Let $N$ be a natural number which can be expressed in the form $a^{2}+b^{2}+c^{2}$, where $a,b,c$ are integers divisible by 3. Prove that $N$ can be expressed in the form $x^{2}+y^{2}+z^{2}$, where $x,y,z$ are integers and any of them are divisible by 3.

Proof that for every primes $p$, $q$ \[p^{q^2-q+1}+q^{p^2-p+1}-p-q\] is never a perfect square. [i]Proposed by chengbilly[/i]

Prove that there doesn't exist a function $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $(m+f(n))^2 \geq 3f(m)^2+n^2$ for all $m, n \in \mathbb{N}$.

A steak temperature $5^\circ$ is put into an oven. After $15$ minutes, it has temperature $45^\circ$. After another $15$ minutes it has temperature $77^\circ$. The oven is at a constant temperature. The steak changes temperature at a rate proportional to the difference between its temperature and that of the oven. Find the oven temperature.

In this diagram a scheme is indicated for associating all the points of segment $ \overline{AB}$ with those of segment $ \overline{A'B'}$, and reciprocally. To described this association scheme analytically, let $ x$ be the distance from a point $ P$ on $ \overline{AB}$ to $ D$ and let $ y$ be the distance from the associated point $ P'$ of $ \overline{A'B'}$ to $ D'$. Then for any pair of associated points, if $ x \equal{} a,\, x \plus{} y$ equals: [asy]defaultpen(linewidth(.8pt)); unitsize(.8cm); pair D= (0,9); pair E = origin; pair A = (3,9); pair P = (3.6,9); pair B = (4,9); pair F = (1,0); pair G = (2.6,0); pair H = (5,0); dot((0,0));dot((1,0));dot((2,0));dot((3,0));dot((4,0));dot((5,0)); dot((0,9));dot((1,9));dot((2,9));dot((3,9));dot((4,9));dot((5,9)); draw((D+(0,0.5))--(0,-0.5)); draw(A--H); draw(P--G); draw(B--F); draw(F--H); draw(A--B); label("$D$",D,NW); label("$D'$",E,NW); label("0",(0,0),SE); label("1",(1,0),SE); label("2",(2,0),SE); label("3",(3,0),SE); label("4",(4,0),SE); label("5",(5,0),SE); label("0",(0,9),SE); label("1",(1,9),SE); label("2",(2,9),SE); label("3",(3,9),SW); label("4",(4,9),SE); label("5",(5,9),SE); label("$B'$",F,NW); label("$P'$",G,S); label("$A'$",H,NE); label("$A$",A,NW); label("$P$",P,N); label("$B$",B,NE);[/asy] $ \textbf{(A)}\ 13a\qquad \textbf{(B)}\ 17a \minus{} 51\qquad \textbf{(C)}\ 17 \minus{} 3a\qquad \textbf{(D)}\ \frac {17 \minus{} 3a}{4}\qquad \textbf{(E)}\ 12a \minus{} 34$

Let $X$ be a finite set with $n\geqslant 3$ elements and let $A_1,A_2,\ldots, A_p$ be $3$-element subsets of $X$ satisfying $|A_i\cap A_j|\leqslant 1$ for all indices $i,j$. Show that there exists a subset $A{}$ of $X$ so that none of $A_1,A_2,\ldots, A_p$ is included in $A{}$ and $|A|\geqslant\lfloor\sqrt{2n}\rfloor$.

Let $p\neq 3$ be a prime number. Show that there is a non-constant arithmetic sequence of positive integers $x_1,x_2,\ldots ,x_p$ such that the product of the terms of the sequence is a cube.

Prove that there exists a real $c<\frac{3}{4}$, such that for each sequence $x_1, x_2, \ldots$ satisfying $0 \leq x_i \leq 1$ for all $i$, there exist infinitely many $(m, n)$ with $m>n$, such that $$|x_m-x_n|\leq \frac{c} {m}.$$

Let $X = \{1, 2, 3, 4, 5, 6, 7, 8\}$. We want to color, using $k$ colors, all subsets of $3$ elements of $X$ in such a way that, two disjoint subsets have distinct colors. Prove that: (a) $4$ colors are sufficient; (b) $3$ colors are not sufficient.

