Let $[n]=\{1,2,...,n\}$.Let $p$ be any prime number. Find how many finite non-empty sets $S\in [p] \times [p]$ are such that $$\displaystyle \large p | \sum_{(x,y) \in S}{x},p | \sum_{(x,y) \in S}{y}$$

nic$\kappa$y is drawing kappas in the cells of a square grid. However, he does not want to draw kappas in three consecutive cells (horizontally, vertically, or diagonally). Find all real numbers $d>0$ such that for every positive integer $n,$ nic$\kappa$y can label at least $dn^2$ cells of an $n\times n$ square. [i]Proposed by Mihir Singhal and Michael Kural[/i]

Find a pair of coprime positive integers $(m,n)$ other than $(41,12)$ such that $m^2-5n^2$ and $m^2+5n^2$ are both perfect squares.

The number 104,060,465 is divisible by a five-digit prime number. What is that prime number?

The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at $O$. The circumcircles of triangles $AOB$ and $COD$ intersect again at $K$. Point $L$ is such that the triangles $BLC$ and $AKD$ are similar and equally oriented. Prove that if the quadrilateral $BLCK$ is convex, then it is tangent [has an incircle].

A cylindrical container has height $6 cm$ and radius $4 cm$. It rests on a circular hoop, also of radius $4 cm$, fixed in a horizontal plane with its axis vertical and with each circular rim of the cylinder touching the hoop at two points. The cylinder is now moved so that each of its circular rims still touches the hoop in two points. Find with proof the locus of one of the cylinder’s vertical ends.

Six regular octagons each with sides of length 2 are placed in a two-by-three array and inscribed in a square as shown. The area of the square can be written in the form $m+n\sqrt{2}$ where $m$ and $n$ are positive integers. Find $m+n$. [asy] size(175); pair A=dir(-45/2); real h=A.x; path P=A; for(int k=1;k<8;++k) P=P--rotate(45*k)*A; P=P--cycle; for(real x=-2*h;x<3*h;x+=2*h) for(real y:new real[]{-h,h} ) draw (shift((x,y))*P); pair C=dir(45/2)+(2*h,h), X=(C.x+C.y,0), Y=(0,C.x+C.y); draw(X--Y--(-X)--(-Y)--cycle);[/asy]

Given a positive integer $n$, define $\tau(n)$ as the number of positive divisors of $n$ and $\sigma(n)$ as the sum of those divisors. For example, $\tau(12) = 6$ and $\sigma(12) = 28$. Find all positive integers $n$ that satisfy: \[ \sigma(n) = \tau(n) \cdot \lceil \sqrt{n} \rceil \]

Let $f(x)=x^{n}+5x^{n-1}+3$, where $n>1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

Joy has a square board of size $n \times n$. At every step, he colours a cell of the board. He cannot colour any cell more than once. He also counts points while colouring the cells. At first, he has $0$ points. Every step, after colouring a cell $c$, he takes the largest possible set $S$ that creates a "$+$" sign where all cells are coloured and $c$ lies in the centre. Then, he gets the size of set $S$ as points. After colouring the whole $n \times n$ board, what is the maximum possible amount of points he can get?

Define $a_{n}$ as satisfying: $\left(1+\frac{1}{n}\right)^{n+a_{n}}=e$. Find $\lim_{n\rightarrow\infty}a_{n}$.

A $tetromino$ is a figure made up of four unit squares connected by common edges. [List=i] [*] If we do not distinguish between the possible rotations of a tetromino within its plane, prove that there are seven distinct tetrominos. [*]Prove or disprove the statement: It is possible to pack all seven distinct tetrominos into $4\times 7$ rectangle without overlapping. [/list]

We call a positive integer [i]heinersch[/i] if it can be written as the sum of a positive square and positive cube. Prove: There are infinitely many heinersch numbers $h$, such that $h-1$ and $h+1$ are also heinersch.

We want to partition the integers $1, 2, 3, . . . , 100$ into several groups such that within each group either any two numbers are coprime or any two are not coprime. At least how many groups are needed for such a partition? [i]We call two integers coprime if they have no common divisor greater than $1$.[/i]

Determine all non-constant polynomials $X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0$ with integer coefficients for which the roots are exactly the numbers $a_0,a_1,\ldots ,a_{n-1}$ (with multiplicity).

Consider all segments dividing the area of a triangle $ABC$ in two equal parts. Find the length of the shortest segment among them, if the side lengths $a,$ $b,$ $c$ of triangle $ABC$ are given. How many of these shortest segments exist ?

