An interstellar hotel has $100$ rooms with capacities $101,102,\ldots, 200$ people. These rooms are occupied by $n$ people in total. Now a VIP guest is about to arrive and the owner wants to provide him with a personal room. On that purpose, the owner wants to choose two rooms $A$ and $B$ and move all guests from $A$ to $B$ without exceeding its capacity. Determine the largest $n$ for which the owner can be sure that he can achieve his goal no matter what the initial distribution of the guests is.

Show that for any positive real numbers $a,b,c$ the following inequality is true: $$4(a^3+b^3+c^3+3)\geq 3(a+1)(b+1)(c+1)$$ When does equality hold?

Let positive integers $m,n$ satisfy $n=2^m-1$. $P_n =\{1,2,\cdots ,n\}$ is a set that contains $n$ points on an axis. A grasshopper on the axis can leap from one point to another adjacent point. Find the maximal value of $m$ satisfying following conditions: (a) $x, y$ are two arbitrary points in $P_n$; (b) starting at point $x$, the grasshopper leaps $2012$ times and finishes at point $y$; (the grasshopper is allowed to travel $x$ and $y$ more than once) (c) there are even number ways for the grasshopper to do (b).

For any real numbers $x , y ,z$ prove that : $$(x+y+z)^2 + \sum_{cyc}{\frac{(x+y)(y+z)}{1+|x-z|}} \ge xy+yz+zx$$ [i]Proposed by Navid Safaei[/i]

Let be a ring that has the property that all its elements are the product of two idempotent elements of it. Show that: [b]a)[/b] $ 1 $ is the only unit of this ring. [b]b)[/b] this ring is Boolean.

Let $a_1,a_2,a_3,a_4$ be real numbers such that $a_1^2 + a_2^2 + a_3^2 + a_4^2 = 1$. Show that there exist $i,j$ with $ 1 \leq i < j \leq 4$, such that $(a_i - a_j)^2 \leq \frac{1}{5}$.

Let $M$ be an arbitrary point on side $BC$ of triangle $ABC$. $W$ is a circle which is tangent to $AB$ and $BM$ at $T$ and $K$ and is tangent to circumcircle of $AMC$ at $P$. Prove that if $TK||AM$, circumcircles of $APT$ and $KPC$ are tangent together.

Let $ABC$ be a triangle with $AB : AC : BC =5:5:6$. Denote by $M$ the midpoint of $BC$ and by $N$ the point on the segment $BC$ such that $BN = 5 \cdot CN$. Prove that the circumcenter of triangle $ABN$ is the midpoint of the segment connecting the incenters of triangles $ABC$ and $ABM$. Slovakia

Let $ABCD$ be a convex quadrilateral such that $BC=AD$. Let $M$ and $N$ be the midpoints of $AB$ and $CD$, respectively. The lines $AD$ and $BC$ meet the line $MN$ at $P$ and $Q$, respectively. Prove that $CQ=DP$.

Let $ABC$ be an equilateral triangle, and let $K, L,M$ be points respectively on $BC, CA, AB$ such that $BK/KC = CL/LA = AM/MB =\lambda $. Find all values of $\lambda$ such that the circle with $BC$ as a diameter completely covers the triangle bounded by the lines $AK,BL,CM$.

Assume that the complex numbers $a_1 , a_2, \ldots$ are all different from $0$, and that $|a_r - a_s| >1$ for $r\ne s.$ Show that the series $$\sum_{n=1}^{\infty} \frac{1}{a_{n}^{3}}$$ converges.

Let $ABCD$ be an isosceles trapezoid, with a large base $CD$ and a small base $AB$. Let $M$ be any point on side $AB$ and $(d)$ be the line through $M$ and perpendicular to $AB$. Two rays $Mx$ and $My$ are said to satisfy the condition $(T)$ if they are symmetric about each other through $(d)$ and intersect the two rays $AD$ and $BC$ at $E$ and $F$ respectively. Find the locus of the midpoint of the segment $EF$ when the two rays $Mx$ and $My$ change and satisfy condition $(T)$.

In triangle $ABC$ the bisector of the angle at $B$ meets $AC$ at $D$ and the bisector of the angle at $C$ meets $AB$ at $E$. These bisectors intersect at $O$ and the lengths of $OD$ and $OE$ are equal. Prove that either $\angle BAC = 60^o$ or triangle $ABC$ is isosceles.

Let $n$ and $k$ be integers, $1\le k\le n$. Find an integer $b$ and a set $A$ of $n$ integers satisfying the following conditions: (i) No product of $k-1$ distinct elements of $A$ is divisible by $b$. (ii) Every product of $k$ distinct elements of $A$ is divisible by $b$. (iii) For all distinct $a,a'$ in $A$, $a$ does not divide $a'$.

