Distinct prime numbers $p,q,r$ satisfy the equation $$2pqr+50pq=7pqr+55pr=8pqr+12qr=A$$ for some positive integer $A.$ What is $A$?

Two circles $C_1$ and $C_2$ intersect in points $A$ and $B$. A circle $C$ with center in $A$ intersect $C_1$ in $M$ and $P$ and $C_2$ in $N$ and $Q$ so that $N$ and $Q$ are located on different sides wrt $MP$ and $AB> AM$. Prove that $\angle MBQ = \angle NBP$.

Find the sum of the numbers written with two digits $\overline{ab}$ for which the equation $3^{x + y} =3^x + 3^y + \overline{ab}$ has at least one solution $(x, y)$ in natural numbers.

Let $N$ be the number of functions $f:\{1,2,3,4,5,6,7,8,9,10\} \rightarrow \{1,2,3,4,5\}$ that have the property that for $1\leq x\leq 5$ it is true that $f(f(x))=x$. Given that $N$ can be written in the form $5^a\cdot b$ for positive integers $a$ and $b$ with $b$ not divisible by $5$, find $a+b$. [i]Proposed by Nathan Ramesh

Prove that\[(1+\frac{1}{3})(1+\frac{1}{3^2})\cdots(1+\frac{1}{3^n})< 2.\]

Find any polynomial with integer coefficients, the smallest value of which on the entire line is equal to : a) $-\sqrt2$ b) $\sqrt2$

Prove that there exists a partition of the set $A = \{1^3, 2^3, \ldots , 2000^3\}$ into $19$ nonempty subsets such that the sum of elements of each subset is divisible by $2001^2$.

Prove that if $a,b,c$ are positive numbers with $abc=1$, then \[\frac{a}{b} +\frac{b}{c} + \frac{c}{a} \ge a + b + c. \]

[b]a)[/b] Determine the center of the ring of square matrices of a certain dimensions with elements in a given field, and prove that it is isomorphic with the given field. [b]b)[/b] Prove that $$ \left(\mathcal{M}_n\left( \mathbb{R} \right) ,+, \cdot\right)\not\cong \left(\mathcal{M}_n\left( \mathbb{C} \right) ,+,\cdot\right) , $$ for any natural number $ n\ge 2. $ [i]Marian Andronache, Ion Sava[/i]

For $n\in\mathbb{N}$, let $a_{n}$ denote the closest integer to $\sqrt{n}$. Evaluate \[\sum_{n=1}^\infty{\frac{1}{a_{n}^{3}}}.\]

(a) Let $a, b, c, d$ be integers such that $ad\ne bc$. Show that is always possible to write the fraction $\frac{1}{(ax+b)(cx+d)}$in the form $\frac{r}{ax+b}+\frac{s}{cx+d}$ (b) Find the sum $$\frac{1}{1 \cdot 4}+\frac{1}{4 \cdot 7}+\frac{1}{7 \cdot 10}+...+\frac{1}{1995 \cdot 1996}$$

There is a population $P$ of $10000$ bacteria, some of which are friends (friendship is mutual), so that each bacterion has at least one friend and if we wish to assign to each bacterion a coloured membrane so that no two friends have the same colour, then there is a way to do it with $2021$ colours, but not with $2020$ or less. Two friends $A$ and $B$ can decide to merge in which case they become a single bacterion whose friends are precisely the union of friends of $A$ and $B$. (Merging is not allowed if $A$ and $B$ are not friends.) It turns out that no matter how we perform one merge or two consecutive merges, in the resulting population it would be possible to assign $2020$ colours or less so that no two friends have the same colour. Is it true that in any such population $P$ every bacterium has at least $2021$ friends?

A cup with a volume of $8$ fluid ounces is filled at the rate of $0.5$ ounces per second. However, a hole at the bottom of the cup also drains it at the rate of $0.3$ ounces per second. Once the cup is full, how many ounces of water will have drained out of the cup?

Consider an orange and black coloring of a $20 \times 14$ square grid. Let $n$ be the number of colorings such that every row and column has an even number of orange squares. Evaluate $\log_2 n$.

