What is the value of $9901\cdot101-99\cdot10101?$ $\textbf{(A) }2\qquad\textbf{(B) }20\qquad\textbf{(C) }21\qquad\textbf{(D) }200\qquad\textbf{(E) }2020$

Let $B$ and $D$ be points on segments $[AE]$ and $[AF]$ respectively. Excircles of triangles $ABF$ and $ADE$ touching sides $BF$ and $DE$ is the same, and its center is $I$. $BF$ and $DE$ intersects at $C$. Let $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3, Q_4$ be the circumcenters of triangles $IAB, IBC, ICD, IDA, IAE, IEC, ICF, IFA$ respectively. [b]a) [/b] Show that points $P_1, P_2, P_3, P_4$ concylic and points $Q_1, Q_2, Q_3, Q_4$ concylic. [b]b) [/b] Denote centers of theese circles as $O_1$ and $O_2$. Prove that $O_1, O_2$ and $I$ are collinear.

The sweeties shop called "Olympiad" sells boxes of $6,9$ or $20$ chocolates. Groups of students from a school that is near the shop collect money to buy a chocolate for each student; to make this they buy a box and than give to everybody a chocolate. Like this students can create groups of $15=6+9$ students, $38=2*9+20$ students, etc. The seller has promised to the students that he can satisfy any group of students, and if he will need to open a new box of chocolate for any group (like groups of $4,7$ or $10$ students) than he will give all the chocolates for free to this group. Can there be constructed the biggest group that profits free chocolates, and if so, how many students are there in this group?

Let $a,b,c$ be non-negative real numbers such that \[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]Prove that $a^2+b^2+c^2\leqslant 2abc+1.$

$ABCDE$ is a convex pentagon such that the triangles $ABC, BCD, CDE, DEA$ and $EAB$ have equal areas. Show that $(1/4)$ area $(ABCDE) <$ area $(ABC) < (1/3)$ area $(ABCDE)$.

Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc \equal{} 1$. Prove that \[ \frac {1}{a^{3}\left(b \plus{} c\right)} \plus{} \frac {1}{b^{3}\left(c \plus{} a\right)} \plus{} \frac {1}{c^{3}\left(a \plus{} b\right)}\geq \frac {3}{2}. \]

Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee? $ \textbf{(A) } \frac 67 \qquad \textbf{(B) } \frac {13}{14} \qquad \textbf{(C) } 1 \qquad \textbf{(D) } \frac {14}{13} \qquad \textbf{(E) } \frac 76$

Find the smallest positive value of $x$ such that $x^3-9x^2+22x-16=0$.

A square is divided into $25$ small squares. We draw diagonals of some of the small squares so that no two diagonals share a common point (not even a common endpoint). What is the largest possible number of diagonals that we can draw? (I Rubanov)

There is an Equilateral trapezoid $ ABCD. $ $ \bar{AB} =60, \bar{BC}=\bar{DA}= 36, \bar{CD}=108. $ $ M $ is the middle point of $ \bar {AB} $, and point $P$ on $ \bar{AM} $ follows that $ \bar {AP} $ =10. The foot of perpendicular dropped from $P$ to $ \bar {BD} $ is $E$. $ \bar{AC} \cap \bar{BD} $ is $ F $. Point $X$ is on $ \bar {AF} $ which follows $ \bar{MX}=\bar{ME} $ Find $ \bar{AX} \times \bar{AF} $

$a_{n}$ ($n$ is integer) is a sequence from positive reals that \[a_{n}\geq \frac{a_{n+2}+a_{n+1}+a_{n-1}+a_{n-2}}4\] Prove $a_{n}$ is constant.

Determine all pairs $ (m,n)$ of natural numbers for which $ m^2\equal{}nk\plus{}2$ where $ k\equal{}\overline{n1}$. EDIT. [color=#FF0000]It has been discovered the correct statement is with $ k\equal{}\overline{1n}$.[/color]

Let $a > 0$ and $0 < \alpha <\pi$ be given. Let $ABC$ be a triangle with $BC = a$ and $\angle BAC = \alpha$ , and call the cicumcentre $O$, and the orthocentre $H$. The point $P$ lies on the ray from $A$ through $O$. Let $S$ be the mirror image of $P$ through $AC$, and $T$ the mirror image of $P$ through $AB$. Assume that $SATH$ is cyclic. Show that the length $AP$ depends only on $a$ and $\alpha$.

