The two legs of a right triangle, which are altitudes, have lengths $2\sqrt3$ and $6$. How long is the third altitude of the triangle? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $

Suppose that $a, b,$ and $c$ are positive real numbers such that $$a + b + c \ge \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$ Find the largest possible value of the expression $$\frac{a + b - c}{a^3 + b^3 + abc} + \frac{b + c - a}{b^3 + c^3 + abc} + \frac{c + a - b}{c^3 + a^3 + abc}.$$

If $0 < v <\frac{\pi}{2}$ and $\tan v = 2v$, decide whether $sinv < \frac{20}{21}$.

The pressure $ (P)$ of wind on a sail varies jointly as the area $ (A)$ of the sail and the square of the velocity $ (V)$ of the wind. The pressure on a square foot is $ 1$ pound when the velocity is $ 16$ miles per hour. The velocity of the wind when the pressure on a square yard is $ 36$ pounds is: $ \textbf{(A)}\ 10\frac {2}{3} \text{ mph} \qquad\textbf{(B)}\ 96 \text{ mph} \qquad\textbf{(C)}\ 32\text{ mph} \qquad\textbf{(D)}\ 1\frac {2}{3} \text{ mph} \qquad\textbf{(E)}\ 16 \text{ mph}$

Let $D$ be the set of all pairs $(i,j)$, $1\le i,j\le n$. Prove there exists a subset $S \subset D$, with $|S|\ge\left \lfloor\frac{3n(n+1)}{5}\right \rfloor$, such that for any $(x_1,y_1), (x_2,y_2) \in S$ we have $(x_1+x_2,y_1+y_2) \not \in S$. (Peter Cameron)

The segments connecting the interior point of a convex non-sided $n$-gon with its vertices divide the $n$-gon into $n$ congruent triangles. For what is the smallest $n$ that is possible?

Let $A_1A_2... A_{2n + 1}$ be a convex polygon, $a_1 = A_1A_2$, $a_2 ​​= A_2A_3$, $...$, $a_{2n} = A_{2n}A_{2n + 1}$, $a_{2n + 1} = A_{2n + 1}A_1$. Denote by: $\alpha_i = \angle A_i$, $1 \le i \le 2n + 1$, $\alpha_{k + 2n + 1} = \alpha_k$, $k \ge 1$, $ \beta_i = \alpha_{i + 2} + \alpha_{i + 4} +... + \alpha_{i + 2n}$, $1 \le i \le 2n + 1$. Prove what if $$\frac{\alpha_1}{\sin \beta_1}=\frac{\alpha_2}{\sin \beta_2}=...=\frac{\alpha_{2n+1}}{\sin \beta_{2n+1}}$$ then a circle can be circumscribed around this polygon. Does the inverse statement hold a place?

In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?

In a multiple choice test there were 4 questions and 3 possible answers for each question. A group of students was tested and it turned out that for any three of them there was a question which the three students answered differently. What is the maximum number of students tested?

A teacher gives a multiple choice test to $15$ students and that each student answered each question. Each question had $5$ choices, but remarkably, no pair of students had more than $2$ answers in common. What is the maximum number of questions that could have been on the quiz?

Let $ABCD$ be a convex quadrilateral and let $M$, $N$, $P$ and $Q$ be the midpoints of the sides $AB$, $CD$, $BC$ and $DA$ respectively. The line $MN$ intersects the segments $AP$ and $CQ$ at points $X$ and $Y$, respectively. Suppose that $MX = NY$. Prove that $\text{area}(ABCD) = 4 \cdot \text{area}(BXDY).$

Given a triangle $ABC$ with its incircle touching sides $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Let the median from $A$ intersects $B_1C_1$ at $M$. Show that $A_1M\perp BC$.