How many three-digit numbers $\underline{abc}$ have the property that when it is added to $\underline{cba}$, the number obtained by reversing its digits, the result is a palindrome? (Note that $\underline{cba}$ is not necessarily a three-digit number since before reversing, $c$ may be equal to $0$.)

In a class with $n$ students in the span of $k$ days, each day are chosen three to be tested. Each two students can be taken in such triple only once. Prove that for the greatest $k$ satisfying these conditions, the following inequalities are true: $\frac{n(n-3)}{6}\leq k\leq \frac{n(n-1)}{6}$.

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Squirrels $A$ and $B$ have $360$ nuts. $A$ divides these nuts into five non-empty heaps and $B$ chooses three heaps. If the total number of nuts in these heaps is divisible by the total number of nuts in other two heaps then $A$ wins. Otherwise $B$ wins. Which of the squirrels has a winning strategy?

Inside the acute-angled triangle $ABC$, mark the point $O$ so that $\angle AOB=90^o$, a point $M$ on the side $BC$ such that $\angle COM=90^o$, and a point $N$ on the segment $BO$ such that $\angle OMN = 90^o$. Let $P$ be the point of intersection of the lines $AM$ and $CN$, and let $Q$ be a point on the side $AB$ that such $\angle POQ = 90^o$. Prove that the lines $AN, CO$ and $MQ$ intersect at one point.

All pages of a magazine are numbered and printed on both sides. One sheet with two sides is missing. The numbers of the remaining pages sum to $963$. How many pages did the magazine have originally and which pages are missing?

Prove that there are two natural numbers $ p,q, $ satisfying $$ p<q<n\bigg|p+(p+1)+\cdots +(q-1) +q, $$ if and only if $ n $ is not a power of $ 2. $

As shown in the figure, line segment $\overline{AD}$ is trisected by points $B$ and $C$ so that $AB=BC=CD=2.$ Three semicircles of radius $1,$ $\overarc{AEB},\overarc{BFC},$ and $\overarc{CGD},$ have their diameters on $\overline{AD},$ and are tangent to line $EG$ at $E,F,$ and $G,$ respectively. A circle of radius $2$ has its center on $F. $ The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form \[\frac{a}{b}\cdot\pi-\sqrt{c}+d,\] where $a,b,c,$ and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d$? [asy] size(6cm); filldraw(circle((0,0),2), gray(0.7)); filldraw(arc((0,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((-2,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((2,-1),1,0,180) -- cycle, gray(1.0)); dot((-3,-1)); label("$A$",(-3,-1),S); dot((-2,0)); label("$E$",(-2,0),NW); dot((-1,-1)); label("$B$",(-1,-1),S); dot((0,0)); label("$F$",(0,0),N); dot((1,-1)); label("$C$",(1,-1), S); dot((2,0)); label("$G$", (2,0),NE); dot((3,-1)); label("$D$", (3,-1), S); [/asy] $\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16\qquad\textbf{(E) } 17$

Find the greatest integer less than or equal to $\sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1}.$

It is known that each vertex of the convex polyhedron $P$ belongs to three different faces, and each vertex of $P$ can be dyed black and white, so that the two endpoints of each edge of $P$ are different colors. Proof: The interior of each edge of $P$ can be dyed red, yellow, and blue, so that the colors of the three edges connected to each vertex are different, and each face contains two colors of edges. [i]Created by Liang Xiao[/i]

Let $ABC$ be a triangle with sides $AB = 19$, $BC = 21$ and $AC = 20$. Let $\omega$ be the incircle of $ABC$ with center $I$. Extend $BI$ so that it intersects $AC$ at $E$. If $\omega$ is tangent to $AC$ at the point $D$, then find the length of $DE$.

Determine all functions $f:\mathbb{Z}^+\to \mathbb{Z}$ which satisfy the following condition for all pairs $(x,y)$ of [i]relatively prime[/i] positive integers: \[f(x+y) = f(x+1) + f(y+1).\]