Let $ABC$ be an acute triangle with orthocenter $H$, and $P$ a point interior to the triangle. Points $D,E,F$ are the reflections of $P$ about $BC,CA,AB$. If $Q$ is the intersection of $HD$ and $EF$, prove that the ratio $HQ/HD$ is independent of the choice of $P$. Luis González

Determine the largest real number $z$ such that \begin{align*} x + y + z = 5 \\ xy + yz + xz = 3 \end{align*} and $x$, $y$ are also real.

Find the largest value of the real number $ \lambda$, such that as long as point $ P$ lies in the acute triangle $ ABC$ satisfying $ \angle{PAB}\equal{}\angle{PBC}\equal{}\angle{PCA}$, and rays $ AP$, $ BP$, $ CP$ intersect the circumcircle of triangles $ PBC$, $ PCA$, $ PAB$ at points $ A_1$, $ B_1$, $ C_1$ respectively, then $ S_{A_1BC}\plus{} S_{B_1CA}\plus{} S_{C_1AB} \geq \lambda S_{ABC}$.

Real numbers $x,y$ are such that $x^2\ge y$ and $y^2\ge x.$ Prove that $\frac{x}{y^2+1}+\frac{y}{x^2+1}\le1.$

Consider a $(2m-1)\times(2n-1)$ rectangular region, where $m$ and $n$ are integers such that $m,n\ge 4.$ The region is to be tiled using tiles of the two types shown: \[ \begin{picture}(140,40) \put(0,0){\line(0,1){40}} \put(0,0){\line(1,0){20}} \put(0,40){\line(1,0){40}} \put(20,0){\line(0,1){20}} \put(20,20){\line(1,0){20}} \put(40,20){\line(0,1){20}} \multiput(0,20)(5,0){4}{\line(1,0){3}} \multiput(20,20)(0,5){4}{\line(0,1){3}} \put(80,0){\line(1,0){40}} \put(120,0){\line(0,1){20}} \put(120,20){\line(1,0){20}} \put(140,20){\line(0,1){20}} \put(80,0){\line(0,1){20}} \put(80,20){\line(1,0){20}} \put(100,20){\line(0,1){20}} \put(100,40){\line(1,0){40}} \multiput(100,0)(0,5){4}{\line(0,1){3}} \multiput(100,20)(5,0){4}{\line(1,0){3}} \multiput(120,20)(0,5){4}{\line(0,1){3}} \end{picture} \] (The dotted lines divide the tiles into $1\times 1$ squares.) The tiles may be rotated and reflected, as long as their sides are parallel to the sides of the rectangular region. They must all fit within the region, and they must cover it completely without overlapping. What is the minimum number of tiles required to tile the region?

Let $P$ and $Q$ be two convex polygons. Let $h$ be the length of the projection of $Q$ onto a line perpendicular to a side of $P$ which is of length $p$. Define $f(P,Q)$ to be the sum of the products $hp$ over all sides of $P$. Prove that $f(P,Q) = f(Q, P)$.

If it is possible to find six elements, whose sum are divisible by $6$, from every set with $n$ elements, what is the least $n$ ? $\textbf{(A)}\ 13 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 9$

Let $n>1$ be a positive integer and $d_1<d_2<\dots<d_m$ be its $m$ positive divisors, including $1$ and $n$. Lalo writes the following $2m$ numbers on a board: \[d_1,d_2\dots, d_m,d_1+d_2,d_2+d_3,\dots,d_{m-1}+d_m,N \] where $N$ is a positive integer. Afterwards, Lalo erases any number that is repeated (for example, if a number appears twice, he erases one of them). Finally, Lalo realizes that the numbers left on the board are exactly all the divisors of $N$. Find all possible values that $n$ can take.

The equation $ 3y^2 \plus{} y \plus{} 4 \equal{} 2(6x^2 \plus{} y \plus{} 2)$ where $ y \equal{} 2x$ is satisfied by: $ \textbf{(A)}\ \text{no value of }x \qquad\textbf{(B)}\ \text{all values of }x \qquad\textbf{(C)}\ x \equal{} 0\text{ only}$ $ \textbf{(D)}\ \text{all integral values of }x\text{ only} \qquad\textbf{(E)}\ \text{all rational values of }x\text{ only}$

Monic quadratic polynomials $ P(x)$ and $ Q(x)$ have the property that $ P(Q(x))$ has zeroes at $ x\equal{}\minus{}23,\minus{}21,\minus{}17, \text{and} \minus{}15$, and $ Q(P(x))$ has zeroes at $ x\equal{}\minus{}59, \minus{}57, \minus{}51, \text{and} \minus{}49$. What is the sum of the minimum values of $ P(x)$ and $ Q(x)$? $ \textbf{(A)}\ \text{\minus{}100} \qquad \textbf{(B)}\ \text{\minus{}82} \qquad \textbf{(C)}\ \text{\minus{}73} \qquad \textbf{(D)}\ \text{\minus{}64} \qquad \textbf{(E)}\ 0$

Let $e$ be the identity of monoid $(M,\cdot)$ and $a\in M$ an invertible element. Prove that [list=a] [*]The set $M_a:=\{x\in M:ax^2a=e\}$ is nonempty; [*]If $b\in M_a$ is invertible, then $b^{-1}\in M_a$ if and only if $a^4=e$; [*]If $(M_a,\cdot)$ is a monoid, then $x^2=e$ for all $x\in M_a.$ [/list] [i]Mathematical Gazette[/i]

Consider all sums that add up to $2015$. In each sum, the addends are consecutive positive integers, and all sums have less than $10$ addends. How many such sums are there?

