Let $ABC$ be an acute triangle. Let $D, E,$ and $F$ be the feet of altitudes from $A, B,$ and $C$ to sides $BC, CA,$ and $AB$, respectively, and let $Q$ be the foot of altitude from A to line $EF$ . Given that $AQ = 20, BC = 15,$ and $AD = 24$, compute the perimeter of triangle $DEF.$

All edges of a regular right pyramid are equal to $1$, and all vertices lie on the side surface of a (infinite) right circular cylinder of radius $R$. Find all possible values of $R$.

Determine the values of $n \in N$ such that a square of side $n$ can be split into a square of side $1$ and five rectangles whose side measures are $10$ distinct natural numbers and all greater than $1$.

$n$ is a natural number. We shall call a permutation $a_1,\dots,a_n$ of $1,\dots,n$ a quadratic(cubic) permutation if $\forall 1\leq i \leq n-1$ we have $a_ia_{i+1}+1$ is a perfect square(cube). $(a)$ Prove that for infinitely many natural numbers $n$ there exists a quadratic permutation. $(b)$ Prove that for no natural number $n$ exists a cubic permutation.

Is it possible to cover a $13\times 13$ chessboard with forty-two pieces of dimensions $4\times 1$ such that only the central square of the chessboard remains uncovered?

Thanos establishes $5$ settlements on a remote planet, randomly choosing one of them to stay in, and then he randomly builds a system of roads between these settlements such that each settlement has exactly one outgoing (unidirectional) road to another settlement. Afterwards, the Avengers randomly choose one of the $5$ settlements to teleport to. Then, they (the Avengers) must use the system of roads to travel from one settlement to another. The probability that the Avengers can find Thanos can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m+n$.

Altitudes $AH_A, BH_B, CH_C$ of triangle $ABC$ intersect at $H$, and let $M$ be the midpoint of the side $AC$. The bisector $BL$ of $\triangle ABC$ intersects $H_AH_C$ at point $K$. The line through $L$ parallel to $HM$ intersects $BH_B$ in point $T$. Prove that $TK = TL$. [i]Proposed by Anton Trygub[/i]

If $ y \equal{} \log_{a}{x}$, $ a > 1$, which of the following statements is incorrect? $\textbf{(A)}\ \text{If }x=1,y=0 \qquad\\ \textbf{(B)}\ \text{If }x=a,y=1 \qquad\\ \textbf{(C)}\ \text{If }x=-1,y\text{ is imaginary (complex)} \qquad\\ \textbf{(D)}\ \text{If }0<x<z,y\text{ is always less than 0 and decreases without limit as }x\text{ approaches zero} \qquad\\ \textbf{(E)}\ \text{Only some of the above statements are correct}$

Find the remainder when $8^{2014}$ + $6^{2014}$ is divided by 100.

The area of a convex polygon in the plane is equally shared by the four standard quadrants, and all non-zero lattice points lie outside the polygon. Show that the area of the polygon is less than $4$.

Find the least $n\in N$ such that among any $n$ rays in space sharing a common origin there exist two which form an acute angle.

$a_{0}=2,a_{1}=1$ and for $n\geq 1$ we know that : $a_{n+1}=a_{n}+a_{n-1}$ $m$ is an even number and $p$ is prime number such that $p$ divides $a_{m}-2$. Prove that $p$ divides $a_{m+1}-1$.

Find all natural numbers $n$ for which the equation $\frac1x+\frac1y=\frac1n$ has exactly five solutions $(x,y)$ in the set of natural numbers.

A number N is inserted into the list 2, 6, 7, 7, 28. The mean is now twice as great as the median. What is N? $\textbf{(A) } 7\qquad\textbf{(B) } 14\qquad\textbf{(C) } 20\qquad\textbf{(D) } 28\qquad\textbf{(E) } 34$

Equilateral triangles $\triangle{ABC}$ and $\triangle{DEF}$ are drawn such that points $B, E, F,$ and $C$ lie on a line in this order, and point $D$ lies inside triangle $\triangle{ABC}.$ If $BE=14, EF=15,$ and $FC=16,$ compute $AD.$

