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 $r$.

[asy]pair A=(-2,0), O=origin, C=(2,0); path X=Arc(O,2,0,180), Y=Arc((-1,0),1,180,0), Z=Arc((1,0),1,180,0), N=X..Y..Z..cycle; filldraw(N, black, black); draw(reflect(A,C)*N); draw(A--C, dashed); label("A",A,W); label("C",C,E); label("O",O,SE); dot((-1,0)); dot(O); dot((1,0)); label("1",(-1,0),NE); label("1",(1,0),NW);[/asy] The large circle has diameter $ \overline{AC}$. The two small circles have their centers on $ \overline{AC}$ and just touch at $ O$, the center of the large circle. If each small circle has radius $ 1$, what is the value of the ratio of the area of the shaded region to the area of one of the small circles? \[ \textbf{(A)}\ \text{between }\frac{1}{2} \text{ and }1 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ \text{between 1 and }\frac{3}{2} \qquad \textbf{(D)}\ \text{between }\frac{3}{2} \text{ and }2 \\ \textbf{(E)}\ \text{cannot be determined from the information given} \]

We call a subset $B$ of natural numbers [i]loyal[/i] if there exists natural numbers $i\le j$ such that $B=\{i,i+1,\ldots,j\}$. Let $Q$ be the set of all [i]loyal[/i] sets. For every subset $A=\{a_1<a_2<\ldots<a_k\}$ of $\{1,2,\ldots,n\}$ we set \[f(A)=\max_{1\le i \le k-1}{a_{i+1}-a_i}\qquad\text{and}\qquad g(A)=\max_{B\subseteq A, B\in Q} |B|.\] Furthermore, we define \[F(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} f(A)\qquad\text{and}\qquad G(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} g(A).\] Prove that there exists $m\in \mathbb N$ such that for each natural number $n>m$ we have $F(n)>G(n)$. (By $|A|$ we mean the number of elements of $A$, and if $|A|\le 1$, we define $f(A)$ to be zero). [i]Proposed by Javad Abedi[/i]

A circle is contained in a quadrilateral with successive sides of lengths $3,6,5$ and $8$. Prove that the length of its radius is less than $3$. [i]Proposed by K. Kokhas[/i]

Let $ABC$ be a right-angled triangle ($\angle C = 90^\circ$) and $D$ be the midpoint of an altitude from C. The reflections of the line $AB$ about $AD$ and $BD$, respectively, meet at point $F$. Find the ratio $S_{ABF}:S_{ABC}$. Note: $S_{\alpha}$ means the area of $\alpha$.

Find the number of binary sequences $S$ of length $2015$ such that for any two segments $I_1, I_2$ of $S$ of the same length, we have • The sum of digits of $I_1$ differs from the sum of digits of $I_2$ by at most $1$, • If $I_1$ begins on the left end of S then the sum of digits of $I_1$ is not greater than the sum of digits of $I_2$, • If $I_2$ ends on the right end of S then the sum of digits of $I_2$ is not less than the sum of digits of $I_1$. Lê Anh Vinh

There is a collection of $25$ indistinguishable black chips and $25$ indistinguishable white chips. Find the number of ways to place some of these chips in $25$ unit cells of a $5 \times 5$ grid so that: [list] [*]each cell contains at most one chip, [*]all chips in the same row and all chips in the same column have the same color, [*]any additional chip placed on the grid would violate one or more of the previous two conditions. [/list]

Let $x,y$ be real numbers such that $(x+1)(y+2)=8.$ Prove that $$(xy-10)^2\ge 64.$$

Let $G$ be a tournoment such that it's edges are colored either red or blue. Prove that there exists a vertex of $G$ like $v$ with the property that, for every other vertex $u$ there is a mono-color directed path from $v$ to $u$.

Two players $A$ and $B$ play the following game taking alternate moves. In each move, a player writes one digit on the blackboard. Each new digit is written either to the right or left of the sequence of digits already written on the blackboard. Suppose that $A$ begins the game and initially the blackboard was empty. $B$ wins the game if ,after some move of $B$, the sequence of digits written in the blackboard represents a perfect square. Prove that $A$ can prevent $B$ from winning.

Let $ ABC$ be a triangle, line $ l$ cuts its sides $ BC,CA,AB$ at $ D,E,F$, respectively. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ AEF,BFD,CDE$, respectively. Prove that the orthocenter of triangle $ O_{1}O_{2}O_{3}$ lies on line $ l$.

In a certain Zoom meeting, there are $4$ students. How many ways are there to split them into any number of distinguishable breakout rooms, each with at least $ 1$ student?

Find the largest positive real number \( c \) such that for any positive integer \( n \), satisfies \(\{ \sqrt{7n} \} \geq \frac{c}{\sqrt{7n}}\).

Points $K$ and $L$ are inner for $AB$ for an acute $\Delta ABC$, where $K$ is between $A$ and $L$. Let $P,Q$, and $H$ be the feet of the perpendiculars from $A$ to $CK$, from $B$ to $CL$, and from $C$ to $AB$, respectively. Point $M$ is the middle point of $AB$. If $PH\cap AC=X$ and $QH\cap BC=Y$, prove that points $H,P,M$, and $Q$ lie on one circle, if and only if the lines $AY,BX$, and $CH$ intersect in one point.