Let $x=\frac pq$ for $p$, $q$ coprime. Find $p+q$. [asy] import olympiad; size(200); pen qq=font("phvb"); defaultpen(linewidth(0.6)+fontsize(10pt)); pair A=(-2.25,7),B=(-5,0),C=(5,0),D=waypoint(A--B,3/7), E=waypoint(A--C,1/2),F=intersectionpoint(C--D, B--E); draw(A--B--C--cycle^^B--E^^C--D); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NW); label("$E$",E,NE); label("$F$",F,N); label(scale(2.5)*"X",centroid(A,D,E),qq); label(scale(2.5)*"3",centroid(B,D,F),0.5*N,qq); label(scale(2.5)*"6",centroid(B,F,C),0.25*dir(180),qq); label(scale(2.5)*"2",centroid(C,E,F),dir(140),qq); [/asy]

Let $S$ be the set of the points $(x_1, x_2, . . . , x_{2012})$ in $2012$-dimensional space such that $|x_1|+|x_2|+...+|x_{2012}| \le 1$. Let $T$ be the set of points in $2012$-dimensional space such that $\max^{2012}_{i=1}|x_i| = 2$. Let $p$ be a randomly chosen point on $T$. What is the probability that the closest point in $S$ to $p$ is a vertex of $S$?

Let $a_n = \frac{\pi}{2n}$, where $n$ is a natural number. Prove that for any $k = 1$,$2$,$...$, $n$ holds the equality $$\frac{\sin ka_n}{1-\cos a_n}+\frac{\sin 5ka_n}{1-\cos 5a_n}+\frac{\sin 9ka_n}{1-\cos 9a_n}+...+\frac{\sin (4n-3)a_n}{1-\cos (4n-3)a_n}=kn$$

Let $c$ be a fixed real number. Show that a root of the equation \[x(x+1)(x+2)\cdots(x+2009)=c\] can have multiplicity at most $2$. Determine the number of values of $c$ for which the equation has a root of multiplicity $2$.

Let $\triangle ABC$ have median $CM$ ($M\in AB$) and circumcenter $O$. The circumcircle of $\triangle AMO$ bisects $CM$. Determine the least possible perimeter of $\triangle ABC$ if it has integer side lengths.

Let $ n\ge 3$ be a natural number. A set of real numbers $ \{x_1,x_2,\ldots,x_n\}$ is called [i]summable[/i] if $ \sum_{i\equal{}1}^n \frac{1}{x_i}\equal{}1$. Prove that for every $ n\ge 3$ there always exists a [i]summable[/i] set which consists of $ n$ elements such that the biggest element is: a) bigger than $ 2^{2n\minus{}2}$ b) smaller than $ n^2$

What is the sum of all the solutions of $ x \equal{} |2x \minus{} |60\minus{}2x\parallel{}$? $ \textbf{(A)}\ 32\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 92\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ 124$

Let $a, b$ and $c$ be real numbers such that $\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}= 1$. Prove that $\frac{a^2}{b + c}+\frac{b^2}{c + a}+\frac{c^2}{a + b}= 0$.

[b](a) [/b] Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$. [b](b)[/b] Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$.

Let $m,n$ be positive integers with $m \le n$, and let $F$ be a family of $m$-element subsets of $\{1,2,...,n\}$ satisfying $A \cap B \ne \varnothing$ for all $A,B \in F$. Determine the maximum possible number of elements in $F$.

A rectangular piece of paper is folded along its diagonal (as depicted below) to form a non-convex pentagon that has an area of $\tfrac{7}{10}$ of the area of the original rectangle. Find the ratio of the longer side of the rectangle to the shorter side of the rectangle. [asy] size(150); pair A = (-5,0); pair B = (5,0); pair C = (-3,4); pair D = (3,4); pair E = intersectionpoint(B--C,A--D); draw(A--B--D--cycle); draw(A--C); draw(C--E); draw(E--B,dashed); markscalefactor=0.06; draw(rightanglemark(A,C,B)); [/asy]

Find all natural numbers $n$ such that $5^n+3$ is a power of $2$

Let $K$ be a finite field with $q$ elements, $q \ge 3.$ We denote by $M$ the set of polynomials in $K[X]$ of degree $q-2$ whose coefficients are nonzero and pairwise distinct. Find the number of polynomials in $M$ that have $q-2$ distinct roots in $K.$ [i]Marian Andronache[/i]

In triangle $\vartriangle ABC$, $D$ is the midpoint of $BC$. Points $E$ and $F$ are chosen on side $AC$ so that $AF = F E = EC$. Let $AD$ intersect $BE$ and $BF$ and $G$ and $H$, respectively. Find the ratio of the areas of $\vartriangle BGH$ and $\vartriangle ABC$.

