Let $ABCD$ be a square, and let $M$ be the midpoint of side $BC$. Points $P$ and $Q$ lie on segment $AM$ such that $\angle BPD=\angle BQD=135^\circ$. Given that $AP<AQ$, compute $\tfrac{AQ}{AP}$.

Let $2006$ be expressed as the sum of five positive integers $x_1, x_2, x_3, x_4, x_5$, and $S=\sum_{1\le i<j\le 5}x_ix_j$. $ \textbf{(A)}$ What value of $x_1, x_2, x_3, x_4, x_5$ maximizes $S$? $ \textbf{(A)}$ Find, with proof, the value of $x_1, x_2, x_3, x_4, x_5$ which minimizes of $S$ if $|x_i-x_j|\le 2$ for any $1\le i$, $j\le 5$.

Let $ n\ge 2$ be an integer. What is the minimal and maximal possible rank of an $ n\times n$ matrix whose $ n^{2}$ entries are precisely the numbers $ 1, 2, \ldots, n^{2}$?

Let $\psi (n)$ be the number of integers $0 \le r < n$ such that there exists an integer $x$ that satises $x^2 + x \equiv r$ (mod $n$). Find the sum of all distinct prime factors of $$\sum^4_{i=0}\sum^4_{j=0} \psi(3^i5^j).$$

Let $\mathbb{N}$ be the set of positive integers. For every $n \in \mathbb{N}$, define $d(n)$ as the number of positive divisors of $n$. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that: a) $d(f(x)) = x$ for all $x \in \mathbb{N}$ b) $f(xy)$ divides $(x-1)y^{xy-1}f(x)$ for all $x,y \in \mathbb{N}$

Let $S$ be the set of $n$-uples $\left( x_{1}, x_{2}, \ldots, x_{n}\right)$ such that $x_{i}\in \{ 0, 1 \}$ for all $i \in \overline{1,n}$, where $n \geq 3$. Let $M(n)$ be the smallest integer with the property that any subset of $S$ with at least $M(n)$ elements contains at least three $n$-uples \[\left( x_{1}, \ldots, x_{n}\right), \, \left( y_{1}, \ldots, y_{n}\right), \, \left( z_{1}, \ldots, z_{n}\right) \] such that \[\sum_{i=1}^{n}\left( x_{i}-y_{i}\right)^{2}= \sum_{i=1}^{n}\left( y_{i}-z_{i}\right)^{2}= \sum_{i=1}^{n}\left( z_{i}-x_{i}\right)^{2}. \] (a) Prove that $M(n) \leq \left\lfloor \frac{2^{n+1}}{n}\right\rfloor+1$. (b) Compute $M(3)$ and $M(4)$.

Find the locus of points $M$ inside the rhombus $ABCD$ such that the sum of angles $AMB$ and $CMD$ equals $180^o$ .

There are three boxes, one blue, one white and one red, and $8$ balls. Each of the balls has a number from $1$ to $8$ written on it, without repetitions. The $8$ balls are distributed in the boxes, so that there are at least two balls in each box. Then, in each box, add up all the numbers written on the balls it contains. The three outcomes are called the blue sum, the white sum, and the red sum, depending on the color of the corresponding box. Find all possible distributions of the balls such that the red sum equals twice the blue sum, and the red sum minus the white sum equals the white sum minus the blue sum.

Let $I$ be the incentre of acute-angled triangle $ABC$. Let the incircle meet $BC, CA$, and $AB$ at $D, E$, and $F,$ respectively. Let line $EF$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle DPA + \angle AQD =\angle QIP$. (Slovakia)

A chess king is said to ''attack'' all squares one step away from it (basically any square right next to it in any direction), horizontally, vertically, or diagonally. For instance, a king on the center square of a 3 x 3 grid attacks all 8 other squares, as shown below. Suppose a white king and a black king are placed on different squares of 3 x 3 grid so that they do not attack each other. In how many ways can this be done? [asy] /* AMC8 P17 2024, revised by Teacher David */ unitsize(29pt); import math; add(grid(3,3)); pair [] a = {(0.5,0.5), (0.5, 1.5), (0.5, 2.5), (1.5, 2.5), (2.5,2.5), (2.5,1.5), (2.5,0.5), (1.5,0.5)}; for (int i=0; i<a.length; ++i) { pair x = (1.5,1.5) + 0.4*dir(225-45*i); draw(x -- a[i], arrow=EndArrow()); } label("$K$", (1.5,1.5)); [/asy] $\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 32$

Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.