Fifteen distinct points are designated on $\triangle ABC$: the 3 vertices $A$, $B$, and $C$; $3$ other points on side $\overline{AB}$; $4$ other points on side $\overline{BC}$; and $5$ other points on side $\overline{CA}$. Find the number of triangles with positive area whose vertices are among these $15$ points.

Are there positive integers $m,n$ such that there exist at least $2012$ positive integers $x$ such that both $m-x^2$ and $n-x^2$ are perfect squares? [i]David Yang.[/i]

Let $p$ be a prime number. Find all pairs of coprime positive integers $(m,n)$ such that $$ \frac{p+m}{p+n}=\frac{m}{n}+\frac{1}{p^2}.$$

A lattice point in the plane is a point whose coordinates are both integers. Given a set of 100 distinct lattice points in the plane, find the smallest number of line segments $ \overline{AB}$ for which $ A$ and $ B$ are distinct lattice points in this set and the midpoint of $ \overline{AB}$ is also a lattice point (not necessarily in the set).

Find all pairs of complex numbers $(z,w) \in \mathbb{C}^2$ such that the relation \[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \] holds for all positive integers $n$.

In acute triangle $ABC$, point $H$ is the intersection point of heights $CE$ on side $AB$ and $BD$ on side $AC$. A circle with diameter $DE$ intersects $AB$ and $AC$ at $F$ and $G$ respectively. $FG$ and $AH$ intersect at $K$. If $BC=25,BD=20, BE=7$, find the length of $AK$.

The three medians of a triangle has lengths $3, 4, 5$. What is the length of the shortest side of this triangle?

Solve the system equations \[\left\{\begin{array}{cc}x^{4}-y^{4}=240\\x^{3}-2y^{3}=3(x^{2}-4y^{2})-4(x-8y)\end{array}\right.\]

Determine all positive integers $x$, $y$ and $z$ such that $x^5 + 4^y = 2013^z$. ([i]Serbia[/i])

$Abdulqodir$ cut out $2024$ congruent regular $n-$gons from a sheet of paper and placed these $n-$gons on the table such that some parts of each of these $n-$gons may be covered by others. We say that a vertex of one of the afore-mentioned $n-$gons is $visible$ if it is not in the interior of another $n-$gon that is placed on top of it. For any $n>2$ determine the minimum possible number of visible vertices. \\ Proposed by David Hrushka, Slovakia

Three real numbers satisfy the following statements: (1) the square of their sum equals to the sum their squares. (2) the product of the first two numbers is equal to the square of the third number. Find these numbers.

Prove that, if there exist natural numbers $a_1,a_2…a_{2017}$ for which the product $(a_1^{2017}+a_2 )(a_2^{2017}+a_3 )…(a_{2016}^{2017}+a_{2017})(a_{2017}^{2017}+a_1)$ is a $k$-th power of a prime number, then $k=2017$ or $k\geq 2017.2018$.

There are a number of arcs on the edge of a circular disk. Each pair of arcs has the least one point in common. Show that on the circle you can choose two diametrical opposites points such that each arc contains at least one of these two points.

Determine whether the polynomial $P(x)=(x^2-2x+5)(x^2-4x+20)+1$ is irreducible over $\mathbb{Z}[X]$.

Each detail of the “Young Solderer” instructor is a bracket in the shape of the letter $\Pi$, consisting of three single segments. Is it possible from the parts of this constructor are soldered together, a complete wire frame of the cube $2 \times 2 \times 2$, divided into $1 \times 1 \times 1$ cubes? (The frame consists of 27 points, connected by single segments; any two adjacent points must be connected by exactly one piece of wire.) [hide]=original wording]Каждая деталько нструктора ''Юный паяльщик'' — это скобка в виде буквы П, остоящая из трех единичных отрезков. Можно ли издеталей этого конструктора спаятьполный роволочный каркас куба 2 × × 2 × 2, разбитого на кубики 1 × 1 × 1? (Каркас состоит из 27 точек,соединенных единичными отрезками; любые две соседние точки должны бытьсоединены ровно одним проволочным отрезком.)[/hide]

Suppose you started at the origin on the number line in a coin-flipping game. Every time you flip a heads, you move forward one step, otherwise you move back one step. However, there are walls at positions $8$ and $-8$; if you are at these positions and your coin flip dictates that you should move past them, instead you must stay. What is the expected number of coin flips needed to have visited both walls?