Let $a, b$, and $c$ be positive real numbers. Prove that \[\frac{a^4 + 3}{b} + \frac{b^4 + 3}{c} + \frac{c^4 + 3}{a} \ge 12.\] When does equality hold? [i]Proposed by Petar Filipovski[/i]

Determine all real numbers $x$ which satisfy the inequality: \[ \sqrt{3-x}-\sqrt{x+1}>\dfrac{1}{2} \]

A shop advertises everything is "half price in today's sale." In addition, a coupon gives a $20\%$ discount on sale prices. Using the coupon, the price today represents what percentage off the original price? $\textbf{(A)}\hspace{.05in}10 \qquad \textbf{(B)}\hspace{.05in}33 \qquad \textbf{(C)}\hspace{.05in}40 \qquad \textbf{(D)}\hspace{.05in}60 \qquad \textbf{(E)}\hspace{.05in}70 $

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

24) A ball of mass m moving at speed $v$ collides with a massless spring of spring constant $k$ mounted on a stationary box of mass $M$ in free space. No mechanical energy is lost in the collision. If the system does not rotate, what is the maximum compression $x$ of the spring? A) $x = v\sqrt{\frac{mM}{(m + M)k}}$ B) $x = v\sqrt{\frac{m}{k}}$ C) $x = v\sqrt{\frac{M}{k}}$ D) $x = v\sqrt{\frac{m + M}{k}}$ E) $x = v\sqrt{\frac{(m + M)^3}{mMk}}$

Let $M$ be a finite set and $P=\{ M_1,M_2,\ldots ,M_l\}$ a partition of $M$ (i.e., $\bigcup_{i=1}^k M_i, M_i\not=\emptyset, M_i\cap M_j =\emptyset$ for all $i,j\in\{1,2, \ldots ,k\} ,i\not= j)$. We define the following elementary operation on $P$: Choose $i,j\in\{1,2,\ldots ,k\}$, such that $i=j$ and $M_i$ has a elements and $M_j$ has $b$ elements such that $a\ge b$. Then take $b$ elements from $M_i$ and place them into $M_j$, i.e., $M_j$ becomes the union of itself and a $b$-element subset of $M_i$, while the same subset is subtracted from $M_i$ (if $a=b$, $M_i$ is thus removed from the partition). Let a finite set $M$ be given. Prove that the property “for every partition $P$ of $M$ there exists a sequence $P=P_1,P_2,\ldots ,P_r$ such that $P_{i+1}$ is obtained from $P_i$ by an elementary operation and $P_r=\{M\}$” is equivalent to “the number of elements of $M$ is a power of $2$.”

$M$ and $N$ are the midpoints of the sides $AB$ and $AC$ of a triangle ABC respectively. $P$ and $Q$ are points on the sides $AB$ and $AC$ respectively such that the bisector of the angle $ACB$ also bisects the angle $MCP$, and the bisector of the angle $ABC$ also bisects the angle $NBQ$. If $AP = AQ$, does it follow that $ABC$ is isosceles? (V Senderov)

In a quadrilateral $ABCD,$ $AB=BC=DA/\sqrt{2},$ and $\angle ABC$ is a right angle. The midpoint of $BC$ is $E,$ the orthogonal projection of $C$ on $AD$ is $F,$ and the orthogonal projection of $B$ on $CD$ is $G.$ The second intersection point of circle $(BCF)$ (with center $H$) and line $BG$ is $K,$ and the second intersection point of circle $(BCF)$ and line $HK$ is $L.$ The intersection of lines $BL$ and $CF$ is $M.$ The center of the Feurbach circle of triangle $BFM$ is $N.$ Prove that $\angle BNE$ is a right angle. [i]Proposed by Zsombor Fehér, Cambridge[/i]

In acute-angled triangle $ABC, I$ is the center of the inscribed circle and holds $| AC | + | AI | = | BC |$. Prove that $\angle BAC = 2 \angle ABC$.

On the extension of the heights $AH_1$ and $BH_2$ of an acute $\triangle ABC$, after points $H_1$ and $H_2$, are chosen points $M$ and $N$ in such way that $\angle MCB = \angle NCA = 30^\circ$. We denote with $C_1$ the intersection point of the lines $MB$ and $NA$. Analogously we define $A_1$ and $B_1$. Prove that the straight lines $AA_1$, $BB_1$, and $CC_1$ intersect in one point.

Fix a set of integers $S$. An integer is [i]clean[/i] if it is the sum of distinct elements of $S$ in exactly one way, and [i]dirty[/i] otherwise. Prove that the set of dirty numbers is either empty or infinite. [i]Note:[/i] We consider the empty sum to equal \(0\). [i]Proposed by Tony Wang and Ethan Tan[/i]

It is known that a polynomial $P$ with integer coefficients has degree $2022$. What is the maximum $n$ such that there exist integers $a_1, a_2, \cdots a_n$ with $P(a_i)=i$ for all $1\le i\le n$? [Extra: What happens if $P \in \mathbb{Q}[X]$ and $a_i\in \mathbb{Q}$ instead?]

Consider all parabolas of the form $ y\equal{}x^2\plus{}2px\plus{}q$ for $ p,q \in \mathbb{R}$ which intersect the coordinate axes in three distinct points. For such $ p,q$, denote by $ C_{p,q}$ the circle through these three intersection points. Prove that all circles $ C_{p,q}$ have a point in common.