Let $ \mathbb{Z}[x]$ be the ring of polynomials with integer coefficients, and let $ f(x), g(x) \in\mathbb{Z}[x]$ be nonconstant polynomials such that $ g(x)$ divides $ f(x)$ in $ \mathbb{Z}[x]$. Prove that if the polynomial $ f(x)\minus{}2008$ has at least 81 distinct integer roots, then the degree of $ g(x)$ is greater than 5.

Given a set $S$, of integers, an [i]optimal partition[/i] of S into sets T, U is a partition which minimizes the value $|t - u|$, where $t$ and $u$ are the sum of the elements of $T$ and U respectively. Let $P$ be a set of distinct positive integers such that the sum of the elements of $P$ is $2k$ for a positive integer $k$, and no subset of $P$ sums to $k$. Either show that there exists such a $P$ with at least $2020$ different optimal partitions, or show that such a $P$ does not exist.

A and B play a game on a square board consisting of $n \times n$ white tiles, where $n \ge 2$. A moves first, and the players alternate taking turns. A move consists of picking a square consisting of $2\times 2$ or $3\times 3$ white tiles and colouring all these tiles black. The first player who cannot find any such squares has lost. Show that A can always win the game if A plays the game right.

Prove that for all positive integers $n$, \[\lfloor \sqrt{n}+\sqrt{n+1}+\sqrt{n+2}\rfloor =\lfloor \sqrt{9n+8}\rfloor.\]

[b]p1[/b]. Prove that there is no interger $n$ such that $n^2 +1$ is divisible by $7$. [b]p2.[/b] Find all solutions of the system $$x^2-yz=1$$ $$y^2-xz=2$$ $$z^2-xy=3$$ [b]p3.[/b] A triangle with long legs is an isoceles triangle in which the length of the two equal sides is greater than or equal to the length of the remaining side. What is the maximum number, $n$ , of points in the plane with the property that every three of them form the vertices of a triangle with long legs? Prove all assertions. [b]p4.[/b] Prove that the area of a quadrilateral of sides $a, b, c, d$ which can be inscribed in a circle and circumscribed about another circle is given by $A=\sqrt{abcd}$ [b]p5.[/b] Prove that all of the squares of side length $$\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},...,\frac{1}{n},...$$ can fit inside a square of side length $1$ without overlap. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In $\vartriangle ABC, \angle A= 2 \angle B$. Let $a,b,c$ be the lengths of its sides $BC,CA,AB$, respectively. Prove that $a^2 = b(b + c)$.

The center of square $ABCD$ is $K$. The point $P$ is chosen such that $P \ne K$ and the angle $\angle APB$ is right . Prove that the line $PK$ bisects the angle between the lines $AP$ and $BP$.

Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by the numbers of the form $\ z_{k_1,k_2} = k_1r_1 + k_2r_2$ integers, i.e. for every number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found so that $|x - (k_1r_1 + k_2r_2)| < p$ holds.

Let $\omega$ be the incircle of scalene triangle $ABC$. Let $\omega$ be tangent to $AB$ and $AC$ at points $X$ and $Y$. Construct points $X^\prime$ and $Y^\prime$ on line segments $AB$ and $AC$ respectively such that $AX^\prime=XB$ and $AY^\prime=YC$. Let line $CX^\prime$ intersects $\omega$ at points $P,Q$ such that $P$ is closer to $C$ than $Q$. Also let $R^\prime$ be the intersection of lines $CX^\prime$ and $BY^\prime$. Prove that $CP=RX^\prime$.

In triangle $ABC$ medians from $B$ and $C$ are perpendicular. Prove that $\frac{\sin(B+C)}{\sin B \cdot \sin C} \geq \frac{2}{3}.$

Find all the functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the functional equation $$f(xy^2)+f(x^2y)=y^2f(x)+x^2f(y)$$ for every $x,y\in\mathbb{R}$ and $f(2008) =f(-2008)$ [i](nooonuii)[/i]

Compute the product of positive integers $n$ such that $n^2 + 59n + 881$ is a perfect square.

Determine all infinite sequences $(a_1,a_2,\dots)$ of positive integers satisfying \[a_{n+1}^2=1+(n+2021)a_n\] for all $n \ge 1$.

The diagonals of the convex quadrilateral $ABCD$ are mutually perpendicular. Through the midpoint of the sides $AB$ and $AD$ draw lines, which are perpendicular to the opposite sides. Prove that they intersect on line $AC$.