Prove that for $\forall$ $a,b,c\in [\frac{1}{3},3]$ the following inequality is true: $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\geq \frac{7}{5}$.

Find the least natural number $n$, which has at least 6 different divisors $1=d_1<d_2<d_3<d_4<d_5<d_6<...$, for which $d_3+d_4=d_5+6$ and $d_4+d_5=d_6+7$.

In an acute $\Delta ABC$, $AH_a$ and $BH_b$ are altitudes and $M$ is the middle point of $AB$. The circumscribed circles of $\Delta AMH_a$ and $\Delta BMH_b$ intersect for a second time in $P$. Prove that point $P$ lies on the circumscribed circle of $\Delta ABC$.

A square with a side 1 is colored in 3 colors. What’s the greatest real number $a$ such that there can always be found 2 points of the same color at a distance $a$?

Let $f(x)$ be a polynomial with integer coefficients, for which there exist $a,b\in \mathbb{Z}$ ($a\neq b$), such that $f(a)$ and $f(b)$ are coprime. Prove that there exist infinitely many values for $x$, such that each $f(x)$ is coprime with any other.

$x_1,...,x_n$ are non-negative reals and $n \geq 3$. Prove that at least one of the following inequalities is true: \[ \sum_{i=1} ^n \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}, \] \[ \sum_{i=1} ^n \frac{x_i}{x_{i-1}+x_{i-2}} \geq \frac{n}{2} . \]

If $AG_a,BG_b$, and $CG_c$ are symmedians in $\Delta ABC$ ($G_a\in BC,G_b\in AC,G_c\in AB$), is it possible for $\Delta G_a G_b G_c$ to be equilateral when $\Delta ABC$ is not equilateral?

In a class with $n$ students in the span of $k$ days, each day are chosen three to be tested. Each two students can be taken in such triple only once. Prove that for the greatest $k$ satisfying these conditions, the following inequalities are true: $\frac{n(n-3)}{6}\leq k\leq \frac{n(n-1)}{6}$.

Each of the cells of a table 2014 x 2014 is colored in white or black. It is known that each square 2 x 2 contains an even number of black cells and each cross (3 x 3 square without its corner cells) contains an odd number of black cells. Prove that the 4 corner cells of the table are in the same color.

The radius $r$ of a circle with center at the origin is an odd integer. There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$.

Nikolai and Peter are dividing a cake in the shape of a triangle. Firstly, Nikolai chooses one point $P$ inside the triangle and after that Peter cuts the cake by any line he chooses through $P$, then takes one of the pieces and leaves the other one for Nikolai. What’s the greatest portion of the cake Nikolai can be sure he could take, if he chooses $P$ in the best way possible?

Find all polynomials $P,Q\in \mathbb{R}[x]$, such that $P(2)=2$ , $Q(x)$ has no negative roots, and $(x-2)P(x^2-1)Q(x+1)=P(x)Q(x^2 )+Q(x+1)$.

Let $ABCD$ be a convex quadrilateral. The rays $AB$ and $DC$ intersect in point $E$. Rays $AD$ and $BC$ intersect in point $F$. The angle bisector of $\angle DCF$ intersects $EF$ in point $K$. Let $I_1$ and $I_2$ be the centers of the inscribed circles in $\Delta ECB$ and $\Delta FCD$. $M$ is the projection of $I_2$ on line $CF$ and $N$ is the projection of $I_1$ on line $BC$. Let $P$ be the reflection of $N$ in $I_1$. If $P,M,K$ are colinear, prove that $ABCD$ is tangential.

The positive real numbers $a,b,c$ are such that $21ab+2bc+8ca\leq 12$. Find the smallest value of $\frac{1}{a}+\frac{2}{b}+\frac{3}{c}$.

In a convex quadrilateral $ABCD$, $\angle DAB=\angle BCD$ and the angle bisector of $\angle ABC$ passes through the middle point of $CD$. If $CD=3AD$, determine the ratio $\frac{AB}{BC}$.

We will call a rectangular table filled with natural numbers “good”, if for each two rows, there exist a column for which its two cells that are also in these two rows, contain numbers of different parity. Prove that for $\forall$ $n>2$ we can erase a column from a good $n$ x $n$ table so that the remaining $n$ x $(n-1)$ table is also good.

A line $l$ passes through the center $O$ of an equilateral triangle $\Delta ABC$, which intersects $CA$ in $N$ and $BC$ in $M$. Prove that we can construct a triangle with $AM$,$BN$, and $MN$ such that the altitude to $MN$ (in this triangle) is constant when $l$ changes.

We define the following sequence: $a_0=a_1=1$, $a_{n+1}=14a_n-a_{n-1}$. Prove that $2a_n-1$ is a perfect square.

The graph $G$ with 2014 vertices doesn’t contain any 3-cliques. If the set of the degrees of the vertices of $G$ is $\{1,2,...,k\}$, find the greatest possible value of $k$.

Prove that for $\forall$ $x,y,z\in \mathbb{R}^+$ the following inequality is true: $\frac{x}{y+z}+\frac{25y}{z+x}+\frac{4z}{x+y}>2$.

Let $\Delta ABC$ be an acute triangle. Points $P,Q\in AB$ so that $P$ is between $A$ and $Q$. Let $H_1$ and $H_2$ be the feet of the perpendiculars from $A$ to $CP$ and $CQ$ respectively. Let $H_3$ and $H_4$ be the feet of the perpendiculars from $B$ to $CP$ and $CQ$ respectively. Let $H_3 H_4\cap BC=X$ and $H_1 H_2\cap AC=Y$, so that $X$ is after $B$ and $Y$ is after $A$. If $XY\parallel AB$, prove that $CP$ and $CQ$ are isogonal to $\Delta ABC$.