In the plane containing a triangle $ABC$, points $A'$, $B'$ and $C'$ distinct from the vertices of $ABC$ lie on the lines $BC$, $AC$ and $AB$ respectively such that $AA'$, $BB'$ and $CC'$ are concurrent at $G$ and $AG/GA' = BG/GB' = CG/GC'$. Prove that $G$ is the centroid of $ABC$.

The polynomial $A(x) = x^2 + ax + b$ with integer coefficients has the following property: for each prime $p$ there is an integer $k$ such that $A(k)$ and $A(k + 1)$ are both divisible by $p$. Proof that there is an integer $m$ such that $A(m) = A(m + 1) = 0$.

Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$. [i]Carl Lian and Brian Hamrick.[/i]

Which digit must be substituted instead of the star so that the following large number $$\underbrace{66...66}_{2023} \star \underbrace{55...55}_{2023}$$ is divisible by $7$?

(a) Let $u$, $v$, and $w$ be the real solutions to the equation $x^3 - 7x + 7 = 0$. Show that there exists a quadratic polynomial $f$ with rational coefficients such that $u = f(v)$, $v = f(w)$, and $w = f(u)$. (b) Let $u$, $v$, and $w$ be the real solutions to the equation $x^3 -7x+4 = 0$. Show that there does not exist a quadratic polynomial $f $with rational coefficients such that $u = f(v)$, $v = f(w)$, and $w = f(u)$.

Let $n \ge 2$ be a positive integer. Each square of an $n\times n$ board is coloured red or blue. We put dominoes on the board, each covering two squares of the board. A domino is called [i]even [/i] if it lies on two red or two blue squares and [i]colourful [/i] if it lies on a red and a blue square. Find the largest positive integer $k$ having the following property: regardless of how the red/blue-colouring of the board is done, it is always possible to put $k$ non-overlapping dominoes on the board that are either all [i]even [/i] or all [i]colourful[/i].

Given convex hexagon $ABCDEF$ with $AB \parallel DE$, $BC \parallel EF$, and $CD \parallel FA$ . The distance between the lines $AB$ and $DE$ is equal to the distance between the lines $BC$ and $EF$ and to the distance between the lines $CD$ and $FA$. Prove that the sum $AD+BE+CF$ does not exceed the perimeter of hexagon $ABCDEF$.

During a school year 44 competitions were held. Exactly 7 students won in each of the competition. For any two competitions, there exists exactly 1 student who won both competitions. Is it true that there exists a student who won all the competitions?

Let $ABC$ be a triangle inscribed in a circle $(O)$. d is the tangent at $A$ of $(O), P$ is an arbitrary point in the plane. $D,E, F$ are the projections of $P$ on $BC,CA,AB$. Let $DE,DF$ intersect the line $d$ at $M,N$ respectively. The circumcircle of triangle $DEF$ meets $CA,AB$ at $K,L$ distinct from $E, F$. Prove that $KN$ meets $LM$ at a point on the circumcircle of triangle $DEF$. Trần Quang Hùng

A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form \[ a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6}, \] where $a$, $b$, $c$, and $d$ are positive integers. Find $a + b + c + d$.

Let $ABC$ be a triangle with side lengths $13,14,15$. The points on the interior of $ABC$ with distance at least $1$ from each side are shaded. The area of the shaded region can be written in simplest form as $\tfrac{m}{n}$. Find $m+n$.

Two players $A$ and $B$ have one stone each on a $100 \times 100$ chessboard. They move their stones one after the other, and a move means moving one's stone to a neighbouring field (horizontally or vertically, not diagonally). At the beginning of the game, the stone of $A$ lies in the lower left corner, and the one of $B$ in the lower right corner. Player $A$ starts. Prove: Player $A$ is, independently from that what $B$ does, able to reach, after finitely many steps, the field $B$'s stone is lying on at that moment.

A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly 1/1985. [asy] size(200); pair A=(0,1), B=(1,1), C=(1,0), D=origin; draw(A--B--C--D--A--(1,1/6)); draw(C--(0,5/6)^^B--(1/6,0)^^D--(5/6,1)); pair point=( 0.5 , 0.5 ); //label("$A$", A, dir(point--A)); //label("$B$", B, dir(point--B)); //label("$C$", C, dir(point--C)); //label("$D$", D, dir(point--D)); label("$1/n$", (11/12,1), N, fontsize(9));[/asy]