Four $L$s are equivalent to three $M$s. Nine $M$s are equivalent to fourteen $T$ s. Seven $T$ s are equivalent to two $W$ s. If Kevin has thirty-six $L$s, how many $W$ s would that be equivalent to?

Mr. and Mrs. Zeta want to name their baby Zeta so that its monogram (first, middle, and last initials) will be in alphabetical order with no letter repeated. How many such monograms are possible? $\text{(A)} \ 276 \qquad \text{(B)} \ 300 \qquad \text{(C)} \ 552 \qquad \text{(D)} \ 600 \qquad \text{(E)} \ 15600$

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Find all positive integers $a, b, c$ such that $3abc+11(a+b+c)=6(ab+bc+ac)+18.$

This graph depicts the torque output of a hypothetical gasoline engine as a function of rotation frequency. The engine is incapable of running outside of the graphed range. [asy] size(200); draw((0,0)--(10,0)--(10,7)--(0,7)--cycle); for (int i=1;i<10;++i) { draw((i,0)--(i,7),dashed+linewidth(0.5)); } for (int j=1;j<7;++j) { draw((0,j)--(10,j),dashed+linewidth(0.5)); } draw((0,0)--(0,-0.3)); draw((4,0)--(4,-0.3)); draw((8,0)--(8,-0.3)); draw((0,0)--(-0.3,0)); draw((0,2)--(-0.3,2)); draw((0,4)--(-0.3,4)); draw((0,6)--(-0.3,6)); label("0",(0,-0.5),S); label("1000",(4,-0.5),S); label("2000",(8,-0.5),S); label("0",(-0.5,0),W); label("10",(-0.5,2),W); label("20",(-0.5,4),W); label("30",(-0.5,6),W); label("I",(1,-1.5),S); label("II",(6,-1.5),S); label("III",(9,-1.5),S); label(scale(0.95)*"Engine Revolutions per Minute",(5,-3.5),N); label(scale(0.95)*rotate(90)*"Output Torque (Nm)",(-1.5,3),W); path A=(0.9,2.7)--(1.213, 2.713)-- (1.650, 2.853)-- (2.087, 3)-- (2.525, 3.183)-- (2.963, 3.471)-- (3.403, 3.888)-- (3.823, 4.346)-- (4.204, 4.808)-- (4.565, 5.277)-- (4.945, 5.719)-- (5.365, 6.101)-- (5.802, 6.298)-- (6.237, 6.275)-- (6.670, 6.007)-- (7.101, 5.600)-- (7.473, 5.229)-- (7.766, 4.808)-- (8.019, 4.374)-- (8.271, 3.894)-- (8.476, 3.445)-- (8.568, 2.874)-- (8.668, 2.325)-- (8.765, 1.897)-- (8.794, 1.479)--(8.9,1.2); draw(shift(0.1*right)*shift(0.2*down)*A,linewidth(3)); [/asy] At what engine RPM (revolutions per minute) does the engine produce maximum power? (A) $\text{I}$ (B) At some point between $\text{I}$ and $\text{II}$ (C) $\text{II}$ (D) At some point between $\text{II}$ and $\text{III}$ (E) $\text{III}$

[asy] for(int a=0; a<4; ++a) { draw((a,0)--(a,3)); } for(int b=0; b<4; ++b) { draw((0,b)--(3,b)); }[/asy] Placing no more than one $x$ in each small square, what is the greatest number of $x$'s that can be put on the grid shown without getting three $x$'s in a row vertically, horizontally, or diagonally? $ \text{(A)}\ 2\qquad\text{(B)}\ 3\qquad\text{(C)}\ 4\qquad\text{(D)}\ 5\qquad\text{(E)}\ 6 $

([b]8[/b]) Evaluate the integral $ \int_0^1\ln x \ln(1\minus{}x)\ dx$.

Let $ABCDEF$ be a regular hexagon with side length $2$. A circle with radius $3$ and center at $A$ is drawn. Find the area inside quadrilateral $BCDE$ but outside the circle. [i]Proposed by Carl Joshua Quines.[/i]

Let $m$ and $n$ be arbitrary non-negative integers. Prove that \[\frac{(2m)!(2n)!}{m! n!(m+n)!}\] is an integer.

$101$ people, sitting at a round table in any order, had $1,2,... , 101$ cards, respectively. A transfer is someone give one card to one of the two people adjacent to him. Find the smallest positive integer $k$ such that there always can through no more than $ k $ times transfer, each person hold cards of the same number, regardless of the sitting order.