Find all triples of non-negative integer numbers $(E, C, U)$ such that $EC \ge 1$ and: $$2^{3^E}+3^{2^C}=593 \cdot 5^U$$

Given any positive real number $\varepsilon$, prove that, for all but finitely many positive integers $v$, any graph on $v$ vertices with at least $(1+\varepsilon)v$ edges has two distinct simple cycles of equal lengths. (Recall that the notion of a simple cycle does not allow repetition of vertices in a cycle.) [i]Fedor Petrov, Russia[/i]

Given a triangle $ABC$. Let $S$ be the circle passing through $C$, centered at $A$. Let $X$ be a variable point on $S$ and let $K$ be the midpoint of the segment $CX$ . Find the locus of the midpoints of $BK$, when $X$ moves along $S$. (I. Gorodnin)

On the circumference of a circle are $21$ points. Prove that among the arcs which join any two of these points, at least $100$ of them must subtend an angle at the centre of the circle not exceeding $120^o$ . ( A . F . Sidorenko)

What is the maximum number of possible points of intersection of a circle and a triangle? $ \textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$

Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$, and let $O$ be the center of its circumcircle. Show that the segments $OA$, $OF$, $OB$, $OD$, $OC$, $OE$ dissect the triangle $ABC$ into three pairs of triangles that have equal areas.

Let $A, B, C, A', B', C'$ be six pairs of different points. Prove that the Circles $BCA'$, $CAB'$ and $ABC'$ have a common point, then the Circles $B'C'A, C'A'B$ and $A'B'C$ also share a common point. Note: For three pairs of different points $X, Y$ and $Z$ we define the [i]Circle [/i] $XYZ$ as the circumcircle of the triangle $XYZ$, or - in the case when the points $X, Y$ and $Z$ lie on a straight line - this straight line.

We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.

In a school there is a person with $l$ friends for all $1 \leq l \leq 99$. If there is no trio of students in this school, all three of whom are friends with each other, what is the minimum number of students in the school?

Find all values of the positive integer $ m$ such that there exists polynomials $ P(x),Q(x),R(x,y)$ with real coefficient satisfying the condition: For every real numbers $ a,b$ which satisfying $ a^m-b^2=0$, we always have that $ P(R(a,b))=a$ and $ Q(R(a,b))=b$.

What's the largest number of elements that a set of positive integers between $1$ and $100$ inclusive can have if it has the property that none of them is divisible by another?

The graph below shows the total accumulated dollars (in millions) spent by the Surf City government during $1988$. For example, about $.5$ million had been spent by the beginning of February and approximately $2$ million by the end of April. Approximately how many millions of dollars were spent during the summer months of June, July, and August? $\text{(A)}\ 1.5 \qquad \text{(B)}\ 2.5 \qquad \text{(C)}\ 3.5 \qquad \text{(D)}\ 4.5 \qquad \text{(E)}\ 5.5$ [asy] unitsize(18); for (int a=1; a<13; ++a) { draw((a,0)--(a,.5)); } for (int b=1; b<6; ++b) { draw((-.5,2b)--(0,2b)); } draw((0,0)--(0,12)); draw((0,0)--(14,0)); draw((0,0)--(1,.9)--(2,1.9)--(3,2.6)--(4,4.3)--(5,4.5)--(6,5.7)--(7,8.2)--(8,9.4)--(9,9.8)--(10,10.1)--(11,10.2)--(12,10.5)); label("J",(.5,0),S); label("F",(1.5,0),S); label("M",(2.5,0),S); label("A",(3.5,0),S); label("M",(4.5,0),S); label("J",(5.5,0),S); label("J",(6.5,0),S); label("A",(7.5,0),S); label("S",(8.5,0),S); label("O",(9.5,0),S); label("N",(10.5,0),S); label("D",(11.5,0),S); label("month F=February",(16,0),S); label("$1$",(-.6,2),W); label("$2$",(-.6,4),W); label("$3$",(-.6,6),W); label("$4$",(-.6,8),W); label("$5$",(-.6,10),W); label("dollars in millions",(0,11.9),N); [/asy]

Let $x$ and $y$ be distinct real numbers such that \[ \sqrt{x^2+1}+\sqrt{y^2+1}=2021x+2021y. \] Find, with proof, the value of \[ \left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right). \]

Find all triangles with integer sidelenghts such that their perimeter and area are equal.

