A wall made of mirrors has the shape of $\triangle ABC$, where $AB = 13$, $BC = 16$, and $CA = 9$. A laser positioned at point $A$ is fired at the midpoint $M$ of $BC$. The shot reflects about $BC$ and then strikes point $P$ on $AB$. If $\tfrac{AM}{MP} = \tfrac{m}{n}$ for relatively prime positive integers $m, n$, compute $100m+n$. [i]Proposed by Michael Tang[/i]

The domain of a function $f$ is $\mathbb{N}$ (The set of natural numbers). The function is defined as follows : $$f(n)=n+\lfloor\sqrt{n}\rfloor$$ where $\lfloor k\rfloor$ denotes the nearest integer smaller than or equal to $k$. Prove that, for every natural number $m$, the following sequence contains at least one perfect square $$m,~f(m),~f^2(m),~f^3(m),\cdots$$ The notation $f^k$ denotes the function obtained by composing $f$ with itself $k$ times.

Find all triples of integers $(a, b, c)$ satisfying $a^2 + b^2 + c^2 =3(ab + bc + ca).$

In the Olympic school the exams are graded with whole numbers, the lowest possible grade is $0$, and the highest is $10$. In the arithmetic class the teacher takes two exams. This year he has $15$ students. When one of his students gets less than $3$ on the first exam and more than $7$ on the second exam, he calls him an overachieving student. The teacher, at the end of correcting the exams, averaged the $30$ grades and obtained $8$. What is the largest number of students who passed this class could have had?

In the following sum: $1 + 2 + 3 + 4 + 5 + 6$, if we remove the first two “+” signs, we obtain the new sum $123 + 4 + 5 + 6 = 138$. By removing three “$+$” signs, we can obtain $1 + 23 + 456 = 480$. Let us now consider the sum $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13$, in which some “$+$” signs are to be removed. What are the three smallest multiples of $100$ that we can get in this way?

Consider a rectangular grid of $ 10 \times 10$ unit squares. We call a [i]ship[/i] a figure made up of unit squares connected by common edges. We call a [i]fleet[/i] a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the least number of squares in a fleet to which no new ship can be added.

The points $A_1$,$B_1$,$C_1$ are middle points of the arcs $\widehat{BC}, \widehat{CA}, \widehat{AB}$ of the circumscribed circle of $\Delta ABC$, respectively. The points $I_a,I_b,I_c$ are the reflections in the middle points of $BC,CA,AB$ of the center $I$ of the inscribed circle in the triangle. Prove that $I_a A_1,I_b B_1$, and $I_c C_1$ are concurrent.

Consider the sequence $(a_n)_{n\ge 1}$ defined by $a_n=n$ for $n\in \{1,2,3.4,5,6\}$, and for $n \ge 7$: $$a_n={\lfloor}\frac{a_1+a_2+...+a_{n-1}}{2}{\rfloor}$$ where ${\lfloor}x{\rfloor}$ is the greatest integer less than or equal to $x$. For example : ${\lfloor}2.4{\rfloor} = 2, {\lfloor}3{\rfloor} = 3$ and ${\lfloor}\pi {\rfloor}= 3$. For all integers $n \ge 2$, let $S_n = \{a_1,a_1,...,a_n\}- \{r_n\}$ where $r_n$ is the remainder when $a_1 + a_2 + ... + a_n$ is divided by $3$. The minus $-$ denotes the ''[i]remove it if it is there[/i]'' notation. For example : $S_4 = {2,3,4}$ because $r_4= 1$ so $1$ is removed from $\{1,2,3,4\}$. However $S_5= \{1,2,3,4,5\}$ betawe $r_5 = 0$ and $0$ is not in the set $\{1,2,3,4,5\}$. 1. Determine $S_7,S_8,S_9$ and $S_{10}$. 2. We say that a set $S_n$ for $n\ge 6$ is well-balanced if it can be partitioned into three pairwise disjoint subsets with equal sum. For example : $S_6 = \{1,2,3,4,5,6\} =\{1,6\}\cup \{2,5\}\cup \{3,4\}$ and $1 +6 = 2 + 5 = 3 + 4$. Prove that $S_7,S_8,S_9$ and $S_{10}$ are well-balanced . 3. Is the set $S_{2019}$ well-balanced? Justify your answer.

In the plane, the segments $AB$ and $CD$ are given, while the lines $AB$ and $CD$ intersect. Prove that the set of all points $P$ in the plane such that triangles $ABP$ and $CDP$ have equal areas , form two lines intersecting at the intersection of the lines $AB$ and $CD$.

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?