In the acute-angled triangle $ABC$, the points $P$, $N$, $ M$ are the feet of the altitudes drawn from the vertices $C$, $A$, $B$, respectively. The lengths of the projections of the sides $AB$, $BC$, $CA$ on straight lines $MN$, $PM$, $NP$ respectively, are equal to each other. Prove that triangle $ABC$ is regular.

A positive integer \( n \), which has at least one proper divisor, is divisible by the arithmetic mean of the smallest and largest of its proper divisors (which may coincide). What can be the number of divisors of \( n \)? [i]A proper divisor of a positive integer \( n \) is any of its divisors other than \( 1 \) and \( n \).[/i] [i]Proposed by Mykhailo Shtandenko[/i]

The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence? $ \textbf{(A)}\ 29\qquad \textbf{(B)}\ 55\qquad \textbf{(C)}\ 85\qquad \textbf{(D)}\ 133\qquad \textbf{(E)}\ 250$

Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence.

Let $ a$ be a positive real number. Find $ \lim_{n\to\infty} \frac{(n\plus{}1)^a\plus{}(n\plus{}2)^a\plus{}\cdots \plus{}(n\plus{}n)^a}{1^{a}\plus{}2^{a}\plus{}\cdots \plus{}n^{a}}$

A white equilateral triangle $T$ with side length $2022$ is divided into equilateral triangles with side $1$ (cells) by lines parallel to the sides of $T$. We'll call two cells $\textit{adjacent}$ if they have a common vertex. Ivan colours some of the cells in black. Without knowing which cells are black, Peter chooses a set $S$ of cells and Ivan tells him the parity of the number of black cells in $S$. After knowing this, Peter is able to determine the parity of the number of $\textit{adjacent}$ cells of different colours. Find all possible cardinalities of $S$ such that this is always possible independent of how Ivan chooses to colour the cells.

Let $O{}$ be the circumcenter of an acute triangle $ABC$. Let $M{}$ be the midpoint of $AC$. The straight line $BO$ intersects the altitudes $AA_1{}$ and $CC_1{}$ at the points $H_a$ and $H_c$ respectively. The circumcircles of the triangles $BH_aA$ and $BH_cC$ have a second point of intersection $K{}$. Prove that $K{}$ lies on the straight line $BM$. [i]Mikhail Evdokimov[/i]

In a cyclic quadrilateral $ABCD$, the extensions of sides $AB$ and $CD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet at point $Q$. Prove that the distance between the orthocenters of triangles $APD$ and $AQB$ is equal to the distance between the orthocenters of triangles $CQD$ and $BPC$.

A trapezoid $ABCD$ and a line $\ell$ perpendicular to its bases $AD$ and $BC$ are given. A point $X$ moves along $\ell$. The perpendiculars from $A$ to $BX$ and from $D$ to $CX$ meet at point $Y$ . Find the locus of $Y$ . by D.Prokopenko

Consider a right-angled triangle with integer-valued sides $a<b<c$ where $a,b,c$ are pairwise co-prime. Let $d=c-b$ . Suppose $d$ divides $a$ . Then [b](a)[/b] Prove that $d\leqslant 2$. [b](b)[/b] Find all such triangles (i.e. all possible triplets $a,b,c$) with perimeter less than $100$ .

Let $d(n)=\sum_{0<d|n}{1}$. Show that, for any natural $n>1$, \[ \sum_{2 \leq i \leq n}{\frac{1}{i}} \leq \sum{\frac{d(i)}{n}} \leq \sum_{1 \leq i \leq n}{\frac{1}{i}} \]

A line through point $P$ outside of circle $O$ meets the said circle at $B,C$ ($PB < PC$). Let $PO$ meet circle $O$ at $Q,D$ (with $PQ < PD$). Let the line passing $Q$ and perpendicular to $BC$ meet circle $O$ at $A$. If $BD^2 = AD\cdot CP$, prove that $PA$ is a tangent to $O$.

Find the smallest natural number $n$ for which the number $n^2-n+11$ has exactly four prime factors (not necessarily distinct).

A [i]pucelana[/i] sequence is an increasing sequence of $16$ consecutive odd numbers whose sum is a perfect cube. How many pucelana sequences are there with $3$-digit numbers only?

Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$.

