Let $ABC$ be a triangle for which there exists an interior point $F$ such that $\angle AFB=\angle BFC=\angle CFA$. Let the lines $BF$ and $CF$ meet the sides $AC$ and $AB$ at $D$ and $E$ respectively. Prove that \[ AB+AC\geq4DE. \]

Let $ k$ be a positive integer. Prove that the number $ (4 \cdot k^2 \minus{} 1)^2$ has a positive divisor of the form $ 8kn \minus{} 1$ if and only if $ k$ is even. [url=http://www.mathlinks.ro/viewtopic.php?p=894656#894656]Actual IMO 2007 Problem, posed as question 5 in the contest, which was used as a lemma in the official solutions for problem N6 as shown above.[/url] [i]Author: Kevin Buzzard and Edward Crane, United Kingdom [/i]

A classroom has $30$ students and $30$ desks arranged in $5$ rows of $6$. If the class has $15$ boys and $15$ girls, in how many ways can the students be placed in the chairs such that no boy is sitting in front of, behind, or next to another boy, and no girl is sitting in front of, behind, or next to another girl?

Show that for every prime number $p$ and every positive integer $n\geq2$ there exists a positive integer $k$ such that the decimal representation of $p^k$ contains $n$ consecutive equal digits.

Let $k$ be a positive integer. Determine the least positive integer $n$, with $n\geq k+1$, for which the game below can be played indefinitely: Consider $n$ boxes, labelled $b_1,b_2,...,b_n$. For each index $i$, box $b_i$ contains exactly $i$ coins. At each step, the following three substeps are performed in order: [b](1)[/b] Choose $k+1$ boxes; [b](2)[/b] Of these $k+1$ boxes, choose $k$ and remove at least half of the coins from each, and add to the remaining box, if labelled $b_i$, a number of $i$ coins. [b](3)[/b] If one of the boxes is left empty, the game ends; otherwise, go to the next step. [i]Proposed by Demetres Christofides, Cyprus[/i]

Let $ABCD$ be a quadrilateral with $AB>AD$ such that the inscribed circle $k_1$ of $\triangle ABC$ with center $O_1$ and the inscribed circle $k_2$ of $\triangle ADC$ with center $O_2$ have a common point on $AC$. If $k_1$ is tangent to $AB$ at $M$ and $k_2$ is tangent to $AD$ at $L$, prove that the lines $BD$, $LM$ and $O_1O_2$ pass through a common point.

Determine all real numbers $a$ and $b$ such that $a+b\in\mathbb{Z}$ and $a^2+b^2=2$.

Prove that if $ n $ is a non-negative integer, then number $$ 2^{n+2} + 3^{2n+1}$$ is divisible by $7$.

Let $n$ be a positive integer. a) Prove that the number $(2n + 1)^3 - (2n - 1)^3$ is the sum of three perfect squares. b) Prove that the number $(2n+1)^3-2$ is the sum of $3n-1$ perfect squares greater than $1$.

Let $ a,b,c$ be real numbers. Prove that $ (ab\plus{}bc\plus{}ca\minus{}1)^2 \le (a^2\plus{}1)(b^2\plus{}1)(c^2\plus{}1)$.

We have an $n \times n$ square divided into unit squares. Each side of unit square is called unit segment. Some isoceles right triangles of hypotenuse $2$ are put on the square so all their vertices are also vertices of unit squares. For which $n$ it is possible that every unit segment belongs to exactly one triangle(unit segment belongs to a triangle even if it's on the border of the triangle)?

Is it possible to find $100$ positive integers not exceeding $25000$ such that all pairwise sums of them are different?

