$a_1,a_2,\cdots,a_n$ are positive real numbers, $a_1+a_2+\cdots,a_n=1$. Prove that $$\sum_{m=1}^n\frac{a_m}{\prod\limits_{k=1}^m(1+a_k)}\leq1-\frac{1}{2^n}.$$

Prove that there exist $0 < \alpha< \beta<1$ numbers have the following properties. (i) for any sufficiently large n, n points can be specified in $\Bbb R^3$ , so that each point is equidistant from at least $n^\alpha$ other points. (ii) the above statement is no longer true with $n^\beta$ instead of $n^\alpha$

Suppose we have $10$ balls and $10$ colors. For each ball, we (independently) color it one of the $10$ colors, then group the balls together by color at the end. If $S$ is the expected value of the square of the number of distinct colors used on the balls, find the sum of the digits of $S$ written as a decimal. [i]Proposed by Michael Kural[/i]

Let $ABC$ be a triangle with $AB<AC$, let $G,H$ be its centroid and otrhocenter. Let $D$ be the otrhogonal projection of $A$ on the line $BC$, and let $M$ be the midpoint of the side $BC$. The circumcircle of $ABC$ crosses the ray $HM$ emanating from $M$ at $P$ and the ray $DG$ emanating from $D$ at $Q$, outside the segment $DG$. Show that the lines $DP$ and $MQ$ meet on the circumcircle of $ABC$.

Let $A=(a_{k,l})_{k,l=1,...,n}$ be a complex $n \times n$ matrix such that for each $m \in \{1,2,...,n\}$ and $1 \leq j_{1} <...<j_{m}$ the determinant of the matrix $(a_{j_{k},j_{l}})_{k,l=1,...,n}$ is zero. Prove that $A^{n}=0$ and that there exists a permutation $\sigma \in S_{n}$ such that the matrix $(a_{\sigma(k),\sigma(l)})_{k,l=1,...,n}$ has all of its nonzero elements above the diagonal.

Let $(a_i)_{i\in \mathbb N}$ be a strictly increasing sequence of positive real numbers such that $\lim_{i \to \infty} a_i = +\infty$ and $a_{i+1}/a_i \leq 10$ for each $i$. Prove that for every positive integer $k$ there are infinitely many pairs $(i, j)$ with $10^k \leq a_i/a_j \leq 10^{k+1}.$

Let $D$ be a unique point on segment $BC$, in $ABC$. If $AD^2 = BD \cdot CD$, show that $AB + AC = \sqrt{2}BC$.

Let $m_1< m_2 < \ldots m_{k-1}< m_k$ be $k$ distinct positive integers such that their reciprocals are in arithmetic progression. 1.Show that $k< m_1 + 2$. 2. Give an example of such a sequence of length $k$ for any positive integer $k$.

[b]p1.[/b] What is the maximum amount of a $12\%$ acid solution that can be obtained from $1$ liter of $5\%$, $10\%$ and $15\%$ solutions? [b]p2.[/b] Which number is greater: $199,719,971,997^2$ or $199,719,971,996 * 19,9719,971,998$ ? [b]p3.[/b] Is there a convex $1998$-gon whose angles are all integer degrees? [b]p4.[/b] Is there a ten-digit number divisible by $11$ that uses all the digits from$ 0$ to $9$? [b]p5.[/b] There are $20$ numbers written in a circle, each of which is equal to the sum of its two neighbors. Prove that the sum of all numbers is $0$. [b]p6.[/b] Is there a convex polygon that has neither an axis of symmetry nor a center of symmetry, but which transforms into itself when rotated around some point through some angle less than $180$ degrees? [b]p7.[/b] In a convex heptagon, draw as many diagonals as possible so that no three of them are sides of the same triangle, the vertices of which are at the vertices of the original heptagon. [b]p8.[/b] Give an example of a natural number that is divisible by $30$ and has exactly $105$ different natural factors, including $1$ and the number itself. [b]p9.[/b] In the writing of the antipodes, numbers are also written with the digits $0, ..., 9$, but each of the numbers has different meanings for them and for us. It turned out that the equalities are also true for the antipodes $5 * 8 + 7 + 1 = 48$ $2 * 2 * 6 = 24$ $5* 6 = 30$ a) How will the equality $2^3 = ...$ in the writing of the antipodes be continued? b) What does the number$ 9$ mean among the Antipodes? Clarifications: a) It asks to convert $2^3$ in antipodes language, and write with what number it is equal and find a valid equality in both numerical systems. b) What does the digit $9$ mean among the antipodes, i.e. with which digit is it equal in our number system? [b]p10.[/b] Is there a convex quadrilateral that can be cut along a straight line into two parts of the same size and shape, but neither the diagonal nor the straight line passing through the midpoints of opposite sides divides it into two equal parts? PS.1. There was typo in problem $9$, it asks for $2^3$ and not $23$. PS.2. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

