Determine all functions $f:(0;\infty)\to \mathbb{R}$ that satisfy $$f(x+y)=f(x^2+y^2)\quad \forall x,y\in (0;\infty)$$

Let $K$ be a point inside an acute triangle $ ABC $, such that $ BC $ is a common tangent of the circumcircles of $ AKB $ and $ AKC$. Let $ D $ be the intersection of the lines $ CK $ and $ AB $, and let $ E $ be the intersection of the lines $ BK $ and $ AC $ . Let $ F $ be the intersection of the line $BC$ and the perpendicular bisector of the segment $DE$. The circumcircle of $ABC$ and the circle $k$ with centre $ F$ and radius $FD$ intersect at points $P$ and $Q$. Prove that the segment $PQ$ is a diameter of $k$.

Find all $(x,y)$ real numbers pairs such that, $$x^7+y^7=x^4+y^4$$

Determine alle positive integers $n>1$ with the following property: For each colouring of the lattice points in the plane with $n$ colours, there are three lattice points of the same colour forming an isosceles right triangle with legs parallel to the coordinate axes.

A list of integers with average $89$ is split into two disjoint groups. The average of the integers in the first group is $73$ while the average of the integers in the second group is $111$. What is the smallest possible number of integers in the original list? [i] Proposed by David Altizio [/i]

Let $AC>BC$ in $\triangle ABC$ and $M$ and $N$ be the midpoints of $AC$ and $BC$ respectively. The angle bisector of $\angle B$ intersects $\overline{MN}$ at $P$. The incircle of $\triangle ABC$ has center $I$ and touches $BC$ at $Q$. The perpendiculars from $P$ and $Q$ to $MN$ and $BC$ respectively intersect at $R$. Let $S=AB\cap RN$. a) Prove that $PCQI$ is cyclic b) Express the length of the segment $BS$ with $a$, $b$, $c$ - the side lengths of $\triangle ABC$ .

Let $n\ge3$ be an integer. Two regular $n$-gons $\mathcal{A}$ and $\mathcal{B}$ are given in the plane. Prove that the vertices of $\mathcal{A}$ that lie inside $\mathcal{B}$ or on its boundary are consecutive. (That is, prove that there exists a line separating those vertices of $\mathcal{A}$ that lie inside $\mathcal{B}$ or on its boundary from the other vertices of $\mathcal{A}$.)

Consider the following inequality for real numbers $x,y,z$: $|x-y|+|y-z|+|z-x| \le a \sqrt{x^2 +y^2 +z^2}$ . (a) Prove that the inequality is valid for $a = 2\sqrt2$ (b) Assuming that $x,y,z$ are nonnegative, show that the inequality is also valid for $a = 2$.

Let $A$ be a $3$-digit positive integer and $B$ be the positive integer that comes from $A$ be replacing with each other the digits of hundreds with the digit of the units. It is also given that $B$ is a $3$-digit number. Find numbers $A$ and $B$ if it is known that $A$ divided by $B$ gives quotient $3$ and remainder equal to seven times the sum of it's digits.

Prove theat for an arbitrary triangle holds the inequality $$a \cos A+ b \cos B + c \cos C \le p ,$$ where $a, b, c$ are the sides of the triangle, $A, B, C$ are the angles, $p$ is the semiperimeter.

A pair of positive integers $(m,n)$ is called [b][i]'steakmaker'[/i][/b] if they maintain the equation 1 + 2$^m$ = n$^2$. For which values of m and n, the pair $(m,n)$ are steakmaker, find the sum of $mn$

In isosceles $\vartriangle ABC, AB = AC, \angle BAC$ is obtuse, and points $E$ and $F$ lie on sides $AB$ and $AC$, respectively, so that $AE = 10, AF = 15$. The area of $\vartriangle AEF$ is $60$, and the area of quadrilateral $BEFC$ is $102$. Find $BC$.

$ABCDEF$ is an equiangular hexagon of perimeter $21$. Given that $AB = 3, CD = 4$, and $EF = 5$, compute the area of hexagon $ABCDEF$.