Let $\triangle ABC$ be equilateral. Let $D$ be the midpoint of side $AC,$ and let $DEFG$ be a square such that $D, F, B$ are collinear and $E,G$ lie on $AB,CB$ respectively. What fraction of the area of $\triangle ABC$ is covered by square $DEFG?$ [i]Proposed by Lohith Tummala[/i]

Consider the triangle $ ABC$ with side-lenghts equal to $ a,b,c$. Let $ p\equal{}\frac{a\plus{}b\plus{}c}{2}$, $ R$-the radius of circumcircle of the triangle $ ABC$, $ r$-the radius of the incircle of the triangle $ ABC$ and let $ l_a,l_b,l_c$ be the lenghts of bisectors drawn from $ A,B$ and $ C$, respectively, in the triangle $ ABC$. Prove that: $ l_al_b\plus{}l_bl_c\plus{}l_cl_a\leq p\sqrt{3r^2\plus{}12Rr}$ [i]Proposer[/i]: [b]Baltag Valeriu[/b]

Let $M=(0,1)\cap \mathbb Q$. Determine, with proof, whether there exists a subset $A\subset M$ with the property that every number in $M$ can be uniquely written as the sum of finitely many distinct elements of $A$.

Let \(c\) be a positive real number. Alice wishes to pick an integer \(n\) and a sequence \(a_1\), \(a_2\), \(\ldots\) of distinct positive integers such that \(a_{i} \leq ci\) for all positive integers \(i\) and \[n, \qquad n + a_1, \qquad n + a_1 - a_2, \qquad n + a_1 - a_2 + a_3, \qquad \cdots\] is a sequence of distinct nonnegative numbers. Find all \(c\) such that Alice can fulfil her wish.

A line that passes through the origin intersects both the line $x=1$ and the line $y=1+\frac{\sqrt{3}}{3}x$. The three lines create an equilateral triangle. What is the perimeter of the triangle? $ \textbf{(A) }2\sqrt{6}\qquad\textbf{(B) }2+2\sqrt{3}\qquad\textbf{(C) }6\qquad\textbf{(D) }3+2\sqrt{3}\qquad\textbf{(E) }6+\frac{\sqrt{3}}{3} $

Prove that for any real numbers $ x_1, x_2, \ldots, x_{1981} $, $ y_1, y_2, \ldots, y_{1981} $ such that $ \sum_{j=1}^{1981} x_j = 0 $, $ \sum_{j=1}^{1981} y_j = 0 $ the inequality occurs $$ \sqrt{\sum_{j=1}^{1981} (x_j^2+y_j^2)} \leq \frac{1}{\sqrt{2}} \sum_{j=1}^{1981} \sqrt{x_j^2+y_j^2}.$$

A positive integer is a non-trivial perfect power if it can be expressed as $n^k$ where $n$ and $k$ are positive integers and $k>1$. Show that there exist arbitrarily large consecutive square numbers with no other non-trivial perfect powers between them.

Let $ m \equal{} 2007^{2008}$, how many natural numbers n are there such that $ n < m$ and $ n(2n \plus{} 1)(5n \plus{} 2)$ is divisible by $ m$ (which means that $ m \mid n(2n \plus{} 1)(5n \plus{} 2)$) ?

Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation $$x(x-y)+y(y-z)+z(z-x) = 1?$$$\textbf{(A)}\: x>y$ and $y=z$ $\textbf{(B)}\: x=y-1$ and $y=z-1$ $\textbf{(C)} \: x=z+1$ and $y=x+1$ $\textbf{(D)} \: x=z$ and $y-1=x$ $\textbf{(E)} \: x+y+z=1$

In triangle $ABC$ let $AD,BE$ and $CF$ be the altitudes. Let the points $P,Q,R$ and $S$ fulfil the following requirements: i) $P$ is the circumcentre of triangle $ABC$. ii) All the segments $PQ,QR$ and $RS$ are equal to the circumradius of triangle $ABC$. iii) The oriented segment $PQ$ has the same direction as the oriented segment $AD$. Similarly, $QR$ has the same direction as $BE$, and $Rs$ has the same direction as $CF$. Prove that $S$ is the incentre of triangle $ABC$.

66 dwarfs have a total of 111 hats. Each of the hats belongs to a dwarf and colored by 66 different colors. Festivities are organized where each of these dwarfs wears their own hat. There is no dwarf pair wearing the same colored hat in any of the festivities. For any two of the festivities, there exist a dwarf wearing a hat of a different color in these festivities. Find the maximum value of the number of festivities that can be organized.

In quadrilateral $ABCD$, $AC = 5$, $CD = 7$, and $AD = 3$. The angle bisector of $\angle CAD$ intersects $CD$ at $E$. If $\angle CBD = 60^o$ and $\angle AED = \angle BEC$, compute the value of $AE + BE$.