Let $A$ and $B$ be two non-infinite sets of natural numbers, each of which contains at least 3 elements. Two numbers $a\in A$ and $b\in B$ are called "harmonious", if they are not coprime. It is known that each element from $A$ is not harmonious with at least one element from $B$ and each element from $B$ is harmonious with at least one from $A$. Prove that there exist $a_1,a_2\in A$ and $b_1,b_2\in B$ such that $(a_1,b_1)$ and $(a_2,b_2)$ are harmonious but $(a_1,b_2)$ and $(a_2,b_1)$ are not.

Find all $f: \mathbb{N}\rightarrow \mathbb{N}$, for which $f(f(n)+m)=n+f(m+2014)$ for $\forall$ $m,n\in \mathbb{N}$.

Prove that, if $a,b,c$ are sides of a triangle, then we have the following inequality: $3(a^3 b+b^3 c+c^3 a)+2(ab^3+bc^3+ca^3 )\geq 5(a^2 b^2+a^2 c^2+b^2 c^2 )$.

Find all pairs of natural numbers $(m,n)$, for which $m\mid 2^{\varphi(n)} +1$ and $n\mid 2^{\varphi (m)} +1$.

Does there exist a natural number $n$, for which $n.2^{2^{2014}}-81-n$ is a perfect square?

Let each sequence of capital Bulgarian letters be a word (Note that the Bulgarian alphabet consists of 30 letters). We say that a word is calm, if there is no sequence of the letter $P$ in it (two letters $P$ next to each other). Find a clear expression for $f(n)$ (closed formula), which represents the number of calm words with length $n$. Example: РНТ and ТТРВ are calm, but ЖГРР and ЖРРРП aren’t.

Let $\Delta ABC$ be a right triangle with $\angle ACB=90^\circ$. The points $P$ and $Q$ on the side $BC$ and $R$ and $S$ on the side $CA$ are such that $\angle BAP=\angle PAQ=\angle QAC$ and $\angle ABS=\angle SBR=\angle RBC$. If $AP\cap BS=T$, prove that $120^\circ<\angle RTB<150^\circ$.

The real function $f$ is defined for $\forall$ $x\in \mathbb{R}$ and $f(0)=0$. Also $f(9+x)=f(9-x)$ and $f(x-10)=f(-x-10)$ for $\forall$ $x\in \mathbb{R}$. What’s the least number of zeros $f$ can have in the interval $[0;2014]$? Does this change, if $f$ is also continuous?

Is it true that for each natural number $n$ there exist a circle, which contains exactly $n$ points with integer coordinates?

On an international conference there are 4 official languages. Each two of the attendees can have a conversation on one of the languages. Prove that at least 60% of the attendees can speak the same language.

Let $c>1$ be a real constant. For the sequence $a_1,a_2,...$ we have: $a_1=1$, $a_2=2$, $a_{mn}=a_m a_n$, and $a_{m+n}\leq c(a_m+a_n)$. Prove that $a_n=n$.

A plane is cut into unit squares, each of which is colored in black or white. It is known that each rectangle 3 x 4 or 4 x 3 contains exactly 8 white squares. In how many ways can this plane be colored?

Polly can do the following operations on a quadratic trinomial: 1) Swapping the places of its leading coefficient and constant coefficient (swapping $a_2$ with $a_0$); 2) Substituting (changing) $x$ with $x-m$, where $m$ is an arbitrary real number; Is it possible for Polly to get $25x^2+5x+2014$ from $6x^2+2x+1996$ with finite applications of the upper operations?

Find the smallest number $n$ such that there exist polynomials $f_1, f_2, \ldots , f_n$ with rational coefficients satisfying \[x^2+7 = f_1\left(x\right)^2 + f_2\left(x\right)^2 + \ldots + f_n\left(x\right)^2.\] Proposed by Mariusz Skałba, Poland

Let $A$ be the set of permutations $a=(a_1,a_2,…,a_n)$ of $M=\{1,2,…n\}$ with the following property: There doesn’t exist a subset $S$ of $M$ such that $a(S)=S$. For $\forall$ such permutation $a$ let $d(a)=\sum_{k=1}^n (a_k-k)^2$ . Determine the smallest value of $d(a)$.

Let $\Delta ABC$ be an acute triangle with $a>b$, center $O$ of its circumscribed circle and middle point $M$ of $AC$. Let $K$ be the reflection of $O$ in $M$. Point $E\in BC$ is such that $EO\perp AB$. Point $F\in MK$ is such that $FK=OE$ and $K$ lies between $F$ and $M$. The altitude through $C$ and the angle bisector of $\angle CAB$ intersect in $D$. Let $BD$ intersect the circumscribed circle of $\Delta ABC$ for a second time in $P$. Prove that $AP\perp CF$.

We have 19 triminos (2 x 2 squares without one unit square) and infinite amount of 2 x 2 squares. Find the greatest odd number $n$ for which a square $n$ x $n$ can be covered with the given figures.

It is known that each two of the 12 competitors, that participated in the finals of the competition “Mathematical duels”, have a common friend among the other 10. Prove that there is one of them that has at least 5 friends among the group.

Some number of coins is firstly separated into 200 groups and then to 300 groups. One coin is special, if on the second grouping it is in a group that has less coins than the previous one, in the first grouping, that it was in. Find the least amount of special coins we can have.