Determine all positive integers $k$, $\ell$, $m$ and $n$, such that $$\frac{1}{k!}+\frac{1}{\ell!}+\frac{1}{m!} =\frac{1}{n!} $$

$10$ teams participated in a football tournament: every two teams played each other exactly once. After the end of the tournament it turned out that all teams got different amount of points and some teams won more games, than the winner of the tournament, call them strong. What is the maximum number of teams that could be strong? (In football the winner of the match gets three points, the loser - 0 points, and if the match ends in a draw both teams get 1 point.)

$ \lim_{n\to\infty }\left( \frac{1}{e}\sum_{i=0}^n \frac{1}{i!} \right)^{n!} $

Consider two coprime integers $p{}$ and $q{}$ which are greater than $1{}$ and differ from each other by more than $1{}$. Prove that there exists a positive integer $n{}$ such that \[\text{lcm}(p+n, q+n)<\text{lcm}(p,q).\]

Let $ABC$ be an equilateral triangle such that length of its altitude is $1$. Circle with center on the same side of line $AB$ as point $C$ and radius $1$ touches side $AB$. Circle rolls on the side $AB$. While the circle is rolling, it constantly intersects sides $AC$ and $BC$. Prove that length of an arc of the circle, which lies inside the triangle, is constant

A four-element set $\{a, b, c, d\}$ of positive integers is called [i]good[/i] if there are two of them such that their product is a mutiple of the greatest common divisor of the remaining two. For example, the set $\{2, 4, 6, 8\}$ is good since the greatest common divisor of $2$ and $6$ is $2$, and it divides $4\times 8=32$. Find the greatest possible value of $n$, such that any four-element set with elements less than or equal to $n$ is good. [i]Proposed by Victor and Isaías de la Fuente[/i]

Given the rational numbers $r$, $q$, and $n$, such that $\displaystyle\frac1{r+qn}+\frac1{q+rn}=\frac1{r+q}$, prove that $\displaystyle\sqrt{\frac{n-3}{n+1}}$ is a rational number.

Prove that for every integer $n$ greater than $1:$ $$\frac{3n+1}{2n+2} < \left( \frac{1}{n} \right)^{n} + \left( \frac{2}{n} \right)^{n}+ \ldots+\left( \frac{n}{n} \right)^{n} <2.$$

Prove that you can select two adjacent sides of any quadrilateral and supplement them in order to create a parallelogram, the resulting parallelogram contains the original quadrilateral .

A positive number, if its fractional part, integeral part, and itself are geometric series, then the number is________.

The point $P$, from the plane in which $\Delta ABC$ lies, is such that if $A_1,B_1$, and $C_1$ are the orthogonal projections of $P$ on the respective altitudes of $ABC$, then $AA_1=BB_1=CC_1=t$. Determine the locus of $P$ and length of $t$.

In triangle $ABC$, angle $B$ is not right. The circle inscribed in $ABC$ touches $AB$ and $BC$ at points $C_1$ and $A_1$, and the feet of the altitudes drawn to the sides $AB$ and $BC$ are points $C_2$ and $A_2$. Prove that the intersection point of the altitudes of triangle $A_1BC_1$ is the center of the circle inscribed in triangle $A_2BC_2$.

Let $a_1,a_2,\ldots ,a_n\ge 0.$ For all $1\le k\le n$ define $$b_k:=\min_{1\le i<j\le n,j-i\le 2}|2a_k-a_i-a_j|.$$ Here the index mod $n.$ Find the maximum value of $\frac{b_1+b_2+\cdots +b_n}{a_1+a_2+\cdots +a_n}.$ [i]Proposed by Zheng Wang[/i]

Let $ a$, $ b$, and $ c$ be positive real numbers such that $ ab + bc + ca = 3$. Prove the inequality \[ 3 + \sum_{\mathrm{\cyc}} (a - b)^2 \ge \frac {a + b^2c^2}{b + c} + \frac {b + c^2a^2}{c + a} + \frac {c + a^2b^2}{a + b} \ge 3. \]

Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \] cannot be divided by $5$.

The positive integers from 1 to 100 are painted into three colors: 50 integers are red, 25 integers are yellow and 25 integers are green. The red and yellow integers can be divided into 25 triples such that each triple includes two red integers and one yellow integer which is greater than one of the red integers and smaller than another one. The same assertion is valid for the red and green integers. Is it necessarily possible to divide all the 100 integers into 25 quadruples so that each quadruple includes two red integers, one yellow integer and one green integer such that the yellow and the green integer lie between the red ones? [i]Alexandr Gribalko[/i]

Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

Given two positive integers $x,y$, we define $z=x\,\oplus\,y$ to be the bitwise XOR sum of $x$ and $y$; that is, $z$ has a $1$ in its binary representation at exactly the place values where $x,y$ have differing binary representations. It is known that $\oplus$ is both associative and commutative. For example, $20 \oplus 18 = 10100_2 \oplus 10010_2 = 110_2 = 6$. Given a set $S=\{a_1, a_2, \dots, a_n\}$ of positive integers, we let $f(S) = a_1 \oplus a_2 \oplus a_3\oplus \dots \oplus a_n$. We also let $g(S)$ be the number of divisors of $f(S)$ which are at most $2018$ but greater than or equal to the largest element in $S$ (if $S$ is empty then let $g(S)=2018$). Compute the number of $1$s in the binary representation of $\displaystyle\sum_{S\subseteq \{1,2,\dots, 2018\}} g(S)$. [i]Proposed by Brandon Wang and Vincent Huang

Find all pair of integers $(m,n)$ and $m \ge n$ such that there exist a positive integer $s$ and a) Product of all divisor of $sm, sn$ are equal. b) Number of divisors of $sm,sn$ are equal.

Let $n$ be a fixed natural number. [b]a)[/b] Find all solutions to the following equation : \[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \] [b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) : \[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]

Each morning of her five-day workweek, Jane bought either a $ 50$-cent muffin or a $ 75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Let $a$ be a real number. Find the minimum value of $\int_0^1 |ax-x^3|dx$. How many solutions (including University Mathematics )are there for the problem? Any advice would be appreciated.

Let $P$ be a convex planar polygon with equal angles. Let $l_1,\cdots, l_n$ be its sides. Show that a necessary and sufficient condition for $P$ to be regular is that the sum of the ratios $\frac{l_i}{l_{i+1}} (i = 1,\cdots, n; l_{n+1}= l_1)$ equals the number of sides.

Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent: [b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$ [b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.

Find all the triples of positive integers $(a,b,c)$ for which the number \[\frac{(a+b)^4}{c}+\frac{(b+c)^4}{a}+\frac{(c+a)^4}{b}\] is an integer and $a+b+c$ is a prime.