For $m\in \mathbb{N}$ define $m?$ be the product of first $m$ primes. Determine if there exists positive integers $m,n$ with the following property : \[ m?=n(n+1)(n+2)(n+3) \] [i]Proposed by Matko Ljulj[/i]

A and B plays the following game: they choose randomly $k$ integers from $\{1,2,\dots,100\}$; if their sum is even, A wins, else B wins. For what values of $k$ does A and B have the same chance of winning?

The net of 20 triangles shown below can be folded to form a regular icosahedron. Inside each of the triangular faces, write a number from 1 to 20 with each number used exactly once. Any pair of numbers that are consecutive must be written on faces sharing an edge in the folded icosahedron, and additionally, 1 and 20 must also be on faces sharing an edge. Some numbers have been given to you. (No proof is necessary.) [asy] unitsize(1cm); pair c(int a, int b){return (a-b/2,sqrt(3)*b/2);} draw(c(0,0)--c(0,1)--c(-1,1)--c(1,3)--c(1,1)--c(2,2)--c(3,2)--c(4,3)--c(4,2)--c(3,1)--c(2,1)--c(2,-1)--c(1,-1)--c(1,-2)--c(0,-3)--c(0,-2)--c(-1,-2)--c(1,0)--cycle); draw(c(0,0)--c(1,1)--c(0,1)--c(1,2)--c(0,2)--c(0,1),linetype("4 4")); draw(c(4,2)--c(3,2)--c(3,1),linetype("4 4")); draw(c(3,2)--c(1,0)--c(1,1)--c(2,1)--c(2,2),linetype("4 4")); draw(c(1,-2)--c(0,-2)--c(0,-1)--c(1,-1)--c(1,0)--c(2,0)--c(0,-2),linetype("4 4")); label("2",(c(0,2)+c(1,2))/2,S); label("15",(c(1,1)+c(2,1))/2,S); label("6",(c(0,1)+c(1,1))/2,N); label("14",(c(0,0)+c(1,0))/2,N);[/asy]

Given two monic polynomials $P(x)$ and $Q(x)$ with degrees 2016. $P(x)=Q(x)$ has no real root. [b]Prove that P(x)=Q(x+1) has at least one real root.[/b]

Given points $P$ and $Q$, Jaqueline has a ruler that allows tracing the line $PQ$. Jaqueline also has a special object that allows the construction of a circle of diameter $PQ$. Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle $ABC$. Show that Jaqueline can construct the incenter of $ABC$.

Indra has a bag for bringing flowers for her grandmother. The first day she brings $n$ flowers. From the second day Indra tries to bring three times plus one with respect to the number of flowers of the previous day. However, if this number is greater or equal to $40$, Indra substracts multiples of $40$ until the remainder is less than this number, since her bag cannot containt so many flowers. For which value of $n$ Indra will bring $30$ flowers the day $2016$?

Find all prime numbers such that the square of the prime number can be written as the sum of cubes of two positive integers.

The points in the plane with integer coordinates are numbered as below. [img]https://cdn.artofproblemsolving.com/attachments/0/2/122cb559c6fb4cb8401ffa215528a035346a3d.png[/img] What are the coordinates of the number $2003$?

In triangle $ABC$ , $\angle A = 45^o$ and $M$ is the midpoint of $\overline{BC}$. $\overline{AM}$ intersects the circumcircle of $ABC$ for the second time at $D$, and $AM = 2MD$. Find $cos\angle AOD$, where $O$ is the circumcenter of $ABC$.

In the diagram below, all circles are tangent to each other as shown. The six outer circles are all congruent to each other, and the six inner circles are all congruent to each other. Compute the ratio of the area of one of the outer circles to the area of one of the inner circles. [img]https://cdn.artofproblemsolving.com/attachments/b/6/4cfbc1df86b8d38e082b7ad0a71b9e366548b3.png[/img]

Given a prime number $p\ge 5$. Find the number of distinct remainders modulus $p$ of the product of three consecutive positive integers.

[b]3.[/b] Let $A$ be a subset of n-dimensional space containing at least one inner point and suppose that, for every point pair $x, y \in A$, the subset $A$ contains the mid point of the line segment beteween $x$ and $y$. Show that $A$ consists of a convex open set and of some of its boundary points. [b](St. 1)[/b]