Let $ABC$ be an acute-angled triangle with circumradius $R$ less than the sides of $ABC$, let $H$ and $O$ be the orthocenter and circuncenter of $ABC$, respectively. The angle bisectors of $\angle ABH$ and $\angle ACH$ intersects in the point $A_1$, analogously define $B_1$ and $C_1$. If $E$ is the midpoint of $HO$, prove that $EA_1+EB_1+EC_1=p-\frac{3R}{2}$ where $p$ is the semiperimeter of $ABC$

Given a triangular pyramid $ABCD$. Sphere $S_1$ passing through points $A$, $B$, $C$, intersects edges $AD$, $BD$, $CD$ at points $K$, $L$, $M$, respectively; sphere $S_2$ passing through points $A$, $B$, $D$ intersects the edges $AC$, $BC$, $DC$ at points $P$, $Q$, $M$ respectively. It turned out that $KL \parallel PQ$. Prove that the bisectors of plane angles $KMQ$ and $LMP$ are the same.

$AB$ is the chord of the circle $S$. Circles $S_1$ and $S_2$ touch the circle $S$ at points $P$ and $Q$, respectively, and the segment $AB$ at point $K$. It turned out that $\angle{PBA}=\angle{QBA}$. Prove that $AB$ is the diameter of the circle $S$.

Into triangle $ABC$ gives point $K$ lies on bisector of $ \angle BAC$. Line $CK$ intersect circumcircle $ \omega$ of triangle $ABC$ at $M \neq C$. Circle $ \Omega$ passes through $A$, touch $CM$ at $K$ and intersect segment $AB$ at $P \neq A$ and $\omega $ at $Q \neq A$. Prove, that $P$, $Q$, $M$ lies at one line.

Square $ABCD$ has side length $5$ and arc $BD$ with center $A$. $E$ is the midpoint of $AB$ and $CE$ intersects arc $BD$ at $F$. $G$ is placed onto $BC$ such that $FG$ is perpendicular to $BC$. What is the length of $FG$?

Two $5\times1$ rectangles have 2 vertices in common as on the picture. (a) Determine the area of overlap (b) Determine the length of the segment between the other 2 points of intersection, $A$ and $B$. [img]https://cdn.artofproblemsolving.com/attachments/9/0/4f1721c7ccdecdfe4d9cc05a17a553a0e9f670.png[/img]

The diagonals of a convex quadrilateral dissect it into four similar triangles. Prove that this quadrilateral can also be dissected into two congruent triangles.

For an $n\times n$ matrix with real entries let $||M||=\sup_{x\in \mathbb{R}^{n}\setminus\{0\}}\frac{||Mx||_{2}}{||x||_{2}}$, where $||\cdot||_{2}$ denotes the Euclidean norm on $\mathbb{R}^{n}$. Assume that an $n\times n$ matrxi $A$ with real entries satisfies $||A^{k}-A^{k-1}||\leq\frac{1}{2002k}$ for all positive integers $k$. Prove that $||A^{k}||\leq 2002$ for all positive integers $k$.

Juku invented an apparatus that can divide any segment into three equal segments. How can you find the midpoint of any segment, using only the Juku made, a ruler and pencil?

Call an $n$-gon to be [i]lattice[/i] if its vertices are lattice points. Prove that inside every lattice convex pentagon there exists a lattice point.

Given that in a triangle $ABC$, $AB=3$, $BC=4$ and the midpoints of the altitudes of the triangle are collinear, find all possible values of the length of $AC$.

Compute the sum of the digits of $1001^{10}$

Find all prime number $p$ such that the number of positive integer pair $(x,y)$ satisfy the following is not $29$. [list] [*]$1\le x,y\le 29$ [*]$29\mid y^2-x^p-26$ [/list]

If $x(y + 1) = 41$ and $x^2(y^2 + 1) = 881$, determine all possible pairs of real numbers $(x,y)$.

