Suppose that $ D,E,F$ are points on sides $ BC,CA,AB$ of a triangle $ ABC$ respectively such that $ BD\equal{}CE\equal{}AF$ and $ \angle BAD\equal{}\angle CBE\equal{}\angle ACF$.Prove that the triangle $ ABC$ is equilateral.

Prove that for an arbitrary $\Delta ABC$ the following inequality holds: $\frac{l_a}{m_a}+\frac{l_b}{m_b}+\frac{l_c}{m_c} >1$, Where $l_a,l_b,l_c$ and $m_a,m_b,m_c$ are the lengths of the bisectors and medians through $A$, $B$, and $C$.

For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n\plus{}1}(k) \equal{} f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$

For all triplets $a,b,c$ of (pairwise) distinct real numbers, prove the inequality $$ \left | \dfrac {a+b} {a-b} \right | + \left | \dfrac {b+c} {b-c} \right | + \left | \dfrac {c+a} {c-a} \right | \geq 2$$ and determine all cases of equality. Prove that if we also impose $a,b,c \geq 0$, then $$ \left | \dfrac {a+b} {a-b} \right | + \left | \dfrac {b+c} {b-c} \right | + \left | \dfrac {c+a} {c-a} \right | > 3,$$ with the value $3$ being the best constant possible. ([i]Dan Schwarz[/i])

Find the number of ordered pairs of integers $(a,b)\in\{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a "quadratic residue modulo $x^2+1$ and $35$", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following $\textit{equivalent}$ conditions holds: [list] [*] there exist polynomials $P$, $Q$ with integer coefficients such that $f(x)^2-(ax+b)=(x^2+1)P(x)+35Q(x)$; [*] or more conceptually, the remainder when (the polynomial) $f(x)^2-(ax+b)$ is divided by (the polynomial) $x^2+1$ is a polynomial with integer coefficients all divisible by $35$. [/list]

Three operations $f,g$ and $h$ are defined on subsets of the natural numbers $\mathbb{N}$ as follows: $f(n)=10n$, if $n$ is a positive integer; $g(n)=10n+4$, if $n$ is a positive integer; $h(n)=\frac{n}{2}$, if $n$ is an [i]even[/i] positive integer. Prove that, starting from $4$, every natural number can be constructed by performing a finite number of operations $f$, $g$ and $h$ in some order. $[$For example: $35=h(f(h(g(h(h(4)))))).]$

The numerator of a fraction is $ 6x \plus{} 1$, then denominator is $ 7 \minus{} 4x$, and $ x$ can have any value between $ \minus{}2$ and $ 2$, both included. The values of $ x$ for which the numerator is greater than the denominator are: $ \textbf{(A)}\ \frac{3}{5} < x \le 2\qquad \textbf{(B)}\ \frac{3}{5} \le x \le 2\qquad \textbf{(C)}\ 0 < x \le 2\qquad \\ \textbf{(D)}\ 0 \le x \le 2\qquad \textbf{(E)}\ \minus{}2 \le x \le 2$

$ n\geq 2$ cars are participating in a rally. The cars leave the start line at different times and arrive at the finish line at different times. During the entire rally each car takes over any other car at most once , the number of cars taken over by each car is different and each car is taken over by the same number of cars. Find all possible values of $ n$

The $A$ set consists of integers only. Its minimal element is $1$ and its maximal element is $100$. Every element of $A$ except $1$ equals to the sum of two (may be equal) numbers being contained in $A$. What is the least possible number of elements in $A$?

Initially, ten consecutive natural numbers are written on the board. In one turn, you may pick any two numbers from the board (call them $a$ and $b$) and replace them with the numbers $a^2-2011b^2$ and $ab$. After several turns, there were no initial numbers left on the board. Could there, at this point, be again, ten consecutive natural numbers?

Find the maximum number of Permutation of set {$1,2,3,...,2014$} such that for every 2 different number $a$ and $b$ in this set at last in one of the permutation $b$ comes exactly after $a$

Find all real numbers $m$ such that $$\frac{1-m}{2m} \in \{x\ |\ m^2x^4+3mx^3+2x^2+x=1\ \forall \ x\in \mathbb{R} \}$$

Consider all the numbers of the form $x+yt+zt^2$, with $x,y,z$ rational numbers and $t=\sqrt[3]{2}$. Prove that if $x+yt+zt^2\not= 0$, then there exist rational numbers $u,v,w$ such that \[(x+yt+z^2)(u+vt+wt^2)=1\]

