Let $ n$ be a positive integer. Prove that $ x^ne^{1\minus{}x}\leq n!$ for $ x\geq 0$,

Determine all pairs of positive integers $(a,b)$ such that the following fraction is an integer: \[\frac{(a+b)^2}{4+4a(a-b)^2}.\]

Consider a square. Find the locus of midpoints of the hypothenuses of rightangled triangles with the vertices lying on three different sides of the square and not coinciding with its vertices. (B.Frenkin)

The number $2023$ is written $2023$ times on a blackboard. On one move, you can choose two numbers $x, y$ on the blackboard, delete them and write $\frac{x+y} {4}$ instead. Prove that when one number remains, it is greater than $1$.

A traffic accident involved three cars: one blue, one green, and one red. Three witnesses spoke to the police and gave the following statements: **Person 1:** The red car was guilty, and either the green or the blue one was involved. **Person 2:** Either the green car or the red car was guilty, but not both. **Person 3:** Only one of the cars was guilty, but it was not the blue one. The police know that at least one car was guilty and that at least one car was not. However, the police do not know if any of the three witnesses lied. Which car(s) were responsible for the accident?

Let $\overline{abcd}$ be one of the 9999 numbers $0001, 0002, 0003, \ldots, 9998, 9999$. Let $\overline{abcd}$ be an [i]special[/i] number if $ab-cd$ and $ab+cd$ are perfect squares, $ab-cd$ divides $ab+cd$ and also $ab+cd$ divides $abcd$. For example 2016 is special. Find all the $\overline{abcd}$ special numbers. [b]Note:[/b] If $\overline{abcd}=0206$, then $ab=02$ and $cd=06$.

On the circle of radius $30$ cm are given $2$ points A,B with $AB = 16$ cm and $C$ is a midpoint of $AB$. What is the perpendicular distance from $C$ to the circle?

Consider any graph with $50$ vertices and $225$ edges. We say that a triplet of its (mutually distinct) vertices is [i]connected[/i] if the three vertices determine at least two edges. Determine the smallest and the largest possible number of connected triples. [i](Jan Mazak, Josef Tkadlec)[/i]

Consider a set of $16$ points arranged in $4 \times 4$ square grid formation. Prove that if any $7$ of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.

For each positive integer $n$, let $I_n$ denote the number of integers $p$ for which $50^n<7^p<50^{n+1}$. (a) Prove that, for each $n$, $I_n$ is either $2$ or $3$. (b) Prove that $I_n=3$ for infinitely many $n\in\mathbb N$, and find at least one such $n$.

Find the prime number $p$ such that $71p + 1$ is a perfect square.

[b]3.[/b] Let $f(x)= x^n +a_1 x^(n-1)+ \dots + a_n$ ($n\geq 1$) be an irreducible polynomial over the field $K$. Show that every non-zero matrix commuting with the matrix $ \begin{bmatrix} 0 & 1 & 0 & \dots & 0 & 0 \\ 0 & 0 & 1 & \dots & 0 & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots \\ 0 & 0 & 0 & \dots & 0 & 1 \\ -a_n & -a_{n-1} & -a_{n-2} & \dots & -a_2 & -a_1 \end{bmatrix} $ is invertible. [b](A. 4)[/b]

Albert and Beatrice play a game. $2021$ stones lie on a table. Starting with Albert, they alternatively remove stones from the table, while obeying the following rule. At the $n$-th turn, the active player (Albert if $n$ is odd, Beatrice if $n$ is even) can remove from $1$ to $n$ stones. Thus, Albert first removes $1$ stone; then, Beatrice can remove $1$ or $2$ stones, as she wishes; then, Albert can remove from $1$ to $3$ stones, and so on. The player who removes the last stone on the table loses, and the other one wins. Which player has a strategy to win regardless of the other player's moves?

Given two relatively prime numbers $p>0$ and $q>0$. An integer $n$ is called "good" if we can represent it as $n = px + qy$ with nonnegative integers $x$ and $y$, and "bad" in the opposite case. a) Prove that there exist integer $c$ such that in a pair $\{n, c-n\}$ always one is "good" and one is "bad". b) How many there exist "bad" numbers?

