Prove that the sum $1^3 + 2^3 +...+ n^3$ is a perfect square for all $n$.

A hinged convex quadrilateral was made of four slats. Then, two points on its opposite sides were connected with another slat, but the structure remained non-rigid. Does it follow from this that this quadrilateral is a parallelogram? [i]Proposed by A. Zaslavsky[/i] [center][img width="40"]https://i.ibb.co/dgqSvLQ/Screenshot-2023-03-09-231327.png[/img][/center]

Let a pair of positive integers $(n, m)$ that are relatively prime be called [i]intertwined[/i] if among any two divisors of $n$ greater than $1$, there exists a divisor of $m$ and among any two divisors of $m$ greater than $1$, there exists a divisor of $n$. For example, pair $(63, 64)$ is intertwined. [b]a)[/b] Find the largest integer $k$ for which there exists an intertwined pair $(n, m)$ such that the product $nm$ is equal to the product of the first $k$ prime numbers. [b]b)[/b] Prove that there does [b]not[/b] exist an intertwined pair $(n, m)$ such that the product $nm$ is the product of $2025$ distinct prime numbers. [b]c)[/b] Prove that there exists an intertwined pair $(n, m)$ such that the number of divisors of $n$ is greater than $2025$. [i]Proposed by Stijn Cambie, Belgium[/i]

\[\int_1^2 \cos(\sin^{-1}(\tan(\cos^{-1}(\sin(\tan^{-1}(x))))))\mathrm dx\] [i]Proposed by Robert Trosten[/i]

We consider a natural number $n$, $n \geq 2$ and the matrices \begin{tabular}{cc} $A= \begin{pmatrix} 1 & 2 & 3 & \cdots & n\\ n & 1 & 2 & \cdots & n - 1\\ n - 1 & n & 1 & \cdots & n - 2\\ \cdots & \cdots & \cdots & \cdots & \cdots\\2 & 3 & 4 & \cdots & 1 \end{pmatrix}$ \end{tabular} Show that$$\epsilon^ndet\left(I_n-A^{2n}\right)+\epsilon^{n-1}det\left(\epsilon I_n-A^{2n}\right)+\epsilon^{n-2}det\left(\epsilon^2 I_n-A^{2n}\right)+\cdots +det\left(\epsilon^n I_n-A^{2n}\right)$$ $$=n(-1)^{n-1}\left[\frac{n^n(n+1)}{2}\right]^{2n^2-4n}\left(1+(n+1)^{2n}\left(2n+(-1)^n{{2n}\choose{n}}\right)\right)$$where $\epsilon \in \mathbb{C}\backslash \mathbb{R}$, $\epsilon^{n+1}=1$

A convex quadrilateral $ABCD$ is given in which $\angle DAB = \angle ABC = 45^o$ and $DA = 3$, $AB = 7\sqrt2$, $BC = 4$. Calculate the length of side $CD$. [img]https://cdn.artofproblemsolving.com/attachments/1/2/046e31a628b3df4d23d3162cb570e1b9cb71e2.png[/img]

Prove the identities: $$C_{n}^{1}+2C_{n}^{2}+3C_{n}^{3}+...+nC_{n}^{n}=n\cdot 2 ^{n-1}$$ $$C_{n}^{1}-2C_{n}^{2}+3C_{n}^{3}+...-(-1)^{n-1}nC_{n}^{n}=0$$

Does there exist an infinite set $S$ consisted of positive integers,so that for any $x,y,z,w\in S,x<y,z<w$,if $(x,y)\ne (z,w)$,then $\gcd(xy+2022,zw+2022)=1$?

