Let $a,b,c$ be positive real numbers such that $2a^2 +b^2=9c^2$.Prove that $\displaystyle \frac{2c}{a}+\frac cb \ge\sqrt 3$.

Let $I\subset \mathbb R$ be a non-empty open interval, $\varepsilon\geq 0$ and $f\colon I\rightarrow\mathbb R$ a function satisfying the $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+\varepsilon t(1-t)|x-y|$$ inequality for all $x,y\in I$ and $t\in [0,1]$. Prove that there exists a convex $g\colon I\rightarrow\mathbb R$ function, such that the function $l :=f-g$ has the $\varepsilon$-Lipschitz property, that is $$|l(x)-l(y)|\leq \varepsilon|x-y|\text{ for all }x,y\in I$$

Let $ABC$ be a non-equilateral triangle and let $M_a$, $M_b$, $M_c$ be the midpoints of the sides $BC$, $CA$, $AB$, respectively. Let $S$ be a point lying on the Euler line. Denote by $X$, $Y$, $Z$ the second intersections of $M_aS$, $M_bS$, $M_cS$ with the nine-point circle. Prove that $AX$, $BY$, $CZ$ are concurrent.

Find all triples (m,n,N) of positive integers numbers m,n and N such that $m^N-n^N=2^{100}$ with N>1

Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ intersect at point $O$. Let a line through $O$ intersect segment $AB$ at $M$ and segment $CD$ at $N$. Prove that the segment $MN$ is not longer than at least one of the segments $AC$ and $BD$.

There are $2014$ airports in the faraway land of Artinia. Each pair of airports is connected by a nonstop flight in one or both directions. Show that there is some airport from which it is possible to reach every other airport in at most two flights.

If $n\ge2$ is an integer, prove the equality $$\lfloor\log_2n\rfloor+\lfloor\log_3n\rfloor+\ldots+\lfloor\log_nn\rfloor=\left\lfloor\sqrt n\right\rfloor+\left\lfloor\sqrt[3]n\right\rfloor+\ldots+\left\lfloor\sqrt[n]n\right\rfloor.$$

Suppose that $f$ is a function on the interval $[1,3]$ such that $-1\le f(x)\le 1$ for all $x$ and $\displaystyle \int_1^3f(x)\,dx=0.$ How large can $\displaystyle\int_1^3\frac{f(x)}x\,dx$ be?

A four-digit number has the property that the units digit equals the tens digit increased by $1$, the hundreds digit equals twice the tens digit, and the thousands digit is at least twice the units. Determine this four-digit number, knowing that it is twice a prime number.

A $100 \times 100$ table is given. At the beginning, every unit square has number $"0"$ written in them. Two players playing a game and the game stops after $200$ steps (each player plays $100$ steps). In every step, one can choose a row or a column and add $1$ to the written number in all of it's squares $\pmod 3.$ First player is the winner if more than half of the squares ($5000$ squares) have the number $"1"$ written in them, Second player is the winner if more than half of the squares ($5000$ squares) have the number $"0"$ written in them. Otherwise, the game is draw. Assume that both players play at their best. What will be the result of the game ? [i]Proposed by Mahyar Sefidgaran[/i]

In acute triangle $ABC$ $(AB < AC)$ let $D$, $E$ and $F$ be foots of perpedicular from $A$, $B$ and $C$ to $BC$, $CA$ and $AB$, respectively. Let $P$ and $Q$ be points on line $EF$ such that $DP \perp EF$ and $BQ=CQ$. Prove that $\angle ADP = \angle PBQ$

In July 1861, $ 366$ inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month? \[ \textbf{(A)}\ \frac{366}{31 \times 24} \qquad \textbf{(B)}\ \frac{366 \times 31}{24} \qquad \textbf{(C)}\ \frac{366 \times 24}{31} \\ \textbf{(D)}\ \frac{31 \times 24}{366} \qquad \textbf{(E)}\ 366 \times 31 \times 24 \]

Recall that an icosahedron is a $3$-dimensional solid characterized by its $20$ congruent faces, each of which is an equilateral triangle. Determine the number of rigid rotations that preserve the symmetry of the icosahedron. (Each vertex moves to the location of another vertex.)

