Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the respective opposite sides are concurrent. [b]Note by Darij:[/b] A [i]cyclic quadrilateral [/i]is a quadrilateral inscribed in a circle.

In a class, some pupils learn German, the other learn French. The number of girls learning French and the number of boys learning German total to 16. There are 11 pupils learning French, and there are 10 girls in the class. In addition to the girls learning French, there are 16 pupils. How many pupils are there in the class? A. 18 B. 21 C. 23 D. 27 E. 31

If $(\log_2 3)^x-(\log_5 3)^x\geq (\log_2 3)^{-y}-(\log_5 3)^{-y}$, then $\text{(A)}x-y\geq0\qquad\text{(B)}x+y\geq0\qquad\text{(C)}x-y\leq0\qquad\text{(D)}x+y\leq0$

We have an a sequence such that $a_n = 2 \cdot 10^{n + 1} + 19$. Determine all the primes $p$, with $p \le 19$, for which there exists some $n \ge 1$ such that $p$ divides $a_n$.

Prove that $ a < b$ implies that $ a^3 \minus{} 3a \leq b^3 \minus{} 3b \plus{} 4.$ When does equality occur?

For which positive integers $k$, is it true that there are infinitely many pairs of positive integers $(m, n)$ such that \[\frac{(m+n-k)!}{m! \; n!}\] is an integer?

Find the largest integer $k$ with the following property: Whenever real numbers $x_1,x_2,\dots,x_{2024}$ satisfy \[x_1^2=(x_1+x_2)^2=\dots=(x_1+x_2+\dots+x_{2024})^2,\] at least $k$ of them are equal.

If $p>2$ is prime number and $m$ and $n$ are positive integers such that $$\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}$$ Prove that $p$ divides $m$

Path graph $U$ is given, and a blindfolded player is standing on one of its vertices. The vertices of $U$ are labeled with positive integers between 1 and $n$, not necessarily in the natural order. In each step of the game, the game master tells the player whether he is in a vertex with degree 1 or with degree 2. If he is in a vertex with degree 1, he has to move to its only neighbour, if he is in a vertex with degree 2, he can decide whether he wants to move to the adjacent vertex with the lower or with the higher number. All the information the player has during the game after $k$ steps are the $k$ degrees of the vertices he visited and his choice for each step. Is there a strategy for the player with which he can decide in finitely many steps how many vertices the path has?

Let $n$ be a positive integer. Let $B_n$ be the set of all binary strings of length $n$. For a binary string $s_1\hdots s_n$, we define it's twist in the following way. First, we count how many blocks of consecutive digits it has. Denote this number by $b$. Then, we replace $s_b$ with $1-s_b$. A string $a$ is said to be a [i]descendant[/i] of $b$ if $a$ can be obtained from $b$ through a finite number of twists. A subset of $B_n$ is called [i]divided[/i] if no two of its members have a common descendant. Find the largest possible cardinality of a divided subset of $B_n$. [i]Remark.[/i] Here is an example of a twist: $101100 \rightarrow 101000$ because $1\mid 0\mid 11\mid 00$ has $4$ blocks of consecutive digits. [i]Viktor Simjanoski[/i]

Let be given an open interval $(\alpha; \beta)$ with $\alpha - \beta = \frac{1}{27}$. Determine the maximum number of irreducible fractions $\frac{a}{b}$ in $(\alpha; \beta)$ with $1 \leq b \leq 2007$?

Determine the value of \[ \sup_{ 1 \leq \xi \leq 2} [\log E \xi\minus{}E \log \xi],\] where $ \xi$ is a random variable and $ E$ denotes expectation. [i]Z. Daroczy[/i]

In the diagram below $ \angle CAB, \angle CBD$, and $\angle CDE$ are all right angles with side lengths $AC = 3$, $BC = 5$, $BD = 12$, and $DE = 84$. The distance from point $E$ to the line $AB$ can be expressed as the ratio of two relatively prime positive integers, $m$ and $n$. Find $m + n$. [asy] size(300); defaultpen(linewidth(0.8)); draw(origin--(3,0)--(0,4)--cycle^^(0,4)--(6,8)--(3,0)--(30,-4)--(6,8)); label("$A$",origin,SW); label("$B$",(0,4),dir(160)); label("$C$",(3,0),S); label("$D$",(6,8),dir(80)); label("$E$",(30,-4),E);[/asy]

Let $A$ be a set of $n$ points in the space. From the family of all segments with endpoints in $A$, $q$ segments have been selected and colored yellow. Suppose that all yellow segments are of different length. Prove that there exists a polygonal line composed of $m$ yellow segments, where $m \geq \frac{2q}{n}$, arranged in order of increasing length.

Prove that $$ a+b+c>\left( \sqrt\alpha +\sqrt\beta +\sqrt\gamma \right)^2, $$ for all positive real numbers $ a,b,c,\alpha ,\beta ,\gamma $ that are under the condition $$ abc>\alpha bc+\beta ac+\gamma ab. $$ [i]Țuțescu Lucian[/i] and [i]Chiriță Aurel[/i]

Let $b$ be the length of the largest diagonal and $c$ be the length of the smallest diagonal of a regular nonagon with side length $a$. Which one of the followings is true? $ \textbf{(A)}\ b=\dfrac{a+c}2 \qquad\textbf{(B)}\ b=\sqrt {ac} \qquad\textbf{(C)}\ b^2=\dfrac{a^2+c^2}2 \\ \textbf{(D)}\ c=a+b \qquad\textbf{(E)}\ c^2=a^2+b^2 $