Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function. Prove that $$\int_0^{\pi/2}f(\sin(2x))\sin x\, dx = \int_0^{\pi/2} f(\cos^2 x)\cos x\, dx.$$

Let $a_1$, $a_2$,...,$a_n$ be constant real numbers and $x$ be variable real number $x$. Let $f(x)=cos(a_1+x)+\frac{cos(a_2+x)}{2}+\frac{cos(a_3+x)}{2^2}+...+\frac{cos(a_n+x)}{2^{n-1}}$. If $f(x_1)=f(x_2)=0$, prove that $x_1-x_2=m\pi$, where $m$ is integer.

What is the product of the real roots of the equation \[x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}\,\,?\]

Find the smallest number among the following numbers: $ \textbf{(A) }\sqrt{55}-\sqrt{52}\qquad\textbf{(B) }\sqrt{56}-\sqrt{53}\qquad\textbf{(C) }\sqrt{77}-\sqrt{74}\qquad\textbf{(D) }\sqrt{88}-\sqrt{85}\qquad\textbf{(E) }\sqrt{70}-\sqrt{67} $

Let $S$ be the domain in the coordinate plane determined by two inequalities: \[y\geq \frac 12x^2,\ \ \frac{x^2}{4}+4y^2\leq \frac 18.\] Denote by $V_1$ the volume of the solid by a rotation of $S$ about the $x$-axis and by $V_2$, by a rotation of $S$ about the $y$-axis. (1) Find the values of $V_1,\ V_2$. (2) Compare the size of the value of $\frac{V_2}{V_1}$ and 1.

Let $ a>0$ and $ f: [0,\infty) \to [0,a]$ be a continuous function on $ (0,\infty)$ and having Darboux property on $ [0,\infty)$. Prove that if $ f(0)\equal{}0$ and for all nonnegative $ x$ we have \[ xf(x) \geq \int^x_0 f(t) dt ,\] then $ f$ admits primitives on $ [0,\infty)$.

Find all pairs of functions $ f, g: \mathbb R \to \mathbb R$ such that [list] (i) if $ x < y$, then $ f(x) < f(y)$; (ii) $ f(xy) \equal{} g(y)f(x) \plus{} f(y)$ for all $ x, y \in \mathbb R$. [/list]

Find the constants $ a,\ b,\ c$ such that a function $ f(x)\equal{}a\sin x\plus{}b\cos x\plus{}c$ satisfies the following equation for any real numbers $ x$. \[ 5\sin x\plus{}3\cos x\plus{}1\plus{}\int_0^{\frac{\pi}{2}} (\sin x\plus{}\cos t)f(t)\ dt\equal{}f(x).\]

For every natural number $ n $, define $ s(n) $ as the smallest natural number so that for every natural number $ a $ relatively prime to $n$, this equation holds: \[ a^{s(n)} \equiv 1 (mod n) \] Find all natural numbers $ n $ such that $ s(n) = 2010 $

Let ${f}$ be a bounded real function defined for all real numbers and satisfying for all real numbers ${x}$ the condition ${ f \Big(x+\frac{1}{3}\Big) + f \Big(x+\frac{1}{2}\Big)=f(x)+ f \Big(x+\frac{5}{6}\Big)}$ . Show that ${f}$ is periodic.

Find all one-to-one mappings $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $n$ the following relation holds: \[ f(f(n)) \leq \frac {n+f(n)} 2 . \]

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)=f(x+y)+xy$ for all $x,y\in \mathbb{R}$.

Let $f:\mathbb{R}\to\mathbb{R}$ be a bijective function. Does there always exist an infinite number of functions $g:\mathbb{R}\to\mathbb{R}$ such that $f(g(x))=g(f(x))$ for all $x\in\mathbb{R}$? [i]Proposed by Daniel Liu[/i]

If $x^2+\frac{1}{x^2}=7,$ find all possible values of $x^5+\frac{1}{x^5}.$

Consider a function $f:\mathbb N\rightarrow \mathbb N$ such that for any two positive integers $x,y$, the equation $f(xf(y))=yf(x)$ holds. Find the smallest possible value of $f(2007)$.

Let $f$ be a function such that \[f(x)=\frac{(x^2-2x+1) \sin \frac{1}{x-1}}{\sin \pi x}.\] Find the limit of $f$ in the point $x_0=1.$

Suppose $S= \{1,2,\dots,n\}$ and $n \geq 3$. There is $f:S^k \longmapsto S$ that if $a,b \in S^k$ and $a$ and $b$ differ in all of elements then $f(a) \neq f(b)$. Prove that $f$ is a function of one of its elements.