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

An equation $P(x)=Q(y)$ is called [b]Interesting[/b] if $P$ and $Q$ are polynomials with degree at least one and integer coefficients and the equations has an infinite number of answers in $\mathbb{N}$. An interesting equation $P(x)=Q(y)$ [b]yields in[/b] interesting equation $F(x)=G(y)$ if there exists polynomial $R(x) \in \mathbb{Q} [x]$ such that $F(x) \equiv R(P(x))$ and $G(x) \equiv R(Q(x))$. (a) Suppose that $S$ is an infinite subset of $\mathbb{N} \times \mathbb{N}$.$S$ [i]is an answer[/i] of interesting equation $P(x)=Q(y)$ if each element of $S$ is an answer of this equation. Prove that for each $S$ there's an interesting equation $P_0(x)=Q_0(y)$ such that if there exists any interesting equation that $S$ is an answer of it, $P_0(x)=Q_0(y)$ yields in that equation. (b) Define the degree of an interesting equation $P(x)=Q(y)$ by $max\{deg(P),deg(Q)\}$. An interesting equation is called [b]primary[/b] if there's no other interesting equation with lower degree that yields in it. Prove that if $P(x)=Q(y)$ is a primary interesting equation and $P$ and $Q$ are monic then $(deg(P),deg(Q))=1$. Time allowed for this question was 2 hours.

In each cell of a $2019\times 2019$ board is written the number $1$ or the number $-1$. Prove that for some positive integer $k$ it is possible to select $k$ rows and $k$ columns so that the absolute value of the sum of the $k^2$ numbers in the cells at the intersection of the selected rows and columns is more than $1000$. [i]Folklore[/i]

David draws a $2 \times 2$ grid of squares in chalk on the sidewalk outside NIMO HQ. He then draws one arrow in each square, each pointing in one of the four cardinal directions (north, south, east, west) parallel to the sides of the grid. In how many ways can David draw his arrows such that no two of the arrows are pointing at each other? [i]Proposed by David Altizio[/i]

Pictured below is part of a large circle with radius $30$. There is a chain of three circles with radius $3$, each internally tangent to the large circle and each tangent to its neighbors in the chain. There are two circles with radius $2$ each tangent to two of the radius $3$ circles. The distance between the centers of the two circles with radius $2$ can be written as $\textstyle\frac{a\sqrt b-c}d$, where $a,b,c,$ and $d$ are positive integers, $c$ and $d$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a+b+c+d$. [asy] size(200); defaultpen(linewidth(0.5)); real r=aCos(79/81); pair x=dir(270+r)*27,y=dir(270-r)*27; draw(arc(origin,30,210,330)); draw(circle(x,3)^^circle(y,3)^^circle((0,-27),3)); path arcl=arc(y,5,0,180), arcc=arc((0,-27),5,0,180), arcr=arc(x,5,0,180); pair centl=intersectionpoint(arcl,arcc), centr=intersectionpoint(arcc,arcr); draw(circle(centl,2)^^circle(centr,2)); dot(x^^y^^(0,-27)^^centl^^centr,linewidth(2)); [/asy]

On a board, Ana writes $a$ different integers, while Ben writes $b$ different integers. Then, Ana adds each of her numbers with with each of Ben’s numbers and she obtains $c$ different integers. On the other hand, Ben substracts each of his numbers from each of Ana’s numbers and he gets $d$ different integers. For each integer $n$ , let $f(n)$ be the number of ways that $n$ may be written as sum of one number of Ana and one number of Ben. [i]a)[/i] Show that there exist an integer $n$ such that, $$f(n)\geq\frac{ab}{c}.$$ [i]b)[/i] Does there exist an integer $n$ such that, $$f(n)\geq\frac{ab}{d}?$$ [i]Proposed by Besfort Shala, Kosovo[/i]

Find all prime numbers $p,q$, for which $p^{q+1}+q^{p+1}$ is a perfect square. [i]Proposed by P. Boyvalenkov[/i]

Let $n \geq 2$ be a natural number. We consider a $(2n - 1) \times (2n - 1)$ table.Ana and Bob play the following game: starting with Ana, the two of them alternately color the vertices of the unit squares, Ana with red and Bob with blue, in $2n^2$ rounds. Then, starting with Ana, each one forms a vector with origin at a red point and ending at a blue point, resulting in $2n^2$ vectors with distinct origins and endpoints. If the sum of these vectors is zero, Ana wins. Otherwise, Bob wins. Show that Bob has a winning strategy.

Suppose 2012 reals are selected independently and at random from the unit interval $[0,1]$, and then written in nondecreasing order as $x_1\le x_2\le\cdots\le x_{2012}$. If the probability that $x_{i+1} - x_i \le \frac{1}{2011}$ for $i=1,2,\ldots,2011$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$, find the remainder when $m+n$ is divided by 1000. [i]Victor Wang.[/i]

In the plane are given $27$ points, no three of which are collinear. Four of this points are vertices of a unit square, while the others lie inside the square. Prove that there are three points in this set forming a triangle with area not exceeding $1/48$.

Find all functions $ f:\mathbb{R^{\ast }}\rightarrow \mathbb{ R^{\ast }}$ satisfying $f(\frac{f(x)}{f(y)})=\frac{1}{y}f(f(x))$ for all $x,y\in \mathbb{R^{\ast }}$ and are strictly monotone in $(0,+\infty )$

Find all positive integers $p$ for which there exists a positive integer $n$ such that $p^n+3^n~|~p^{n+1}+3^{n+1}.$