a) Prove that it is impossible to divide a rectangle into five squares of distinct sizes. b) Prove that it is impossible to divide a rectangle into six squares of distinct sizes.

Find prime numbers $p , q , r$ such that $p+q^2+r^3=200$. Give all the possibilities. Remember that the number $1$ is not prime.

Let $k$ be a given integer, $3 < k \leq n$. Consider a graph $G$ with $n$ vertices satisfying the condition: for any two non-adjacent vertices $x$ and $y$ in graph $G$, the sum of their degrees must satisfy $d(x) + d(y) \geq k$. Please answer the following questions and prove your conclusions. (1) Suppose the length of the longest path in graph $G$ is $l$ satisfying the inequality $3 \leq l < k$, does graph $G$ necessarily contain a cycle of length $l+1$? (The length of a path or cycle refers to the number of edges that make up the path or cycle.) (2) For the case where $3 < k \leq n-1$ and graph $G$ is connected, can we determine that the length of the longest path in graph $G$, $l \geq k$? (3) For the case where $3 < k = n-1$, is it necessary for graph $G$ to have a path of length $n-1$ (i.e., a Hamiltonian path)?

An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.

On the side $CD$ of parallelogram $ABCD$ a point $E$ is chosen. The perpendicular from $C$ to $BE$ and the perpendicular from $D$ to $AE$ intersect at $P$. Point $M$ is the midpoint of $PE$. Prove that the perpendicular from $M$ to $CD$ passes through the center of parallelogram $ABCD$. [i]Matsvei Zorka[/i]

A finite set $S$ of positive integers is given. Show that there is a positive integer $N$ dependent only on $S$, such that any $x_1, \dots, x_m \in S$ whose sum is a multiple of $N$, can be partitioned into groups each of whose sum is exactly $N$. (The numbers $x_1, \dots, x_m$ need not be distinct.)

In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by 30. Find the sum of the four terms.

Prove that for every prime number $p$ there exist infinity many natural numbers $n$ so that they satisfy: $2^{2^{2^{ \dots ^{2^n}}}} \equiv n^{2^{2^{\dots ^{2}}}} (mod p)$ Where in both sides $2$ appeared $1397$ times

Let $\Gamma_1$ and $\Gamma_2$ be externally tangent circles with radii lengths $2$ and $6$, respectively, and suppose that they are tangent to and lie on the same side of line $\ell$. Points $A$ and $B$ are selected on $\ell$ such that $\Gamma_1$ and $\Gamma_2$ are internally tangent to the circle with diameter $AB$. If $AB = a + b\sqrt{c}$ for positive integers $a, b, c$ with $c$ squarefree, then find $a + b + c$. [i]Proposed by Andrew Wu, Deyuan Li, and Andrew Milas[/i]

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]

Let $x$,$y$ be positive real numbers such that $\frac{1}{1+x+x^2}+\frac{1}{1+y+y^2}+\frac{1}{1+x+y}=1$.Prove that $xy=1.$

What is the greatest power of 2 that is a factor of $10^{1002}-4^{501}$? $ \textbf{(A) }2^{1002}\qquad\textbf{(B) }2^{1003}\qquad\textbf{(C) }2^{1004}\qquad\textbf{(D) }2^{1005} \qquad\textbf{(E) }2^{1006} \qquad $

An \textit{island} is a contiguous set of at least two equal digits. Let $b(n)$ be the number of islands in the binary representation of $n$. For example, $2020_{10} = 11111100100_2$, so $b(2020) = 3$. Compute \[b(1) + b(2) + \cdots + b(2^{2020}).\]

The sequence of positive real numbers $(x_n)$ is defined recursively as follows: $x_1 = 1$, and for $n \geq 2$, \[ x_{n} = \begin{cases} nx_{n-1} & \quad \text{when }nx_{n-1}\leq 1, \\ \frac{x_{n-1}}{n} &\quad \text{otherwise}. \end{cases} \] Show that there is an integer $N > 1$ such that $|x_N - 1| < 0.00001$. Thus the elements of the sequence can get very close to 1 for large $N$; however, it is easy to see that they can never be 1 unless $N = 1$.

Find all the polynomials $P(t)$ of one variable that fullfill the following for all real numbers $x$ and $y$: $P(x^2-y^2) = P(x+y)P(x-y)$.

The numbers $1,2,\dots, 49$ are written on unit squares of a $7\times 7$ chessboard such that consequtive numbers are on unit squares sharing a common edge. At most how many prime numbers can a row have? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 3 $

Mary told John her score on the American High School Mathematics Examination (AHSME), which was over 80. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over 80, John could not have determined this. What was Mary's score? (Recall that the AHSME consists of 30 multiple-choice problems and that one's score, $s$, is computed by the formula $s = 30 + 4c - w$, where $c$ is the number of correct and $w$ is the number of wrong answers; students are not penalized for problems left unanswered.)

[size=130]In $\triangle ABC$, $\angle A=60^\circ$. $\odot I$ is the incircle of $\triangle ABC$. $\odot I$ is tangent to sides $AB$, $AC$ at $D$, $E$, respectively. Line $DE$ intersects line $BI$ and $CI$ at $F$, $G$ respectively. Prove that [/size]$FG=\frac{BC}{2}$.

We are given a convex quadrilateral and point $M$ inside it . The perimeter of the quadrilateral has length $L$ while the lengths of the diagonals are $D_1$ and $D_2$. Prove that the sum of the distances from $M$ to the vertices of the quadrilateral are not greater than $L + D_1 + D_2$ . (V. Prasolov)

Given a rectangular grid, split into $m\times n$ squares, a colouring of the squares in two colours (black and white) is called valid if it satisfies the following conditions: [list] [*]All squares touching the border of the grid are coloured black. [*]No four squares forming a $2\times 2$ square are coloured in the same colour. [*]No four squares forming a $2\times 2$ square are coloured in such a way that only diagonally touching squares have the same colour.[/list] Which grid sizes $m\times n$ (with $m,n\ge 3$) have a valid colouring?

Find all non-negative integers $x, y$ and primes $p$ such that $$3^x+p^2=7 \cdot 2^y.$$

Let $\omega$ denote the excircle tangent to side $BC$ of triangle $ABC$. A line $\ell$ parallel to $BC$ meets sides $AB$ and $AC$ at points $D$ and $E$, respectively. Let $\omega'$ denote the incircle of triangle $ADE$. The tangent from $D$ to $\omega$ (different from line $AB$) and the tangent from $E$ to $\omega$ (different from line $AC$) meet at point $P$. The tangent from $B$ to $\omega'$ (different from line $AB$) and the tangent from $C$ to $\omega'$ (different from line $AC$) meet at point $Q$. Prove that, independent of the choice of $\ell$, there is a fixed point that line $PQ$ always passes through.

A ballot box contains the votes for the election of two candidates $A$ and $B$. It is known that candidate $A$ has $6$ votes and candidate $B$ has $9$. Find the probability that, when carrying out the scrutiny, candidate $B$ always goes first.

Are there natural $a, b >1000$ , such that for any $c$ that is a perfect square, the three numbers $a, b$ and $c$ are not the lengths of the sides of a triangle?

let $f(x) = \sum_{n=0}^{\infty} 2^{-n} ||2^n x||$ , where ||x|| is the distance between x and the closest integer to x. Are the level sets $\{ x \in [0,1] : f(x)=y \}$ Lebesgue measurable for almost all $y \in f(R)$?