Prove without using advanced (differential) calculus: (a) For any real numbers a,b,c,d it holds that $a^6+b^6+c^6+d^6-6abcd \ge -2$. When does equality hold? (b) For which natural numbers $k$ does some inequality of the form $a^k +b^k +c^k +d^k -kabcd \ge M_k$ hold for all real $a,b,c,d$? For each such $k$,

Let $ABC$ be a triangle with incenter $I$. The incircle of the triangle is tangent to the sides $BC$ and $AC$ in points $D$ and $E$, respectively. Let $P$ denote the common point of lines $AI$ and $DE$, and let $M$ and $N$ denote the midpoints of sides $BC$ and $AB$, respectively. Prove that points $M, N$ and $P$ are collinear. [i](Proposed by Karl Czakler)[/i]

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

There exist integers $a$ and $b$ such that $(1 +\sqrt2)^{12}= a + b\sqrt2$. Compute the remainder when $ab$ is divided by $13$.

Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$.

Let k,m and n be three different positive integers. Prove that \[ \left( k-\frac{1}{k} \right)\left( m-\frac{1}{m} \right)\left( n-\frac{1}{n} \right) \le kmn-(k+m+n). \]

Prove that there exist infinitely many positive integer triples $(a,b,c)(a,b,c>2015)$ such that $$ a|bc-1, b|ac+1, c|ab+1.$$

Determine if there is a set $S$ of 2011 positive integers so that for every pair $m,n$ of distinct elements of $S$, $|m-n|=(m,n)$. Here $(m,n)$ denotes the greatest common divisor of $m$ and $n$.

When 0.000315 is multiplied by 7,928,564 the product is closest to which of the following? $\textbf{(A) }210\qquad\textbf{(B) }240\qquad\textbf{(C) }2100\qquad\textbf{(D) }2400\qquad\textbf{(E) }24000$

Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.

Let $P_1,P_2,...,P_{2556}$ be distinct points inside a regular hexagon $ABCDEF$ of side $1$. If any three points from the set $S=\{A,B,C,D,E,F,P_1,P_2...,P_{2556}\}$ aren't collinear, prove that there exists a triangle with area smaller than $\frac{1}{1700}$, with vertices from the set $S$.

For how many two-digit integers $n$ is $13 \mid 1 - 2^n - 3^n + 5^n$?

Let $\mathbb Q$ be the set of all rational numbers and $\mathbb R$ be the set of real numbers. Function $f: \mathbb Q \to \mathbb R$ satisfies the following conditions: (i) $f(0) = 0$, and for any nonzero $a \in Q, f(a) > 0.$ (ii) $f(x + y) = f(x)f(y) \qquad \forall x,y \in \mathbb Q.$ (iii) $f(x + y) \leq \max\{f(x), f(y)\} \qquad \forall x,y \in \mathbb Q , x,y \neq 0.$ Let $x$ be an integer and $f(x) \neq 1$. Prove that $f(1 + x + x^2+ \cdots + x^n) = 1$ for any positive integer $n.$

Let be the regular hexagonal prism $ABCDEFA'B C'D'E'F'$ with the base edge of $12$ and the height of $12 \sqrt{3}$. We denote by $N$ the middle of the edge $CC'$. a) Prove that the lines $BF'$ and $ND$ are perpendicular b) Calculate the distance between the lines $BF'$ and $ND$.

How many ways are there to place the integers from $1$ to $8$ on the vertices of a regular octagon such that the sum of the numbers on any $4$ vertices forming a rectangle is even? Rotations and reflections of the same arrangement are considered distinct

A circle with radius $6$ is externally tangent to a circle with radius $24$. Find the area of the triangular region bounded by the three common tangent lines of these two circles.

On each of the $2014^2$ squares of a $2014 \times 2014$-board a light bulb is put. Light bulbs can be either on or off. In the starting situation a number of the light bulbs is on. A move consists of choosing a row or column in which at least $1007$ light bulbs are on and changing the state of all $2014$ light bulbs in this row or column (from on to off or from off to on). Find the smallest non-negative integer $k$ such that from each starting situation there is a finite sequence of moves to a situation in which at most $k$ light bulbs are on.

Given three real numbers $a,b,c\in [0,1)$ such that $a^2+b^2+c^2=1$. Find the smallest possible value of $$\frac{a}{\sqrt{1-a^2}}+\frac{b}{\sqrt{1-b^2}}+\frac{c}{\sqrt{1-c^2}}.$$

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(x+y) - f(x)) + f(x)f(y) = f(x^2) - f(x+y),$ for all real numbers $x, y$.

In the figure, $\overline{AB}$ and $\overline{CD}$ are diameters of the circle with center $O$, $\overline{AB} \perp \overline{CD}$, and chord $\overline{DF}$ intersects $\overline{AB}$ at $E$. If $DE = 6$ and $EF = 2$, then the area of the circle is [asy] size(120); defaultpen(linewidth(0.7)); pair O=origin, A=(-5,0), B=(5,0), C=(0,5), D=(0,-5), F=5*dir(40), E=intersectionpoint(A--B, F--D); draw(Circle(O, 5)); draw(A--B^^C--D--F); dot(O^^A^^B^^C^^D^^E^^F); markscalefactor=0.05; draw(rightanglemark(B, O, D)); label("$A$", A, dir(O--A)); label("$B$", B, dir(O--B)); label("$C$", C, dir(O--C)); label("$D$", D, dir(O--D)); label("$F$", F, dir(O--F)); label("$O$", O, NW); label("$E$", E, SE);[/asy] $\textbf{(A)}\ 23\pi \qquad \textbf{(B)}\ \dfrac{47}{2}\pi \qquad \textbf{(C)}\ 24\pi \qquad \textbf{(D)}\ \dfrac{49}{2}\pi \qquad \textbf{(E)}\ 25\pi$

The altitudes $CC_1$ and $BB_1$ are drawn in the acute triangle $ABC$. The bisectors of angles $\angle BB_1C$ and $\angle CC_1B$ intersect the line $BC$ at points $D$ and $E$, respectively, and meet each other at point $X$. Prove that the intersection points of circumcircles of the triangles $BEX$ and $CDX$ lie on the line $AX$. [i](A. Voidelevich)[/i]

A polygon can be dissected into $100$ rectangles, but it cannot be dissected into $99$ rectangles. Prove that this polygon cannot be dissected into $100$ triangles.

Let $h_1$ and $h_2$ be the altitudes of a triangle drawn to the sides with length $5$ and $2\sqrt 6$, respectively. If $5+h_1 \leq 2\sqrt 6 + h_2$, what is the third side of the triangle? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 2\sqrt 6 \qquad\textbf{(D)}\ 3\sqrt 6 \qquad\textbf{(E)}\ 5\sqrt 3 $

A student concentrates on solving quadratic equations in $\mathbb{R}$. He starts with a first quadratic equation $x^2 + ax + b = 0$ where $a$ and $b$ are both different from 0. If this first equation has solutions $p$ and $q$ with $p \leq q$, he forms a second quadratic equation $x^2 + px + q = 0$. If this second equation has solutions, he forms a third quadratic equation in an identical way. He continues this process as long as possible. Prove that he will not obtain more than five equations.

1) Prove that there are infinitely many positive integers $n$ such that there exists a permutation of $1, 2, 3, . . . , n$ with the property that the difference between any two adjacent numbers is equal to either $2015$ or $2016$. 2) Let $k$ be a positive integer. Is the statement in 1) still true if we replace the numbers $2015$ and $2016$ by $k$ and $k + 2016$, respectively?