Let $n\ge2$ be an integer and let $x_1,x_2,\ldots,x_n$ be real numbers. Consider $N=\binom n2$ sums $x_i+x_j$, $1\le i<j\le n$, and denote them by $y_1,y_2,\ldots,y_N$ (in an arbitrary order). For which $n$ are the numbers $x_1,x_2,\ldots,x_n$ uniquely determined by the numbers $y_1,y_2,\ldots,y_N$?

Let $ABC$ be an isosceles triangle with $AC = BC$ and $I$ is the center of the inscribed circle. The point $P$ lies on the circle circumscribed about the triangle $AIB$ and lies inside the triangle $ABC$. Straight lines passing through point $P$ parallel to $CA$ and $CB$ intersect $AB$ at points $D$ and $E$, respectively. The line through $P$ which is parallel to $AB$ intersects $CA$ and $CB$ at points $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ meet at the circumcircle of triangle $ABC$.

The vertices of two acute-angled triangles lie on the same circle. The Euler circle (nine-point circle) of one of the triangles passes through the midpoints of two sides of the other triangle. Prove that the triangles have the same Euler circle. EDIT by pohoatza (in concordance with Luis' PS): [hide=Alternate/initial version ]Let $ABC$ be a triangle with circumcenter $\Gamma$ and nine-point center $\gamma$. Let $X$ be a point on $\Gamma$ and let $Y$, $Z$ be on $\Gamma$ so that the midpoints of segments $XY$ and $XZ$ are on $\gamma$. Prove that the midpoint of $YZ$ is on $\gamma$.[/hide]

A sequence of numbers $a_n, n = 1,2, \ldots,$ is defined as follows: $a_1 = \frac{1}{2}$ and for each $n \geq 2$ \[ a_n = \frac{2 n - 3}{2 n} a_{n-1}. \] Prove that $\sum^n_{k=1} a_k < 1$ for all $n \geq 1.$

A white equilateral triangle is split into $n^2$ equal smaller triangles by lines that are parallel to the sides of the triangle. Denote a [i]line of triangles[/i] to be all triangles that are placed between two adjacent parallel lines that forms the grid. In particular, a triangle in a corner is also considered to be a line of triangles. We are to paint all triangles black by a sequence of operations of the following kind: choose a line of triangles that contains at least one white triangle and paint this line black (a possible situation with $n=6$ after four operations is shown in Figure 1; arrows show possible next operations in this situation). Find the smallest and largest possible number of operations.

Find all real solutions to the system of equations: $$\begin{cases} (x^2+xy+y^2)\sqrt{x^2+y^2} = 88 \\ (x^2-xy+y^2)\sqrt{x^2+y^2} = 40 \end{cases}$$

Triangle $ABC$ has side lengths $AB = 3$, $BC = 4$, and $CD = 5$. Draw line $\ell_A$ such that $\ell_A$ is parallel to $BC$ and splits the triangle into two polygons of equal area. Define lines $\ell_B$ and $\ell_C$ analogously. The intersection points of $\ell_A$, $\ell_B$, and $\ell_C$ form a triangle. Determine its area.

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

Let $f$ be a function on defined on $|x|<1$ such that $f\left (\tfrac1{10}\right )$ is rational and $f(x)= \sum_{i=1}^{\infty} a_i x^i $ where $a_i\in{\{0,1,2,3,4,5,6,7,8,9\}}$. Prove that $f$ can be written as $f(x)= \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials with integer coefficients.

Let $ABCD$ be a cyclic quadrilateral and let the lines $CD$ and $BA$ meet at $E$. The line through $D$ which is tangent to the circle $ADE$ meets the line $CB$ at $F$. Prove that triangle $CDF$ is isosceles.

Prove that when dividing a prime number with $30$, remainder is always not a composite number

Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, and $S$ be points on the sides $AB$, $BC$, $CD$, and $DA$, respectively. Let the line segment $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS$, $BQOP$, $CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC$, $PQ$, and $RS$ are either concurrent or parallel to each other.

Find all $(x,y,z)\in\mathbb R^3$ such that $$x+y+z=xy+yz+zx=3$$ [i]Proposed by Ezra Guerrero, Francisco Morazan[/i]

Let $p,q$ be positive prime numbers and suppose $q>5$. Prove that if $q \mid 2^{p}+3^{p}$, then $q>p$. [i]Laurentiu Panaitopol[/i]

In an acute-angled triangle $ABC$, point $M$ is the midpoint of side $BC$ and the centers of the $M$- excircles of triangles $AMB$ and $AMC$ are $D$ and $E$, respectively. The circumcircle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. The circumcircle of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF\hspace{0.25mm} = \hspace{0.25mm} CG$ .

[b]1.[/b] Let $p_n$ be the $n$th prime number. Prove that $\sum_{n=2}^{\infty} \frac{1}{np_n-(n-1)p_{n-1}}= \infty$ [b](N.17)[/b]

Let $p,q$ be prime numbers such that their sum isn't divisible by $3$. Find the all $(p,q,r,n)$ positive integer quadruples satisfy: $$p+q=r(p-q)^n$$ [i]Proposed by Şahin Emrah[/i]

Let triangle $ABC$ be an isosceles triangle with $AB = AC$. Suppose that the angle bisector of its angle $\angle B$ meets the side $AC$ at a point $D$ and that $BC = BD+AD$. Determine $\angle A$.

Find all pairs of prime numbers $p$ and $q$ such that $p^3-q^5 = (p+q)^2$.

An arbitrary point $M$ is taken on the basis of a regular triangular pyramid. Let $K$, $L$, $N$ be the projections of $M$ onto the lateral faces of this pyramid, and $P$ be the intersection point of the medians of the triangle $KLN$. Prove that the straight line passing through the points $M$ and$ P$ intersects the height of the pyramid (or its extension). Let us denote this intersection point by $E$. Find $MP: PE$ if the dihedral angles at the base of the pyramid are equal to $a$.

Two quadrilaterals $ABCD$ and $A_1B_1C_1D_1$ are mutually symmetric with respect to the point $P$. It is known that $A_1BCD$, $AB_1CD$ and $ABC_1D$ are cyclic quadrilaterals. Prove that the quadrilateral $ABCD_1$ is also cyclic

How many ways are there to write $657$ as a sum of powers of two where each power of two is used at most twice in the sum? For example, $256+256+128+16+1$ is a valid sum.

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

Find all positive integers $n$ for which a square[b][i] n x n[/i][/b] can be covered with rectangles [b][i]k x 1[/i][/b] and one square [b][i]1 x 1[/i][/b] when: a) $k = 4$ b) $k = 8$

Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.