Consider a circle $C$ with diameter $AB=1$. A point $P_0$ is chosen on $C$, $P_0 \ne A$, and starting in $P_0$ a sequence of points $P_1, P_2, \dots, P_n, \dots$ is constructed on $C$, in the following way: $Q_n$ is the symmetrical point of $A$ with respect of $P_n$ and the straight line that joins $B$ and $Q_n$ cuts $C$ at $B$ and $P_{n+1}$ (not necessary different). Prove that it is possible to choose $P_0$ such that: [b]i[/b] $\angle {P_0AB} < 1$. [b]ii[/b] In the sequence that starts with $P_0$ there are $2$ points, $P_k$ and $P_j$, such that $\triangle {AP_kP_j}$ is equilateral.

Find all polynomials of degree 3, such that for each $x,y\geq 0$: \[p(x+y)\geq p(x)+p(y)\]

Among all triangles having (i) a fixed angle $A$ and (ii) an inscribed circle of fixed radius $r$, determine which triangle has the least minimum perimeter.

Let $ ABC$ be an acute-angled triangle. Let $ E$ be a point such $ E$ and $ B$ are on distinct sides of the line $ AC,$ and $ D$ is an interior point of segment $ AE.$ We have $ \angle ADB \equal{} \angle CDE,$ $ \angle BAD \equal{} \angle ECD,$ and $ \angle ACB \equal{} \angle EBA.$ Prove that $ B, C$ and $ E$ lie on the same line.

Medians $AM_A,BM_B,CM_C$ of triangle $ABC$ intersect at $M$.Let $\Omega_A$ be circumcircle of triangle passes through midpoint of $AM$ and tangent to $BC$ at $M_A$.Define $\Omega_B$ and $\Omega_C$ analogusly.Prove that $\Omega_A,\Omega_B$ and $\Omega_C$ intersect at one point.(A.Yakubov) [hide=P.S]sorry for my mistake in translation :blush: :whistling: .thank you jred for your help :coolspeak: [/hide]

Let $n \geq 2$ be a positive integer. In a mathematics competition, there are $n+1$ students, with one of them being a hacker. The competition is conducted as follows: each receives the same problem with an open-ended answer, has 5 minutes to give their own answer, after which all answers are submitted simultaneously, the correct answer is announced, then they receive a new problem, and so on. The hacker cheats by using spy cameras to see the answers of the other participants. A correct answer gives 1 point, while a wrong answer gives -1 point to everyone except the hacker; for him, it's 0 points because he managed to hack the scoring system. Prove that regardless of the total number of problems, if at some point the hacker is ahead of the second-place contestant by at least $2^{n-2} + 1$ points, then he has a strategy to ensure he will be the sole winner by the end of the competition.

Let $ABC$ be an isosceles triangle $(AB=AC)$ so that $\angle A< 2 \angle B$ . Let $D,Z $ points on the extension of height $AM$ so that $\angle CBD = \angle A$ and $\angle ZBA = 90^\circ$. Let $E$ the orthogonal projection of $M$ on height $BF$, and let $K$ the orthogonal projection of $Z$ on $AE$. Prove that $ \angle KDZ = \angle KDB = \angle KZB$.

You wish to color every vertex, edge, face, and the interior of a cube one color each such that no two adjacent objects are the same color. Faces are adjacent if they share an edge. Edges are adjacent if they share a vertex. The interior is adjacent to all of its faces, edges, and vertices. Each face is adjacent to all of its edges and vertices, but is not adjacent to any other edges or vertices. Each edge is adjacent to both of its vertices, but is not adjacent to any other vertices. What is the minimum number of colors required for a coloring satisfying this property?

For all real numbers $r$, denote by $\{r\}$ the fractional part of $r$, i.e. the unique real number $s\in[0,1)$ such that $r-s$ is an integer. How many real numbers $x\in[1,2)$ satisfy the equation $\left\{x^{2018}\right\} = \left\{x^{2017}\right\}?$

Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ Let $\lambda \geq 1$ be a real number and $n$ be a positive integer with the property that $\lfloor \lambda^{n+1}\rfloor, \lfloor \lambda^{n+2}\rfloor ,\cdots, \lfloor \lambda^{4n}\rfloor$ are all perfect squares$.$ Prove that $\lfloor \lambda \rfloor$ is a perfect square$.$

Let $S$ be a subset of the integers $1,2,\ldots,100$ that has the property that none of its members is $3$ times another. What is the largest number of members $S$ can have? $\text{(A) }67\qquad\text{(B) }71\qquad\text{(C) }72\qquad\text{(D) }76\qquad\text{(E) }77$