles $\Gamma_1$ and $\Gamma_2$ meet at $A$ and $B$. A line through $B$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$ repsectively. Another line through $B$ meets $\Gamma_1$ and $\Gamma_2$ again at $E$ and $F$ repsectively. Line $CF$ meets $\Gamma_1$ and $\Gamma_2$ again at $P$ and $Q$ respectively. $M$ and $N$ are midpoints of arc $PB$ and arc $QB$ repsectively. Show that if $CD = EF$, then $C,F,M,N$ are concyclic.

Find $$\sum^{100}_{i=1}i \gcd(i ,100).$$

Let $n$ be a positive integer and $A$ be a family of subsets of the set $\{1,2,...,n\},$ none of which contains another subset from A . Find the largest possible cardinality of $A$ .