Let $a$ and $b$ be positive integers. Show that if $a^3+b^3$ is the square of an integer, then $a + b$ is not a product of two different prime numbers.

It is known that from segments of lengths $a$, $b$ and $c$, a triangle can be formed. Could it happen that from segments of lengths $$\sqrt{a^2 + \frac{2}{3} bc},\quad \sqrt{b^2 + \frac{2}{3} ca}\quad \text{and} \quad \sqrt{c^2 + \frac{2}{3} ab},$$ a right-angled triangle can be formed?

$ (a)$ Prove that $ \mathbb{N}$ can be partitioned into three (mutually disjoint) sets such that, if $ m,n \in \mathbb{N}$ and $ |m\minus{}n|$ is $ 2$ or $ 5$, then $ m$ and $ n$ are in different sets. $ (b)$ Prove that $ \mathbb{N}$ can be partitioned into four sets such that, if $ m,n \in \mathbb{N}$ and $ |m\minus{}n|$ is $ 2,3,$ or $ 5$, then $ m$ and $ n$ are in different sets. Show, however, that $ \mathbb{N}$ cannot be partitioned into three sets with this property.

Find the limit as below. \[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]

Let $n$ be a positive integer. In how many ways can an $n \times n$ table be filled with integers from $0$ to $5$ such that a) the sum of each row is divisible by $2$ and the sum of each column is divisible by $3$ b) the sum of each row is divisible by $2$, the sum of each column is divisible by $3$ and the sum of each of the two diagonals is divisible by $6$?

Prove that it is impossible to arrange the numbers $1,2,\ldots,1997$ around a circle in such a way that, if $x$ and $y$ are any two neighboring numbers, then $499\le|x-y|\le997$.

Let $I$ be the intersection point of the bisectors of triangle $ABC$. Let us denote by $A', B', C'$ the points symmetrical to $I$ wrt the sides triangle $ABC$. Prove that if a circle circumscribes around triangle $A'B'C'$ passes through vertex $B$, then $\angle ABC = 60^o$.

Pete claims that he can draw $4$ segments of length $1$ and a circle of radius less than $\sqrt3 /3 $ on a piece of paper, such that all segments would lie inside the circle and there would be no line that intersects each of $4$ segments. Is Pete right?

Prove that for each natural number $d$, There is a monic and unique polynomial of degree $d$ like $P$ such that $P(1)$≠$0$ and for each sequence like $a_{1}$,$a_{2}$, $...$ of real numbers that the recurrence relation below is true for them, there is a natural number $k$ such that $0=a_{k}=a_{k+1}= ...$ : $P(n)a_{1} + P(n-1)a_{2} + ... + P(1)a_{n}=0$ $n>1$

Given positive integers $m,n$ and a $m\times n$ matrix $A$ with real entries. (1) Show that matrices $X = I_m + AA^T$ and $Y = I_n + A^T A$ are invertible. ($I_l$ is the $l\times l$ unit matrix.) (2) Evaluate the value of $\text{tr}(X^{-1}) - \text{tr}(Y^{-1})$.

For given positive integer $n$ find all quartets $(x_1,x_2,x_3,x_4)$ such that $x_1^2+x_2^2+x_3^2+x_4^2=4^n$

It is given $n$ a natural number and a circle with circumference $n$. On the circle, in clockwise direction, numbers $0,1,2,\dots n-1$ are written, in this order and in the same distance to each other. Every number is colored red or blue, and there exists a non-zero number of numbers of each color. It is known that there exists a set $S\subsetneq \{0,1,2,\dots n-1\}, |S|\geq 2$, for wich it holds: if $(x,y), x<y$ is a circle sector whose endpoints are of distinct colors, whose distance $y-x$ is in $S$, then $y$ is in $S$. Prove that there is a divisor $d$ of $n$ different from $1$ and $n$ for wich holds: if $(x,y),x<y$ are different points of distinct colors, such that their distance is divisible by $d$, then both $x,y$ are divisible by $d$.

Let $a, b, c$ be the side-lengths of a triangle, $r$ be the inradius and $r_a, r_b, r_c$ be the corresponding exradius. Show that \[ \frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \leq 2 \cdot \frac{\sqrt{{r_a}^2+{r_b}^2+{r_c}^2}}{r_a+r_b+r_c-3r} \]

Prove that the following inequality holds for all positive real numbers $x,y,z$: \[\dfrac{x^3}{y^2+z^2}+\dfrac{y^3}{z^2+x^2}+\dfrac{z^3}{x^2+y^2}\ge \dfrac{x+y+z}{2}.\]

An $n\times n$ chessboard is given, where $n$ is an even positive integer. On every line, the unit squares are to be permuted, subject to the condition that the resulting table has to be symmetric with respect to its main diagonal (the diagonal from the top-left corner to the bottom-right corner). We say that a board is [i]alternative[/i] if it has at least one pair of complementary lines (two lines are complementary if the unit squares on them which lie on the same column have distinct colours). Otherwise, we call the board [i]nonalternative[/i]. For what values of $n$ do we always get from the $n\times n$ chessboard an alternative board?\\ \\ [i](Alexandru Petrescu and Andra Elena Mircea)[/i]

For each $r\in\mathbb{R}$ let $T_r$ be the transformation of the plane that takes the point $(x, y)$ into the point $(2^r x; r2^r x+2^r y)$. Let $F$ be the family of all such transformations (i.e. $F = \{T_r : r\in\mathbb{R}\}$). Find all curves $y = f(x)$ whose graphs remain unchanged by every transformation in $F$.

A [i]trifecta[/i] is an ordered triple of positive integers $(a, b, c)$ with $a < b < c$ such that $a$ divides $b$, $b$ divides $c$, and $c$ divides $ab$. What is the largest possible sum $a + b + c$ over all trifectas of three-digit integers?

Find all $3$-digit positive integers that are $34$ times the sum of their digits.

Let $a_1, a_2, \ldots, a_{2016}$ be real numbers such that $9a_i\ge 11a^2_{i+1}$ $(i=,2,\cdots,2015)$. Find the maximum value of $(a_1-a^2_2)(a_2-a^2_3)\cdots (a_{2015}-a^2_{2016})(a_{2016}-a^2_{1}).$

Bisectors of $\angle BAC$, $\angle CAD$ in a rectangle $ABCD$ , intersect the sides $BC$, $CD$ at points $M$ and $N$ resp. Prove that $\frac{(MB)}{(MC)}+\frac{(ND)}{(NC)}>1$

Construct a square, if you know its center and two points that lie on adjacent sides.

Let $ABCD$ be a square with side length $16$ and center $O$. Let $\mathcal S$ be the semicircle with diameter $AB$ that lies outside of $ABCD$, and let $P$ be a point on $\mathcal S$ so that $OP = 12$. Compute the area of triangle $CDP$. [i]Proposed by Brandon Wang[/i]

Compute the number of five-digit positive integers whose digits have exactly $30$ distinct permutations (the permutations do not necessarily have to be valid five-digit integers). [i]Proposed by David Sun[/i]

On side, $BC, AB$ of a parallelogram $ABCD$ lie points $M,N$ respectively such that $|AM| =|CN|$. Let $P$ be the intersection of $AM$ and $CN$. Prove that the angle bisector of $\angle APC$ passes through $D$.

Let side $BC$ of $\bigtriangleup ABC$ be the diameter of a semicircle which cuts $AB$ and $AC$ at $D$ and $E$ respectively. $F$ and $G$ are the feet of the perpendiculars from $D$ and $E$ to $BC$ respectively. $DG$ and $EF$ intersect at $M$. Prove that $AM \perp BC$.