Consider an isosceles triangle $ABC$ with side lengths $AB = AC = 10\sqrt{2}$ and $BC =10\sqrt{3}$. Construct semicircles $P$, $Q$, and $R$ with diameters $AB$, $AC$, $BC$ respectively, such that the plane of each semicircle is perpendicular to the plane of $ABC$, and all semicircles are on the same side of plane $ABC$ as shown. There exists a plane above triangle $ABC$ that is tangent to all three semicircles $P$, $Q$, $R$ at the points $D$, $E$, and $F$ respectively, as shown in the diagram. Calculate, with proof, the area of triangle $DEF$. [asy] size(200); import three; defaultpen(linewidth(0.7)+fontsize(10)); currentprojection = orthographic(0,4,2.5); // 1.15 x-scale distortion factor triple A = (0,0,0), B = (75^.5/1.15,-125^.5,0), C = (-75^.5/1.15,-125^.5,0), D = (A+B)/2 + (0,0,abs((B-A)/2)), E = (A+C)/2 + (0,0,abs((C-A)/2)), F = (C+B)/2 + (0,0,abs((B-C)/2)); draw(D--E--F--cycle); draw(B--A--C); // approximate guess for r real r = 1.38; draw(B--(r*B+C)/(1+r)^^(B+r*C)/(1+r)--C,linetype("4 4")); draw((B+r*C)/(1+r)--(r*B+C)/(1+r)); // lazy so I'll draw six arcs draw(arc((A+B)/2,A,D)); draw(arc((A+B)/2,D,B)); draw(arc((A+C)/2,E,A)); draw(arc((A+C)/2,E,C)); draw(arc((C+B)/2,F,B)); draw(arc((C+B)/2,F,C)); label("$A$",A,S); label("$B$",B,W); label("$C$",C,plain.E); label("$D$",D,SW); label("$E$",E,SE); label("$F$",F,N);[/asy]

Find all positive solution of system of equation: $ \frac{xy}{2005y\plus{}2004x}\plus{}\frac{yz}{2004z\plus{}2003y}\plus{}\frac{zx}{2003x\plus{}2005z}\equal{}\frac{x^{2}\plus{}y^{2}\plus{}z^{2}}{2005^{2}\plus{}2004^{2}\plus{}2003^{2}}$

Julián chooses $2007$ points of the plane between which there are no $3$ aligned, and draw with red all the segments that join two of those points. Next, Roberto draws several lines. Its objective is for each red segment to be cut inside by (at least) one of the lines. Determine the minor $\ell$ lines such that, no matter how Julián chooses the $2007$ points, with the properly chosen $\ell$ lines, Roberto will achieve his objective with certainty.

Susana and Brenda play a game writing polynomials on the board. Susana starts and they play taking turns. 1) On the preparatory turn (turn 0), Susana choose a positive integer $n_0$ and writes the polynomial $P_0(x)=n_0$. 2) On turn 1, Brenda choose a positive integer $n_1$, different from $n_0$, and either writes the polynomial $$P_1(x)=n_1x+P_0(x) \textup{ or } P_1(x)=n_1x-P_0(x)$$ 3) In general, on turn $k$, the respective player chooses an integer $n_k$, different from $n_0, n_1, \ldots, n_{k-1}$, and either writes the polynomial $$P_k(x)=n_kx^k+P_{k-1}(x) \textup{ or } P_k(x)=n_kx^k-P_{k-1}(x)$$ The first player to write a polynomial with at least one whole whole number root wins. Find and describe a winning strategy.

A polynomial with integer coefficients when divided by $x^2-12x+11$ gives the remainder $990x-889$. Prove that the polynomial has no integer roots.

Are there four positive integers whose sum is $2^{1002}$ and product is $5^{1002}$? Justify your answer.

Let $A$ be a set of finite distinct positive real numbers. Two other sets $B$, $C$ are defined by: \[B = \left\{ {\frac{x}{y};x,y \in A} \right\},\; \; \; C = \left\{ {xy;x,y \in A} \right\}\] Prove that $\left| A \right|.\left| B \right| \le {\left| C \right|^2}$.

Consider the triangular array of numbers with $0,1,2,3,...$ along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows $1$ through $6$ are shown. \begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\ & & & & 1 & & 1 & & & & \\ & & & 2 & & 2 & & 2 & & & \\ & & 3 & & 4 & & 4 & & 3 & & \\ & 4 & & 7 & & 8 & & 7 & & 4 & \\ 5 & & 11 & & 15 & & 15 & & 11 & & 5 \end{tabular} Let $f(n)$ denote the sum of the numbers in row $n$. What is the remainder when $f(100)$ is divided by $100$? $\textbf{(A)}\ 12\qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 74$

Show that if an odd prime number $ p$ can be expressed in the form $ x^5\minus{}y^5$ for some integers $ x,y,$ then: $ \sqrt{\frac{4p\plus{}1}{5}}\equal{}\frac{v^2\plus{}1}{2}$ for some odd integer $ v$.

You have a four-liter jug and a six-liter pot (both of cylindrical shape), and a big barrel of water. Can you measure exactly one liter of water?