Triangle $ABC$ with $AB = 10$ cm ¸and $\angle C= 15^o$, is right at $B$. Point $D \in (AC)$ is the foot of the altitude taken from $B$. Find the distance from point $D$ to the line $AB$.

Show that we can find $\alpha>0$ such that, given any point $P$ inside a regular $2n+1$-gon which is inscribed in a circle radius $1$, we can find two vertices of the polygon whose distance from $P$ differ by less than $\frac1n-\frac\alpha{n^3}$.

Here is a list of the numbers of fish that Tyler caught in nine outings last summer: \[2,0,1,3,0,3,3,1,2.\] Which statement about the mean, median, and mode is true? $\textbf{(A)} \text{median} < \text{mean} < \text{mode} \qquad \textbf{(B)} \text{mean} < \text{mode} < \text{median} \\ \\ \textbf{(C)} \text{mean} < \text{median} < \text{mode} \qquad \textbf{(D)} \text{median} < \text{mode} < \text{mean} \\ \\ \textbf{(E)} \text{mode} < \text{median} < \text{mean}$

Let $p(x)$ be a polynomial with real coefficients and $n$ be a positive integer. Prove that there exists a non-zero polynomial $q(x)$ with real coefficients such that the polynomial $p(x)\cdot q(x)$ has non-zero coefficients only by the powers which are multiples of $n$.

The cells of a $(n^2-n+1)\times(n^2-n+1)$ matrix are coloured using $n$ colours. A colour is called [i]dominant[/i] on a row (or a column) if there are at least $n$ cells of this colour on that row (or column). A cell is called [i]extremal[/i] if its colour is [i]dominant [/i] both on its row, and its column. Find all $n \ge 2$ for which there is a colouring with no [i]extremal [/i] cells. Iurie Boreico (Moldova)

An integer $n$ is chosen uniformly at random from the set $\{1, 2, \ldots, 2023!\}.$ Compute the probability that $$\gcd(n^n+50, n+1)=1.$$

Let $p>2$ be a prime number, and $(a_k)_{k\geqslant 1}$ be a sequence defined by $a_1=p$ and $a_{k+1}=2a_k+1$, $k\geqslant 1$. Show that one of the first $p$ terms of the sequence is not prime. [i]Marcel Țena[/i]

Let $S = \{2,5,8,11,14,17,20,\dots\}$. Given that one can choose $n$ different numbers from $S$, $\{A_1,A_2,\dots,A_n\}$ s.t. $\sum_{i=1}^n \frac{1}{A_i} = 1$, find the minimum possible value of $n$.

Find all polynomials $f(x)$ with real coefficients for which \[f(x)f(2x^2) = f(2x^3 + x).\]

A bug walks all day and sleeps all night. On the first day, it starts at point $O$, faces east, and walks a distance of 5 units due east. Each night the bug rotates $60 ^\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to point $P$. Then $OP^2 = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $ 0 $ be a root for a polynom $ f\in\mathbb{R}[X] $ that has the property that $ f(X^2-X+1) =f^2(X)-f(X)+1. $ Determine this polynom. [i]Nelu Chichirim[/i]

Let $ABCD$ be a cyclic quadrilateral with $\angle ADC = \angle DBA$. Furthermore, let $E$ be the projection of $A$ on $BD$. Show that $BC = DE - BE$ .

Let $\vartriangle ABC$ be a triangle such that $\angle ABC = 30^o$, $\angle ACB = 15^o$. Let $M$ be midpoint of segment $BC$ and point $N$ lies on segment $MC$, such that the length of $NC$ is equal to length of $AB$. Proce that $AN$ is the bisector of the angle $\angle MAC$. [img]https://cdn.artofproblemsolving.com/attachments/2/7/4c554b53f03288ee69931fdd2c6fbd3e27ab13.png[/img]

We delete the four corners of a $8 \times 8$ chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares? [i]Proposed by Evan Chen[/i]

In a rectangle $ABCD, E$ is the midpoint of $AB, F$ is a point on $AC$ such that $BF$ is perpendicular to $AC$, and $FE$ perpendicular to $BD$. Suppose $BC = 8\sqrt3$. Find $AB$.

A diabolical combination lock has $n$ dials (each with $c$ possible states), where $n,c>1$. The dials are initially set to states $d_1, d_2, \ldots, d_n$, where $0\le d_i\le c-1$ for each $1\le i\le n$. Unfortunately, the actual states of the dials (the $d_i$'s) are concealed, and the initial settings of the dials are also unknown. On a given turn, one may advance each dial by an integer amount $c_i$ ($0\le c_i\le c-1$), so that every dial is now in a state $d_i '\equiv d_i+c_i \pmod{c}$ with $0\le d_i ' \le c-1$. After each turn, the lock opens if and only if all of the dials are set to the zero state; otherwise, the lock selects a random integer $k$ and cyclically shifts the $d_i$'s by $k$ (so that for every $i$, $d_i$ is replaced by $d_{i-k}$, where indices are taken modulo $n$). Show that the lock can always be opened, regardless of the choices of the initial configuration and the choices of $k$ (which may vary from turn to turn), if and only if $n$ and $c$ are powers of the same prime. [i]Bobby Shen.[/i]

Given a square cardboard of area $\frac{1}{4}$, and a paper triangle of area $\frac{1}{2}$ such that the square of its sidelength is a positive integer. Prove that the triangle can be folded in some ways such that the squace can be placed inside the folded figure so that both of its faces are completely covered with paper. [i]Proposed by N.Beluhov, Bulgaria[/i]

Given a $\triangle ABC$ with circumcenter $O$ and orthocenter $H(O\ne H)$. Denote the midpoints of $BC, AC$ as $D, E$ and let $D', E'$ be the reflections of $D, E$ w.r.t. point $H$, respectively. If lines $AD'$ and $BE'$ meet at $K$, compute $\frac{KO}{KH}$.

A point $P$ is given inside a triangle $ABC$. Let $D,E,F$ be the midpoints of $AP,BP,CP$, and let $L,M,N$ be the intersection points of $ BF$ and $CE, AF$ and $CD, AE$ and $BD$, respectively. (a) Prove that the area of hexagon $DNELFM$ is equal to one third of the area of triangle $ABC$. (b) Prove that $DL,EM$, and $FN$ are concurrent.

Evaluate $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \left\{\frac{1}{\tan x\ (\ln \sin x)}+\frac{\tan x}{\ln \cos x}\right\}\ dx.$

Given three positive real numbers $ x$, $ y$, $ z$ such that $ x \plus{} y \plus{} z \equal{} 1$, prove that $ \frac {xy}{\sqrt {xy \plus{} yz}} \plus{} \frac {yz}{\sqrt {yz \plus{} zx}} \plus{} \frac {zx}{\sqrt {zx \plus{} xy}} \le \frac {\sqrt {2}}{2}$.

How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one? [i]Proposed by Germany, FR.[/i]

After how much time will the particle first return to the origin? (A) $\text{0.785 s}$ (B) $\text{1.26 s}$ (C) $\text{1.57 s}$ (D) $\text{2.00 s}$ (E) $\text{3.14 s}$

For a positive integer $a$, let $S_{a}$ be the set of primes $p$ for which there exists an odd integer $b$ such that $p$ divides $(2^{2^{a}})^{b}-1.$ Prove that for every $a$ there exist infinitely many primes that are not contained in $S_{a}$.

Points $A$ and $B$ are $5$ units apart. How many lines in a given plane containing $A$ and $B$ are $2$ units from $A$ and $3$ units from $B$? $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }\text{more than }3$