In triangle $ABC$ on side $AC$ are points $K$, $L$ and $M$ such that $BK$ is an angle bisector of $\angle ABL$, $BL$ is an angle bisector of $\angle KBM$ and $BM$ is an angle bisector of $\angle LBC$, respectively. Prove that $4 \cdot LM <AC$ and $3\cdot \angle BAC - \angle ACB < 180^{\circ}$

For the Dürer final results announcement, four loudspeakers are used to provide sound in the hall. However, there are only two sockets in the wall from which the power comes. To solve the problem, Ádám got two extension cords and two power strips. One plug can be plugged into an extension cord, and two plugs can be plugged into a power strip. Gábor, in his haste before the announcement of the results, quickly plugs the $8$ plugs into the $8$ holes. Every possible way of plugging has the same probability, and it is also possible for Gábor to plug something into itself. What is the probability that all $4$ speakers will have sound at the results announcement? For the solution, give the sum of the numerator and the denominator in the simplified form of the probability. A speaker sounds when it is plugged directly or indirectly into the wall.

If an integer of two digits is $ k$ times the sum of its digits, the number formed by interchanging the digits is the sum of the digits multiplied by: $ \textbf{(A)}\ (9 \minus{} k) \qquad\textbf{(B)}\ (10 \minus{} k) \qquad\textbf{(C)}\ (11 \minus{} k) \qquad\textbf{(D)}\ (k \minus{} 1) \qquad\textbf{(E)}\ (k \plus{} 1)$

Determine all integer solutions $(x, y)$ of the equation \[(x - 1)x(x + 1) + (y - 1)y(y + 1) = 24 - 9xy\mbox{.}\]

A finite set $M$ of real numbers has the following properties: $M$ has at least $4$ elements, and there exists a bijective function $f:M\to M$, different from the identity, such that $ab\leq f(a)f(b)$ for all $a\neq b\in M.$ Prove that the sum of the elements of $M$ is $0.$

Let $a \not = 0, b, c$ be real numbers. If $x_{1}$ is a root of the equation $ax^{2}+bx+c = 0$ and $x_{2}$ a root of $-ax^{2}+bx+c = 0$, show that there is a root $x_{3}$ of $\frac{a}{2}\cdot x^{2}+bx+c = 0$ between $x_{1}$ and $x_{2}$.

Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that \[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\] [i]Proposed by Gerhard Wöginger, Austria.[/i]

Prove that the number $ (\underbrace{9999 \dots 99}_{2005}) ^{2009}$ can be obtained by erasing some digits of $ (\underbrace{9999 \dots 99}_{2008}) ^{2009}$ (both in decimal representation). [i]Yudi Satria, Jakarta[/i]

Prove that the inradius of a right angled triangle with integer sides is an integer.

The Fibonacci sequence is the list of numbers that begins $1, 2, 3, 5, 8, 13$ and continues with each subsequent number being the sum of the previous two. Prove that for every positive integer $n$ when the first $n$ elements of the Fibonacci sequence are alternately added and subtracted, the result is an element of the sequence or the negative of an element of the sequence. For example, when $n = 4$ we have $1-2+3-5 = -3$ and $3$ is an element of the Fibonacci sequence.

1. Determine, with proof, the least positive integer $n$ for which there exist $n$ distinct positive integers, $\left(1-\frac{1}{x_1}\right)\left(1-\frac{1}{x_2}\right)......\left(1-\frac{1}{x_n}\right)=\frac{15}{2013}$

The expression $ \sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}}$ is equal to: $ \textbf{(A)}\ \frac{\sqrt{3}}{6} \qquad \textbf{(B)}\ \frac{-\sqrt{3}}{6} \qquad \textbf{(C)}\ \frac{\sqrt{-3}}{6} \qquad \textbf{(D)}\ \frac{5 \sqrt{3}}{6} \qquad \textbf{(E)}\ 1$

All $n+3$ offices of University of Somewhere are numbered with numbers $0,1,2, \ldots ,$ $n+1,$ $n+2$ for some $n \in \mathbb{N}$. One day, Professor $D$ came up with a polynomial with real coefficients and power $n$. Then, on the door of every office he wrote the value of that polynomial evaluated in the number assigned to that office. On the $i$th office, for $i$ $\in$ $\{0,1, \ldots, n+1 \}$ he wrote $2^i$ and on the $(n+2)$th office he wrote $2^{n+2}$ $-n-3$. [list=a] [*] Prove that Professor D made a calculation error [*] Assuming that Professor D made a calculation error, what is the smallest number of errors he made? Prove that in this case the errors are uniquely determined, find them and correct them. [/list]