In a triangle $ ABC$, the bisector of $ \angle ACB$ cuts the side $ AB$ at $ D$. An arbitrary circle $ (O)$ passing through $ C$ and $ D$ meets the lines $ BC$ and $ AC$ at $ M$ and $ N$ (different from $ C$), respectively. (a) Prove that there is a circle $ (S)$ touching $ DM$ at $ M$ and $ DN$ at $ N$. (b) If circle $ (S)$ intersects the lines $ BC$ and $ CA$ again at $ P$ and $ Q$ respectively, prove that the lengths of the segments $ MP$ and $ NQ$ are constant as $ (O)$ varies.

On a real number line, the points $1, 2, 3, \dots, 11$ are marked. A grasshopper starts at point $1$, then jumps to each of the other $10$ marked points in some order so that no point is visited twice, before returning to point $1$. The maximal length that he could have jumped in total is $L$, and there are $N$ possible ways to achieve this maximum. Compute $L+N$. [i]Proposed by Yannick Yao[/i]

It is known that there are $3$ buildings in the same shape which are located in an equilateral triangle. Each building has a $2015$ floor with each floor having one window. In all three buildings, every $1$st floor is uninhabited, while each floor of others have exactly one occupant. All windows will be colored with one of red, green or blue. The residents of each floor of a building can see the color of the window in the other buildings of the the same floor and one floor just below it, but they cannot see the colors of the other windows of the two buildings. Besides that, sresidents cannot see the color of the window from any floor in the building itself. For example, resident of the $10$th floor can see the colors of the $9$th and $10$th floor windows for the other buildings (a total of $4$ windows) and he can't see the color of the other window. We want to color the windows so that each resident can see at lest $1$ window of each color. How many ways are there to color those windows?

Let $m,n,k$ be pairwise relatively prime positive integers greater than $3$. Find the minimal possible number of points on the plane with the following property: there are $x$ of them which are the vertices of a regular $x$-gon for $x = m, x = n, x = k$. (E.Piryutko)

A $20 \times 20$ treasure map is glued to a torus. A treasure is hidden in a cell of this map. We can ask questions about $1\times 4$ or $4 \times 1$ rectangles so that we find out if there is a treasure in this rectangle or not. The answers to all questions are absolutely true, but they are given only after all rectangles we want to ask are set. What is the least amount of questions needed to be asked so that we can be sure to find the treasure? (If you describe the position of the cells in a torus with numbers $(i, j)$ of row and column, $1 \leq i, j \leq 20$, then two cells are neighbors, if and only if two of the coordinates they have are the same, and the other two differ by $1$ mod $20$.)

Let $n\geqslant 2$ and $a_1,a_2,\ldots,a_n$ be non-zero integers such that $a_1+a_2+\cdots+a_n=a_1a_2\cdots a_n.$ Prove that \[(a_1^2-1)(a_2^2-1)\cdots(a_n^2-1)\]is a perfect square.

The inscribed circle $\omega$ of the non-isosceles acute-angled triangle $ABC$ touches the side $BC$ at the point $D$. Suppose that $I$ and $O$ are the centres of inscribed circle and circumcircle of triangle $ABC$ respectively. The circumcircle of triangle $ADI$ intersects $AO$ at the points $A$ and $E$. Prove that $AE$ is equal to the radius $r$ of $\omega$.

Two kids $A$ and $B$ play a game as follows: from a box containing $n$ marbles ($n > 1$), they alternately take some marbles for themselves, such that: 1. $A$ goes first. 2. The number of marbles taken by $A$ in his first turn, denoted by $k$, must be between $1$ and $n - 1$, inclusive. 3. The number of marbles taken in a turn by any player must be between $1$ and $k$, inclusive. The winner is the one who takes the last marble. Determine all natural numbers $n$ for which $A$ has a winning strategy

Show that the only solutions of te equation \[ p^{k} + 1 = q^{m} \], in positive integers $k,q,m > 1$ and prime $p$ are (i) $(p,k,q,m) = (2,3,3,2)$ (ii) $k=1 , q=2,$and $p$ is a prime of the form $2^{m} -1$, $m > 1 \in \mathbb{N}$

Let \( I \) and \( O \) be the incenter and circumcenter of the right triangle \( ABC \) (\( \angle C = 90^\circ \)), and let \( K \) be the tangency point of the incircle with \( AC \). Let \( P \) and \( Q \) be the points where the circumcircle of triangle \( AOK \) intersects \( OC \) and the circumcircle of triangle \( ABC \), respectively. Prove that points \( C, I, P, \) and \( Q \) are concyclic. [i]Proposed by Mykhailo Sydorenko[/i]

Let $M, N, P$ be the midpoints of the sides $AB, BC, CA$ of a triangle $ABC$. Let $X$ be a fixed point inside the triangle $MNP$. The lines $L_1, L_2, L_3$ that pass through point $X$ are such that $L_1$ intersects segment $AB$ at point $C_1$ and segment $AC$ at point $B_2$; $L_2$ intersects segment $BC$ at point $A_1$ and segment $BA$ at point $C_2$; $L_3$ intersects segment $CA$ at point $B_1$ and segment $CB$ at point $A_2$. Indicates how to construct the lines $L_1, L_2, L_3$ in such a way that the sum of the areas of the triangles $A_1A_2X, B_1B_2X$ and $C_1C_2X$ is a minimum.

Let $M$ be the midpoint of a given segment $BC$. Point $A$ is chosen to maximize $\angle ABC$ while subject to the condition that $\angle MAC = 20^o$ . What is the ratio $BC/BA$ ?