For a positive integer $m$ denote by $S(m)$ and $P(m)$ the sum and product, respectively, of the digits of $m$. Show that for each positive integer $n$, there exist positive integers $a_1, a_2, \ldots, a_n$ satisfying the following conditions: \[ S(a_1) < S(a_2) < \cdots < S(a_n) \text{ and } S(a_i) = P(a_{i+1}) \quad (i=1,2,\ldots,n). \] (We let $a_{n+1} = a_1$.) [i]Problem Committee of the Japan Mathematical Olympiad Foundation[/i]

The sequence $(a_n)$ is defined by $a_0 = 0$ and $a_{n+1} = [\sqrt[3]{a_n +n}]^3$ for $n \ge 0$. (a) Find $a_n$ in terms of $n$. (b) Find all $n$ for which $a_n = n$.

If $p$ and $q$ are distinct prime numbers, then there are integers $x_0$ and $y_0$ such that $1 = px_0 + qy_0.$ Determine the maximum value of $b - a$, where $a$ and $b$ are positive integers with the following property: If $a \leq t \leq b$, and $t$ is an integer, then there are integers $x$ and $y$ with $0 \leq x \leq q - 1$ and $0 \leq y \leq p - 1$ such that $t = px + qy.$

Does there exist a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that \[f(f(n-1)=f(n+1)-f(n)\quad\text{for all}\ n\ge 2\text{?} \]

Let $G,A_1,A_2,A_3,A_4,B_1,B_2,B_3,B_4,B_5$ be ten points on a circle such that $GA_1A_2A_3A_4$ is a regular pentagon and $GB-1B_2B_3B_4B_5$ is a regular hexagon, and $B_1$ lies on minor arc $GA_1$. Let $B_5B_3$ intersect $B_1A_2$ at $G_1$, and let $B_5A_3$ intersect $GB_3$ at $G_2$. Determine the degree measure of $\angle GG2G_1$.

You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

Let $x$ and $y$ be odd positive integers. A table $x$ x $y$ is given in which the squares with coordinates $(2,1)$, $(x - 2, y)$, and $(x, y)$ are cut. The remaining part of the table is covered in dominoes and squares [b]2 x 2[/b]. Prove that the dominoes in a valid covering of the table are at least $\frac{3}{2}(x+y)-6$

There are $n>1$ cities in the country, some pairs of cities linked two-way through straight flight. For every pair of cities there is exactly one aviaroute (can have interchanges). Major of every city X counted amount of such numberings of all cities from $1$ to $n$ , such that on every aviaroute with the beginning in X, numbers of cities are in ascending order. Every major, except one, noticed that results of counting are multiple of $2016$. Prove, that result of last major is multiple of $2016$ too.

Semi-circle $AB$ with diameter $AB$, and $AB=2r$. Given line $l$, satisfying that $l \perp BA, l \cap BA=T , |AT|=2a(2a<r)$. $M,N$ are two points on the semi-circle, such that $$d(M,l)=|AM|,d(N,l)=|AN|(M\neq N).$$ Prove: $|AM|+|AN|=|AB|$.

Let $M$ and $N$ be positive integers. Pisut walks from point $(0, N)$ to point $(M, 0)$ in steps so that $\bullet$ each step has unit length and is parallel to either the horizontal or the vertical axis, and $\bullet$ each point ($x, y)$ on the path has nonnegative coordinates, i.e. $x, y > 0$. During each step, Pisut measures his distance from the axis parallel to the direction of his step, if after the step he ends up closer from the origin (compared to before the step) he records the distance as a positive number, else he records it as a negative number. Prove that, after Pisut completes his walk, the sum of the signed distances Pisut measured is zero.

A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$. [asy] import three; import solids; size(5cm); currentprojection=orthographic(1,-1/6,1/6); draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,360)),white,nolight); triple A =(8*sqrt(6)/3,0,8*sqrt(3)/3), B = (-4*sqrt(6)/3,4*sqrt(2),8*sqrt(3)/3), C = (-4*sqrt(6)/3,-4*sqrt(2),8*sqrt(3)/3), X = (0,0,-2*sqrt(2)); draw(X--X+A--X+A+B--X+A+B+C); draw(X--X+B--X+A+B); draw(X--X+C--X+A+C--X+A+B+C); draw(X+A--X+A+C); draw(X+C--X+C+B--X+A+B+C,linetype("2 4")); draw(X+B--X+C+B,linetype("2 4")); draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,240)),white,nolight); draw((-2,-2*sqrt(3),0)..(4,0,0)..(-2,2*sqrt(3),0)); draw((-4*cos(atan(5)),-4*sin(atan(5)),0)--(-4*cos(atan(5)),-4*sin(atan(5)),-10)..(4,0,-10)..(4*cos(atan(5)),4*sin(atan(5)),-10)--(4*cos(atan(5)),4*sin(atan(5)),0)); draw((-2,-2*sqrt(3),0)..(-4,0,0)..(-2,2*sqrt(3),0),linetype("2 4")); [/asy]