In $\triangle ABC$, $AB = 13$, $BC = 14$ and $CA = 15$. Also, $M$ is the midpoint of side $AB$ and $H$ is the foot of the altitude from $A$ to $BC$. The length of $HM$ is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair H=origin, A=(0,6), B=(-4,0), C=(5,0), M=B+3.6*dir(B--A); draw(B--C--A--B^^M--H--A^^rightanglemark(A,H,C)); label("$A$", A, NE); label("$B$", B, W); label("$C$", C, E); label("$H$", H, S); label("$M$", M, dir(M)); [/asy] $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 6.5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 7.5\qquad\textbf{(E)}\ 8 $

Find all polynomials $P$ in two variables with real coefficients satisfying the identity $$P(x,y)P(z,t)=P(xz-yt,xt+yz).$$

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

There is a wolf in the centre of a square field, and four dogs in the corners. The wolf can easily kill one dog, but two dogs can kill the wolf. The wolf can run all over the field, and the dogs -- along the fence (border) only. Prove that if the dog's speed is $1.5$ times more than the wolf's, than the dogs can prevent the wolf escaping.

Solve in $ \mathbb{R}^2 $ the following equation. $$ 9^{\sqrt x} +9^{\sqrt{y}} +9^{1/\sqrt{xy}} =\frac{81}{\sqrt{x} +\sqrt{y} +1/\sqrt{xy}} $$ [i]O. Trofin[/i]

In an archery competition of 30 contestants, the target is divided into two zones, zone 1 and zone 2. Each arrow hitting the zone 1 gets 10 points, when hitting zone 2 will get 5 points and no score for miss. Each contestant throws 16 arrows. At the end of the competition, the statistics show that more than 50% of the arrows hit zone 2. The number of arrows that hit zone 1 is equal to the arrows which are missed. Prove than there are two contestants having equal score.

Positive real numbers $a$, $b$, $c$ satisfy $a+b+c=1$. Show that $$\frac{a+1}{\sqrt{a+bc}}+\frac{b+1}{\sqrt{b+ca}}+\frac{c+1}{\sqrt{c+ab}} \geq \frac{2}{a^2+b^2+c^2}.$$ When does the equality take place?

If a linear transformation $A$ on an $n$-dimensional vector space has $n+1$ eigenvectors such that any $n$ of them are linearly independent, does it follow that $A$ is a scalar multiple of the identity? Prove your answer.

If positive real numbers $x,y,z$ satisfy the following system of equations, compute $x+y+z$. $$xy+yz = 30$$ $$yz+zx = 36$$ $$zx+xy = 42$$ [i]Proposed by Nathan Xiong[/i]

Consider $5$ distinct positive integers. Can their mean be a)Exactly $3$ times larger than their largest common divisor? b)Exactly $2$ times larger than their largest common divisor?

Given a positive integer $n \ge 2$, let $\mathcal{P}$ and $\mathcal{Q}$ be two sets, each consisting of $n$ points in three-dimensional space. Suppose that these $2n$ points are distinct. Show that it is possible to label the points of $\mathcal{P}$ as $P_1,P_2,\dots,P_n$ and the points of $\mathcal{Q}$ as $Q_1,Q_2,\dots,Q_n$ such that for any indices $i$ and $j$, the balls of diameters $P_iQ_i$ and $P_jQ_j$ have at least one common point.

Does there exist a continuously differentiable function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that for every $x \in \mathbb{R}$ we have $f(x) > 0$ and $f'(x) = f(f(x))$?

Given acute triangle $ABC$ with $AB>AC$, let $M$ be the midpoint of $BC$. $P$ is a point in triangle $AMC$ such that $\angle MAB=\angle PAC$. Let $O,O_1,O_2$ be the circumcenters of $\triangle ABC,\triangle ABP,\triangle ACP$ respectively. Prove that line $AO$ passes through the midpoint of $O_1 O_2$.

Let $H$ be the orthocenter of triangle $ABC$, and $AD$, $BE$, $CF$ be the three altitudes of triangle $ABC$. Let $G$ be the orthogonal projection of $D$ onto $EF$, and $DD'$ be the diameter of the circumcircle of triangle $DEF$. Line $AG$ and the circumcircle of triangle $ABC$ intersect again at point $X$. Let $Y$ be the intersection of $GD'$ and $BC$, while $Z$ be the intersection of $AD'$ and $GH$. Prove that $X$, $Y$, and $Z$ are collinear. [i]Proposed by Li4 and Untro368.[/i]

Let $I$ be the center of the circle of the inscribed triangle $ABC$ and $M$ be the center of its side $BC$. If $|AI| = |MI|$, prove that there are two of the sides of triangle $ABC$, of which one is twice of the other.

Find some antiderivative of the function $y = 1/x^3$, the graph of which has exactly three common points with the graph of the function $y = |x|$.