[b]a)[/b] Find all $ 2\times 2 $ complex matrices $ A $ which have the property that there are two complex numbers $ \alpha ,\gamma $ with $ \alpha \neq \text{tr} (A) $ or $ \gamma\neq \det (A) $ such that $ A^2-\alpha A+\gamma I=0. $ [b]b)[/b] Consider $ B\not\in\{ 0,I\} $ as a matrix having the property mentioned at [b]a).[/b] Solve in the complex numbers the system $ xB-yI-B^2=xB^2-yI-B^4=0. $ [i]Adrian Troie[/i]

Determine all functions $f : R \to R$ such that $f(x) \ge 0$ for all positive reals $x$, $f(0) = 0$ and for all reals $x, y$ $$f(x + y -xy) = f(x) + f(y) - f(xy).$$

In the quadrilateral $ABCD$, $AB = BC$ . The point $E$ lies on the line $AB$ is such that $BD= BE$ and $AD \perp DE$. Prove that the perpendicular bisectors to segments $AD, CD$ and $CE$ intersect at one point.

Let $VWXYZ$ be a square pyramid with vertex $V$ with height $1$, and with the unit square as its base. Let $STANFURD$ be a cube, such that face $FURD$ lies in the same plane as and shares the same center as square face $WXYZ$. Furthermore, all sides of $FURD$ are parallel to the sides of $WXY Z$. Cube $STANFURD$ has side length $s$ such that the volume that lies inside the cube but outside the square pyramid is equal to the volume that lies inside the square pyramid but outside the cube. What is the value of $s$?

Let $A$ and $B$ be two points inside a circle $C$. Show that there exists a circle that contains $A$ and $B$ and lies completely inside $C$.

The arithmetic mean of the nine numbers in the set $ \{9,99,999,9999, . . . ,999999999\}$ is a $ 9$-digit number $ M$, all of whose digits are distinct. The number $ M$ does not contain the digit $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$

Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, Elías is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, the person number twenty-five, who is a friend to everyone. How many people is Zoila a friend of, person number twenty-six? Clarification: If $A$ is a friend of $B$ then $B$ is a friend of $A$.

Determine the greatest possible value of $\sum_{i = 1}^{10} \cos(3x_i)$ for real numbers $x_1, x_2, \dots, x_{10}$ satisfying $\sum_{i = 1}^{10} \cos(x_i) = 0$.

Determine all functions $f : R_{\ge 0} \to R_{\ge 0}$ with the following properties: (a) $f(1) = 0$, (b) $f(x) > 0$ for all $x > 1$, (c) For all $x, y\ge 0$ with $x + y > 0$ holds $$f(xf(y))f(y) = f\left( \frac{xy}{x + y}\right)$$

Given $a;b;c$ satisfying $a^{2}+b^{2}+c^{2}=2$ . Prove that: a) $\left | a+b+c-abc \right |\leqslant 2$ . b) $\left | a^{3}+b^{3}+c^{3}-3abc \right |\leqslant 2\sqrt{2}$

Find the set of all $ a \in \mathbb{R}$ for which there is no infinite sequene $ (x_n)_{n \geq 0} \subset \mathbb{R}$ satisfying $ x_0 \equal{} a,$ and for $ n \equal{} 0,1, \ldots$ we have \[ x_{n\plus{}1} \equal{} \frac{x_n \plus{} \alpha}{\beta x_n \plus{} 1}\] where $ \alpha \beta > 0.$

Find the number of ordered pairs of positive integers $(m,n)$, where $m,n\leq 10$, such that $m!+n!$ is a multiple of $10$.

$12$ distinct points are equally spaced around a circle. How many ways can Bryan choose $3$ points (not in any order) out of these $12$ points such that they form an acute triangle (Rotations of a set of points are considered distinct). [i]Proposed by Bryan Guo [/i]

Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.

Let $ABC$ be an obtuse triangle with orthocenter $H$ and centroid $S$. Let $D$, $E$ and $F$ be the midpoints of segments $BC$, $AC$, $AB$, respectively. Show that the circumcircle of triangle $ABC$, the circumcircle of triangle $DEF$ and the circle with diameter $HS$ have two distinct points in common. [i](Josef Greilhuber)[/i]

Two mathematicians discuss two positive integers. One of them states that the square of the ratio between their product and their sum is exactly one more than this ratio. What is the smaller of these two numbers?

Let $k$ be a positive integer. The organising commitee of a tennis tournament is to schedule the matches for $2k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay $1$ coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.