Let $\mathcal M$ be the set of all polynomial functions $f$ of degree at most 3 such that \[\forall x\in[-1,1]:\ |f(x)|\le 1.\] Denote $a$ the (possibly zero) coefficient of $f$ at $x^3.$ Show that there is a positive number $k$ such that \[\forall f\in\mathcal M:\ |a|\le k\] and find the least $k$ with this property.

What is the smallest positive integer $n$ such that $2013^n$ ends in $001$ (i.e. the rightmost three digits of $2013^n$ are $001$?

Find all functions $f,g$:$R \to R$ such that $f(x+yg(x))=g(x)+xf(y)$ for $x,y \in R$.

Let $ k\geq 1 $ and let $ I_{1},\dots, I_{k} $ be non-degenerate subintervals of the interval $ [0, 1] $. Prove that \[ \sum \frac{1}{\left | I_{i}\cup I_{j} \right |} \geq k^{2} \] where the summation is over all pairs $ (i, j) $ of indices such that $I_i\cap I_j\neq \emptyset$.

10) Call a string of letters $S$ an [i]almost-palindrome[/i] if $S$ and the reverse of $S$ differ in exactly $2$ places. Find the number of ways to order the letters in $HMMTTHEMETEAM$ to get an almost-palindrome. 11) Find all integers $n$, not necessarily positive, for which there exist positive integers ${a,b,c}$ satisfying $a^n + b^n = c^n$. 12) Let $a$ and $b$ be positive real numbers. Determine the minimum possible value of $\sqrt{a^2 + b^2} + \sqrt{a^2 + (b-1)^2} + \sqrt{(a-1)^2 + b^2} + \sqrt{(a-1)^2 + (b-1)^2}$. 13) Consider a $4$ x $4$ grid of squares, each originally colored red. Every minute, Piet can jump on any of the squares, changing the color of it and any adjacent squares to blue (two squares are adjacent if they share a side). What is the minimum number of minutes it will take Piet to change the entire grid to blue? 14) Let $ABC$ be an acute triangle with orthocenter $H$. Let ${D,E}$ be the feet of the ${A,B}$-altitudes, respectively. Given that $\overline{AH} = 20$ and $\overline{HD} =16$ and $\overline{BE} = 56$, find the length of $\overline{BH}$. 15) Find the smallest positive integer $b$ such that $1111 _b$ ($1111$ in base $b$) is a perfect square. If no such $b$ exists, write "No Solution" 16) For how many triples $( {x,y,z} )$ of integers between $-10$ and $10$, inclusive, do there exist reals ${a,b,c}$ that satisfy $ab = x$ $ac = y$ $bc = z$? 17) Unit squares $ABCD$ and $EFGH$ have centers $O_1$ and $O_2$, respectively, and are originally oriented so that $B$ and $E$ are at the same position and $C$ and $H$ are at the same position. The squares then rotate clockwise around their centers at a rate of one revolution per hour. After $5$ minutes, what is the area of the intersection of the two squares? 18) A function $f$ satisfies, for all nonnegative integers $x$ and $y$, $f(x,0) = f(0,x) = x$ If $x \ge y \ge 0$, $f(x,y)=f(x-y,y)+1$ If $y \ge x \ge 0$, $f(x,y) = f(x,y-x)+1$ Find the maximum value of $f$ over $0 \le x,y \le 100$.

Find all pairs of function $f : Q \rightarrow R$ and $g : Q \rightarrow R,$ for which equations \begin{align*} f(x+y) &= f(x) f(y) + g(x) g(y) \\ g(x+y) &= f(x)g(y) + g(x)f(y) + g(x)g(y) \end{align*} holds for all rational numbers $x$ and $y.$ [i]Proposed by Gurgen Asatryan, Armenia [/i]

Prove that there is no function from positive real numbers to itself, $f : (0,+\infty)\to(0,+\infty)$ such that: $f(f(x) + y) = f(x) + 3x + yf(y)$ ,for every $x,y \in (0,+\infty)$ by Greece, Athanasios Kontogeorgis (aka socrates)

Let $m,n \geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_1,X_2,\ldots,X_m$ be pairwise distinct non-empty, not necessary disjoint subset of $X$. A function $f \colon X \to \{1,2,\ldots,n+1\}$ is called [i]nice[/i] if there exists an index $k$ such that \[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\] Prove that the number of nice functions is at least $n^n$.