SQUARING OFF: Master Chief and Samus Aran take turns firing rockets at one another from across the Cartesian plane. Master Chief’s movement is restricted to lattice points within the $10\times 10$ square with vertices $(0,0)$, $(0,10)$, $(10,0)$, and $(10,10)$, while Samus Aran’s movement is restricted to lattice points inside the $10\times 10$ square with vertices $(0,0)$, $(-10,0)$, $(0,-10)$, and $(-10,-10)$. Neither player can be located on or beyond the border of his or her square. Both players randomly choose a lattice point at which they begin the game, and do not move the rest of the game (until either they are killed or kill the other player). Each player’s turn consists of firing a rocket, targeted at a specific undestroyed lattice point inside the border of the opponent’s movement square, which hits immediately. When a rocket hits its intended lattice point, it explodes, destroying the surrounding $3\times 3$ square ($8$ additional adjacent lattice points). The game ends when one player is hit by a rocket (when the player is located within the $3\times 3$ grid hit by a rocket). If the highest possible probability that Samus Aran wins the game in three turns or less, assuming Master Chief goes first, is expressed as $a/b$, where $a$ and $b$ are relatively prime integers, find $a+b$.

$(BEL 6)$ Evaluate $\left(\cos\frac{\pi}{4} + i \sin\frac{\pi}{4}\right)^{10}$ in two different ways and prove that $\dbinom{10}{1}-\dbinom{10}{3}+\frac{1}{2}\dbinom{10}{5}=2^4$

Let $ f(x)$ be a monic polynomial of degree $ 1991$ with integer coefficients. Define $ g(x) \equal{} f^2(x) \minus{} 9.$ Show that the number of distinct integer solutions of $ g(x) \equal{} 0$ cannot exceed $ 1995.$

The value of $\frac{1998- 998}{1000}$ is $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 1000 \qquad \textbf{(C)}\ 0.1 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 0.001$

Three circles $\omega_1$, $\omega_2$, $\omega_3$ are given. Let $A_0$ and $A_1$ be the common points of $\omega_1$ and $\omega_2$, $B_0$ and $B_1$ be the common points of $\omega_2$ and $\omega_3$, $C_0$ and $C_1$ be the common points of $\omega_3$ and $\omega_1$. Let $O_{i,j,k}$ be the circumcenter of triangle $A_iB_jC_k$. Prove that the four lines of the form $O_{ijk}O_{1 - i,1 - j,1 - k}$ are concurrent or parallel.

A convex pentagon $ABCDE$ is given in the coordinate plane with all vertices in lattice points. Prove that there must be at least one lattice point in the pentagon determined by the diagonals $AC$, $BD$, $CE$, $DA$, $EB$ or on its boundary.

Prove the inequality [b]a.)[/b] $ \left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left( a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) , $ where $k\geq 1$ is a natural number and $a_{1},$ $a_{2},$ $...,$ $a_{k}$ are arbitrary real numbers. [b]b.)[/b] Using the inequality (1), show that if the real numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ satisfy the inequality \[ a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left( a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) }, \] then all of these numbers $a_{1},$ $a_{2},$ $\ldots,$ $a_{n}$ are non-negative.

Let $ P$ be a point in triangle $ ABC$, and define $ \alpha,\beta,\gamma$ as follows: \[ \alpha\equal{}\angle BPC\minus{}\angle BAC, \quad \beta\equal{}\angle CPA\minus{}\angle \angle CBA, \quad \gamma\equal{}\angle APB\minus{}\angle ACB.\] Prove that \[ PA\dfrac{\sin \angle BAC}{\sin \alpha}\equal{}PB\dfrac{\sin \angle CBA}{\sin \beta}\equal{}PC\dfrac{\sin \angle ACB}{\sin \gamma}.\]

a) Show that you can divide an angle $\theta$ to three equal parts using compass and ruler if and only if the polynomial $4t^3-3t-\cos (\theta)$ is reducible over $\mathbb{Q}(\cos (\theta))$. b) Is it always possible to divide an angle into five equal parts?

When finding the sum $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}$, the least common denominator used is $\text{(A)}\ 120 \qquad \text{(B)}\ 210 \qquad \text{(C)}\ 420 \qquad \text{(D)}\ 840 \qquad \text{(E)}\ 5040$