A large wedge rests on a horizontal frictionless surface, as shown. A block starts from rest and slides down the inclined surface of the wedge, which is rough. During the motion of the block, the center of mass of the block and wedge [asy] draw((0,0)--(10,0),linewidth(1)); filldraw((2.5,0)--(6.5,2.5)--(6.5,0)--cycle, gray(.9),linewidth(1)); filldraw((5, 12.5/8)--(6,17.5/8)--(6-5/8, 17.5/8+1)--(5-5/8,12.5/8+1)--cycle, gray(.2)); [/asy] $\textbf{(A)}\ \text{does not move}$ $\textbf{(B)}\ \text{moves horizontally with constant speed}$ $\textbf{(C)}\ \text{moves horizontally with increasing speed}$ $\textbf{(D)}\ \text{moves vertically with increasing speed}$ $\textbf{(E)}\ \text{moves both horizontally and vertically}$

Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and \[ \angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \] Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.

Consider the infinite, strictly increasing sequence of positive integer $(a_n)$ such that i. All terms of sequences are pairwise coprime. ii. The sum $\frac{1}{\sqrt{a_1a_2}} +\frac{1}{\sqrt{a_2a_3}}+ \frac{1}{\sqrt{a_3a_4}} + ..$ is unbounded. Prove that this sequence contains infinitely many primes.

Suppose $a$, $b$, and $c$ are positive integers such that \[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\] Which of the following statements are necessarily true? I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$. II. If $\gcd(c,210)=1$, then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both. III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$. $\textbf{(A)}~\text{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II and III only}$

On a party, there are 6 boys and a number of girls. Two of the girls know exactly four boys each and the remaining girls know exactly two boys each. None of the boys know more than three girls. (We assume that if $ A$ knows $ B$, then $ B$ will also know $ A$). Then, the greatest possible number of girls on the party is $ \text{(A)}\ 6 \qquad \text{(B)}\ 7 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ \text{10 or more}$

Let $ S$ be a set of all points of a plane whose coordinates are integers. Find the smallest positive integer $ k$ for which there exists a 60-element subset of set $ S$ with the following condition satisfied for any two elements $ A,B$ of the subset there exists a point $ C$ contained in $ S$ such that the area of triangle $ ABC$ is equal to k .

Let $n$, $(n \geq3)$ be a positive integer and the polynomial $f(x)=(1+x) \cdot (1+2x) \cdot (1+3x) \cdot ... \cdot (1+nx)$ $= a_0+a_1 \cdot x+a_2 \cdot x^2+a_3 \cdot x^3+...+a_n \cdot x^n$. Show that the number $a_3$ divides the number $k=C^2_{n+1} \cdot (2 \cdot C^2_n \cdot C^2_{n+1}-3 \cdot a_2).$

Find all integer triples $(p,m,n)$ that satisfy: $p^m-n^3=27$ where $p$ is a prime number.

Let $n$ be a positive integer, and let $1=d_1<d_2<\dots < d_k=n$ denote all positive divisors of $n$, If the following conditions are satisfied: $$ 2d_2+d_4+d_5=d_7$$ $$ d_3 d_6 d_7=n$$ $$ (d_6+d_7)^2=n+1$$ find all possible values of $n$.

Suppose that a function $f : R^+ \to R^+$ satisfies $$f(f(x))+x = f(2x).$$ Prove that $f(x) \ge x$ for all $x >0$

Pasha and Vova play the game crossing out the cells of the $3\times 101$ board by turns. At the start, the central cell is crossed out. By one move the player chooses the diagonal (there can be $1, 2$ or $3$ cells in the diagonal) and crosses out cells of this diagonal which are still uncrossed. At least one new cell must be crossed out by any player's move. Pasha begins, the one who can not make any move loses. Who has a winning strategy?

Consider the transformation $$T(x,y,z) = (\sin y + \sin z - \sin x,\sin z + \sin x - \sin y,\sin x +\sin y -\sin z).$$ Determine all the points $(x,y,z) \in [0,1]^3$ such that $T^n(x,y,z) \in [0,1]^3,$ for every $n \geq 1$.

Let $ ABC$ be a triangle. Points $ A',B',C'$ are on the sides $ BC, CA, AB,$ respectively such that we have \[ \overline{A'B'} \equal{} \overline{B'C'} \equal{} \overline{C'A'}\] and \[ \overline{AB'} \equal{} \overline{BC'} \equal{} \overline{CA'}.\] Prove that triangle $ ABC$ is equilateral.