Determine all sequences of equal ratios of the form \[ \frac{a_1}{a_2} = \frac{a_3}{a_4} = \frac{a_5}{a_6} = \frac{a_7}{a_8} \] which simultaneously satisfy the following conditions: $\bullet$ The set $\{ a_1, a_2, \ldots , a_8 \}$ represents all positive divisors of $24$. $\bullet$ The common value of the ratios is a natural number.

Let $[ABC]$ be a acute-angled triangle and its circumscribed circle $\Gamma$. Let $D$ be the point on the line $AB$ such that $A$ is the midpoint of the segment $[DB]$ and $P$ is the point of intersection of $CD$ with $\Gamma$. Points $W$ and $L$ lie on the smaller arcs $\overarc{BC}$ and $\overarc{AB}$, respectively, and are such that $\overarc{BW} = \overarc{LA }= \overarc{AP}$. The $LC$ and $AW$ lines intersect at $Q$. Shows that $LQ = BQ$.

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]

There are $456$ people around a circle, denoted as $X_1, X_2, \dots, X_{456}$, and each one of them thought of a number. Every time Laura says an integer $k$ with $2 \leqslant k \leqslant 100$, the announcer announces all the numbers $p_1, p_2, \dots, p_{456}$, which are the averages of the numbers thought by the people in all the groups of $k$ consecutive people: $p_1$ is the average of the numbers thought by the people from $X_1$ to $X_k$, $p_2$ is the average of the numbers thought by the people from $X_2$ to $X_{k+1}$, and so on until $p_{456}$, the average of the numbers thought by the people from $X_{456}$ to $X_{k-1}$. Determine how many numbers $k$ Laura must say at a minimum so that, with certainty, the announcer can know the number thought by the person $X_{456}$.

The geometric mean (G.M.) of $k$ positive integers $a_1$, $a_2$, $\dots$, $a_k$ is defined to be the (positive) $k$-th root of their product. For example, the G.M. of 3, 4, 18 is 6. Show that the G.M. of a set $S$ of $n$ positive numbers is equal to the G.M. of the G.M.'s of all non-empty subsets of $S$.

Let $ABC$ be a triangle with $AB=13$, $BC=14$, $CA=15$. Let $O$ be the circumcenter of $ABC$. Find the distance between the circumcenters of triangles $AOB$ and $AOC$.

Find the maximal value of the following expression, if $a,b,c$ are nonnegative and $a+b+c=1$. \[ \frac{1}{a^2 -4a+9} + \frac {1}{b^2 -4b+9} + \frac{1}{c^2 -4c+9} \]

Let $ ABC$ be a triangle inscribed in a circle of radius $ R$, and let $ P$ be a point in the interior of triangle $ ABC$. Prove that \[ \frac {PA}{BC^{2}} \plus{} \frac {PB}{CA^{2}} \plus{} \frac {PC}{AB^{2}}\ge \frac {1}{R}. \] [i]Alternative formulation:[/i] If $ ABC$ is a triangle with sidelengths $ BC\equal{}a$, $ CA\equal{}b$, $ AB\equal{}c$ and circumradius $ R$, and $ P$ is a point inside the triangle $ ABC$, then prove that $ \frac {PA}{a^{2}} \plus{} \frac {PB}{b^{2}} \plus{} \frac {PC}{c^{2}}\ge \frac {1}{R}$.

Incircle of triangle $ABC$ tangent to $AB$ and $AC$ on $E$ and $F$ respectively. If $X$ is the midpoint of $EF$, prove $\angle BXC > 90^{\circ}$

Calculate the following indefinite integrals. [1] $\int \frac{dx}{1+\cos x}$ [2] $\int x\sqrt{x^2-1}dx$ [3] $\int a^{-\frac{x}{2}}dx\ \ (a>0,a\neq 1)$ [4] $\int \frac{\sin ^ 3 x}{1+\cos x}dx$ [5] $\int e^{4x}\sin 2x dx$

Is it possible to partition the set $S=\{1,2,\ldots,21\}$ into subsets that in each of these subsets the largest number is equal to the sum of the other numbers?