For a sequence of integers $a_1 < a_2 < \cdot\cdot\cdot < a_n$, a pair $(a_i,a_j)$ where $1 \leq i < j \leq n$ is said to be [i]balanced[/i] if the number $\frac{a_i+a_j}{2}$ belongs to the sequence. For every natural number $n \geq 3$, find the maximum possible number of balanced pairs in a sequence with $n$ numbers.

Consider the sets $A_1=\{1\}$, $A_2=\{2,3,4\}$, $A_3=\{5,6,7,8,9\}$, etc. Let $b_n$ be the arithmetic mean of the smallest and the greatest element in $A_n$. Show that the number $\frac{2000}{b_1-1}+\frac{2000}{b_2-1}+\ldots+\frac{2000}{b_{2000}-1}$ is a prime integer.

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Parallelogram $ ABCD$ has area $ 1,\!000,\!000$. Vertex $ A$ is at $ (0,0)$ and all other vertices are in the first quadrant. Vertices $ B$ and $ D$ are lattice points on the lines $ y\equal{}x$ and $ y\equal{}kx$ for some integer $ k>1$, respectively. How many such parallelograms are there? $ \textbf{(A)}\ 49\qquad \textbf{(B)}\ 720\qquad \textbf{(C)}\ 784\qquad \textbf{(D)}\ 2009\qquad \textbf{(E)}\ 2048$

What is $ ( \minus{} 1)^1 \plus{} ( \minus{} 1)^2 \plus{} \cdots \plus{} ( \minus{} 1)^{2006}$? $ \textbf{(A) } \minus{} 2006 \qquad \textbf{(B) } \minus{} 1 \qquad \textbf{(C) } 0 \qquad \textbf{(D) } 1 \qquad \textbf{(E) } 2006$

Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

Inside the convex polygon $A_1A_2...A_n$ , there is a point $M$ such that $\sum_{k=1}^n \overrightarrow {A_kM} = \overrightarrow{0}$. Prove that $nP\ge 4d$, where $P$ is the perimeter of the polygon, and $d=\sum_{k=1}^n |\overrightarrow {A_kM}|$ . Investigate the question of the achievement of equality in this inequality.

Let $ p$ be a prime number and let $ a_1,a_2,\ldots,a_{p \minus{} 2}$ be positive integers such that $ p$ doesn't $ a_k$ or $ {a_k}^k \minus{} 1$ for any $ k$. Prove that the product of some of the $ a_i$'s is congruent to $ 2$ modulo $ p$.

Let $\mathcal{G}$ be the set of all points $(x,y)$ in the Cartesian plane such that $0\le y\le 8$ and $$(x-3)^2+31=(y-4)^2+8\sqrt{y(8-y)}.$$ There exists a unique line $\ell$ of [b]negative slope[/b] tangent to $\mathcal{G}$ and passing through the point $(0,4)$. Suppose $\ell$ is tangent to $\mathcal{G}$ at a [b]unique[/b] point $P$. Find the coordinates $(\alpha, \beta)$ of $P$.

Let $ M$ be the set of those positive integers which are not divisible by $ 3$. The sum of $ 2n$ consecutive elements of $ M$ is $ 300$. Determine $ n$.

Let the points $A,B,C$ lie on a line in this order. $AB$ is the diameter of semicircle $\omega_1$, $AC$ is the diameter of semicircle $\omega_2$. Assume both $\omega_1$ and $\omega_2$ lie on the same side of $AC$. $D$ is a point on $\omega_2$ such that $BD\perp AC$. A circle centered at $B$ with radius $BD$ intersects $\omega_1$ at $E$. $F$ is on $AC$ such that $EF\perp AC$. Prove that $BC=BF$.

We need not restrict our number system radix to be an integer. Consider the phinary numeral system in which the radix is the golden ratio $\phi = \frac{1+\sqrt{5}}{2}$ and the digits $0$ and $1$ are used. Compute $1010100_{\phi}-.010101_{\phi}$