Find all positive integers $k > 1$ for which there exists a positive integer $n$ such that $\tbinom{n}{k}$ is divisible by $n$, and $\tbinom{n}{m}$ is not divisible by $n$ for $2\leq m < k$. [i]Merlijn Staps[/i]

In a school, more than $90\% $ of the students know both English and German, and more than $90\%$ percent of the students know both English and French. Prove that more than $90\%$ percent of the students who know both German and French also know English.

All sets of $49$ distinct positive integers less than or equal to $100$ are considered. Leandro assigned each of these sets a positive integer less than or equal to $100$. Prove that there is a set $L$ of $50$ distinct positive integers less than or equal to $100$, such that for each number $x$ of $L$ the number that Leandro assigned to the set of $49$ numbers $L-\{ x\}$ is different from $x$. Clarification: $L-\{x\}$ denotes the set that results from removing the number $x$ from $L$.

Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

The vertex $E$ of a square $EFGH$ is at the center of square $ABCD$. The length of a side of $ABCD$ is $1$ and the length of a side of $EFGH$ is $2$. Side $EF$ intersects $CD$ at $I$ and $EH$ intersects $AD$ at $J$. If angle $EID=60^\circ$, the area of quadrilateral $EIDJ$ is $\textbf{(A) }\dfrac14\qquad\textbf{(B) }\dfrac{\sqrt3}6\qquad\textbf{(C) }\dfrac13\qquad\textbf{(D) }\dfrac{\sqrt2}4\qquad\textbf{(E) }\dfrac{\sqrt3}2$

For non-empty subsets $A,B \subset \mathbb{Z}$ define \[A+B=\{a+b:a\in A, b\in B\},\ A-B=\{a-b:a\in A, b\in B\}.\] In the sequel we work with non-empty finite subsets of $\mathbb{Z}$. Prove that we can cover $B$ by at most $\frac{|A+B|}{|A|}$ translates of $A-A$, i.e. there exists $X\subset Z$ with $|X|\leq \frac{|A+B|}{|A|}$ such that \[B\subseteq \cup_{x\in X} (x+(A-A))=X+A-A.\]

Find all integers $n > 1$ for which it is possible to fill the cells of an $n \times n$ grid with the integers from $1$ to $n^2$, without repetition, such that the average of the $n$ numbers in each row and each column is an integer.

Let $a_1, a_2, ..., a_n$ be positive integers, not necessarily distinct but with at least five distinct values. Suppose that for any $1 \le i < j \le n$, there exist $k,\ell$, both different from $i$ and $j$ such that $a_i + a_j = a_k + a_{\ell}$. What is the smallest possible value of $n$?

The reals $x, y$ satisfy $x(x-6)\leq y(4-y)+7$. Find the minimal and maximal values of the expression $x+2y$.

Along the route of a bicycle race, $7$ water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also $2$ repair stations evenly spaced between the start and finish lines. The $3$rd water station is located $2$ miles after the $1$st repair station. How long is the race in miles? [asy] size(10cm); filldraw((11,4.5)--(171,4.5)--(171,17.5)--(11,17.5)--cycle,mediumgray); draw((11,11)--(171,11),linetype("4 4")+white+linewidth(1.5)); draw((0,0)--(11,0)--(11,22)--(0,22)--cycle,linewidth(1.125)); draw((171,0)--(182,0)--(182,22)--(171,22)--cycle,linewidth(1.125)); draw((31,4.5)--(31,0)); draw((51,4.5)--(51,0)); draw((151,4.5)--(151,0)); label(scale(.9)*rotate(45)*"Water 1", (23,-13.5)); label(scale(.9)*rotate(45)*"Water 2", (43,-13.5)); label(scale(.9)*rotate(45)*"Water 7", (143,-13.5)); filldraw(circle((101,-13.5),.3)); filldraw(circle((97,-13.5),.3)); filldraw(circle((93,-13.5),.3)); filldraw(circle((89,-13.5),.3)); filldraw(circle((85,-13.5),.3)); label(scale(.9)*rotate(90)*"Start", (5.5,11)); label(scale(.9)*rotate(270)*"Finish", (176.5,11)); [/asy] $\textbf{(A) } 8\qquad\textbf{(B) } 16\qquad\textbf{(C) } 24\qquad\textbf{(D) } 48\qquad\textbf{(E) } 96$

The equation $ x^{n+1} +x=0 $ admits $ 0 $ and $ 1 $ as its unique solutions in a ring of order $ n\ge 2. $ Prove that this ring is a skew field.

Given a sequence of positive integers $$a_1, a_2, a_3, a_4, a_5, \dots$$ such that $a_2 > a_1$ and $a_{n+2} = 3a_{n+1} - 2a_n$ for all $n \geq 1$. Prove that $a_{2021} > 2^{2019}$.

Five points are chosen uniformly at random on a segment of length $1$. What is the expected distance between the closest pair of points?

The distance between any two of five given points exceeds 2. Is it true that the distance between some two of these points exceeds 3 if these five points are in a) the plane; and b) three-dimensional space? [i]Alexey Tolpygo[/i]

Find $f: \mathbb{Z}_+ \rightarrow \mathbb{Z}_+$, such that for any $x,y \in \mathbb{Z}_+$, $$f(f(x)+y)\mid x+f(y).$$

Denote by $B_1$ and $C_1$, the midpoints of the sides $AB$ and $AC$ of the triangle $ABC$. Let the circles circumscribed around the triangles $ABC_1$ and $AB_1C$ intersect at points $A$ and $P$, and let the line $AP$ intersect the circle circumscribed around the triangle $ABC$ at points $A$ and $Q$. Find the ratio $\frac{AQ}{AP}$.