[b]p1.[/b] Two proper ordinary fractions are given. The first has a numerator that is $5$ less than the denominator, and the second has a numerator that is $1998$ less than the denominator. Can their sum have a numerator greater than its denominator? [b]p2.[/b] On New Year's Eve, geraniums, crocuses and cacti stood in a row (from left to right) on the windowsill. Every morning, Masha, wiping off the dust, swaps the places of the flower on the right and the flower in the center. During the day, Tanya, while watering flowers, swaps places between the one in the center and the one on the left. In what order will the flowers be in $365$ days on the next New Year's Eve? [b]p3.[/b] The number $x$ is such that $15\%$ of it and $33\%$ of it are positive integers. What is the smallest number $x$ (not necessarily an integer!) with this property? [b]p4.[/b] In the quadrilateral $ABCD$, the extensions of opposite sides $AB$ and $CD$ intersect at an angle of $20^o$; the extensions of opposite sides $BC$ and $AD$ also intersect at an angle of $20^o$. Prove that two angles in this quadrilateral are equal and the other two differ by $40^o$. [b]p5.[/b] Given two positive integers $a$ and $b$. Prove that $a^ab^b\ge a^ab^a.$ [b]p6.[/b] The square is divided by straight lines into $25$ rectangles (fig.). The areas of some of They are indicated in the figure (not to scale). Find the area of the rectangle marked with a question mark. [img]https://cdn.artofproblemsolving.com/attachments/0/9/591c93421067123d50382744f9d28357acf83a.png[/img] [b]p7.[/b] A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly $ 17^o$ (relative to the previous direction of movement). What is the smallest number of commands required for the toy to pass through the starting point again? [b]p8.[/b] In expression $$(a-b+c)(d+e+f)(g-h-k)(\ell +m- n)(p + q)$$ opened the brackets. How many members will there be? How many of them will be preceded by a minus sign? [b]p9.[/b] In some countries they decided to hold popular elections of the government. Two-thirds of voters in this country are urban and one-third are rural. The President must propose for approval a draft government of $100$ people. It is known that the same percentage of urban (rural) residents will vote for the project as there are people from the city (rural) in the proposed project. What is the smallest number of city residents that must be included in the draft government so that more than half of the voters vote for it? [b]p10.[/b] Vasya and Petya play such a game on a $10 \times 10 board$. Vasya has many squares the size of one cell, Petya has many corners of three cells (fig.). They are walking one by one - first Vasya puts his square on the board, then Petya puts his corner, then Vasya puts another square, etc. (You cannot place pieces on top of others.) The one who cannot make the next move loses. Vasya claims that he can always win, no matter how hard Petya tries. Is Vasya right? [img]https://cdn.artofproblemsolving.com/attachments/f/1/3ddec7826ff6eb92471855322e3b9f01357116.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Let $\mathbb{N} = \{1, 2, 3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^2< f(n)f(f(n)) < n^2+n$ for every positive integer $n$.

Let $ABC$ be an equilateral triangle with centre $O$. A line through $C$ meets the circumcircle of triangle $AOB$ at points $D$ and $E$. Prove that points $A, O$ and the midpoints of segments $BD, BE$ are concyclic.

In triangle $ABC$, points $A_1,B_1,C_1$ are bases of altitudes from vertices $A,B,C$, and points $C_A,C_B$ are the projections of $C_1$ to $AC$ and $BC$ respectively. Prove that line $C_AC_B$ bisects the segments $C_1A_1$ and $C_1B_1$.

We call $A_{1},A_{2},A_{3}$ [i]mangool[/i] iff there is a permutation $\pi$ that $A_{\pi(2)}\not\subset A_{\pi(1)},A_{\pi(3)}\not\subset A_{\pi(1)}\cup A_{\pi(2)}$. A good family is a family of finite subsets of $\mathbb N$ like $X,A_{1},A_{2},\dots,A_{n}$. To each goo family we correspond a graph with vertices $\{A_{1},A_{2},\dots,A_{n}\}$. Connect $A_{i},A_{j}$ iff $X,A_{i},A_{j}$ are mangool sets. Find all graphs that we can find a good family corresponding to it.

Six points are equally spaced around a circle of radius 1. Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle? $ \textbf{(A) }\frac{\sqrt3}3\qquad\textbf{(B) }\frac{\sqrt3}2\qquad\textbf{(C) }1\qquad\textbf{(D) }\sqrt2\qquad\textbf{(E) }2$

Prove that every selection of $1325$ integers from $M=\{1, 2, \cdots, 1987 \}$ must contain some three numbers $\{a, b, c\}$ which are pairwise relatively prime, but that it can be avoided if only $1324$ integers are selected.

Construct a convex quadrilateral given two opposite angles and sides.

The sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=1, \; a_{2}=12, \; a_{3}=20, \; a_{n+3}= 2a_{n+2}+2a_{n+1}-a_{n}.\] Prove that $1+4a_{n}a_{n+1}$ is a square for all $n \in \mathbb{N}$.

Suppose $a_1,a_2,a_3,b_1,b_2,b_3$ are distinct positive integers such that \[(n \plus{} 1)a_1^n \plus{} na_2^n \plus{} (n \minus{} 1)a_3^n|(n \plus{} 1)b_1^n \plus{} nb_2^n \plus{} (n \minus{} 1)b_3^n\] holds for all positive integers $n$. Prove that there exists $k\in N$ such that $ b_i \equal{} ka_i$ for $ i \equal{} 1,2,3$.

Let $n \ge 2$ be an even integer. Find the maximum integer $k$ (in terms of $n$) such that $2^k$ divides $\binom{n}{m}$ for some $0 \le m \le n$.