The grid below is to be filled with integers in such a way that the sum of the numbers in each row and the sum of the numbers in each column are the same. Four numbers are missing. The number $x$ in the lower left corner is larger than the other three missing numbers. What is the smallest possible value of $x$? [asy] unitsize(0.5cm); draw((3,3)--(-3,3)); draw((3,1)--(-3,1)); draw((3,-3)--(-3,-3)); draw((3,-1)--(-3,-1)); draw((3,3)--(3,-3)); draw((1,3)--(1,-3)); draw((-3,3)--(-3,-3)); draw((-1,3)--(-1,-3)); label((-2,2),"$-2$"); label((0,2),"$9$"); label((2,2),"$5$"); label((2,0),"$-1$"); label((2,-2),"$8$"); label((-2,-2),"$x$"); [/asy] $\textbf{(A) } -1 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9 \qquad$

The pupil is training in the square equation solution. Having the recurrent equation solved, he stops, if it doesn't have two roots, or solves the next equation, with the free coefficient equal to the greatest root, the coefficient at $x$ equal to the least root, and the coefficient at $x^2$ equal to $1$. Prove that the process cannot be infinite. What maximal number of the equations he will have to solve?

In a convex quadrilateral $ABCD$, $BD$ is the angle bisector of $\angle{ABC}$. The circumcircle of $ABC$ intersects $CD,AD$ in $P,Q$ respectively and the line through $D$ parallel to $AC$ cuts $AB,AC$ in $R,S$ respectively. Prove that point $P,Q,R,S$ lie on a circle.

Calculate the product of all positive integers less than $100$ and having exactly three positive divisors. Show that this product is a square.

Consider a right -angle triangle $ABC$ with $A=90^{o}$, $AB=c$ and $AC=b$. Let $P\in AC$ and $Q\in AB$ such that $\angle APQ=\angle ABC$ and $\angle AQP = \angle ACB$. Calculate $PQ+PE+QF$, where $E$ and $F$ are the projections of $B$ and $Q$ onto $BC$, respectively.

In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: \[ (AB+BC+CA)^2\le6(AD^2+BD^2+CD^2). \] When do we have equality?

Let $R^+$ be the set of positive real numbers. Determine all functions $f:R^+$ $\rightarrow$ $R^+$ such that for all positive real numbers $x$ and $y:$ \[f(x+f(xy))+y=f(x)f(y)+1\] [i]Ukraine[/i]

A $9 \times 9$ square is divided into $81$ unit cells. Some of the cells are coloured. The distance between the centres of any two coloured cells is more than $2$. (a) Give an example of colouring with $17$ coloured cells. (b) Prove that the numbers of coloured cells cannot exceed $17$. (S. Fomin, Leningrad)

Consider a right-angled triangle with $AB=1,\ AC=\sqrt{3},\ \angle{BAC}=\frac{\pi}{2}.$ Let $P_1,\ P_2,\ \cdots\cdots,\ P_{n-1}\ (n\geq 2)$ be the points which are closest from $A$, in this order and obtained by dividing $n$ equally parts of the line segment $AB$. Denote by $A=P_0,\ B=P_n$, answer the questions as below. (1) Find the inradius of $\triangle{P_kCP_{k+1}}\ (0\leq k\leq n-1)$. (2) Denote by $S_n$ the total sum of the area of the incircle for $\triangle{P_kCP_{k+1}}\ (0\leq k\leq n-1)$. Let $I_n=\frac{1}{n}\sum_{k=0}^{n-1} \frac{1}{3+\left(\frac{k}{n}\right)^2}$, show that $nS_n\leq \frac {3\pi}4I_n$, then find the limit $\lim_{n\to\infty} I_n$. (3) Find the limit $\lim_{n\to\infty} nS_n$.

A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. THe remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $ 15$ and $ 25$ meters. What fraction of the yard is occupied by the flower beds? [asy]unitsize(2mm); defaultpen(linewidth(.8pt)); fill((0,0)--(0,5)--(5,5)--cycle,gray); fill((25,0)--(25,5)--(20,5)--cycle,gray); draw((0,0)--(0,5)--(25,5)--(25,0)--cycle); draw((0,0)--(5,5)); draw((20,5)--(25,0));[/asy]$ \textbf{(A)}\ \frac18\qquad \textbf{(B)}\ \frac16\qquad \textbf{(C)}\ \frac15\qquad \textbf{(D)}\ \frac14\qquad \textbf{(E)}\ \frac13$

Let $r$ be a natural number. Prove that the quadratic trinomial $x^2 - rx- 1$ does not divide any nonzero polynomial whose coefficients are integers with absolute values less than $r$.