The square polynomial $x^2+ax+b+1$ has natural roots. Prove that $(a^2+b^2)$ is a composite number.

How many triples $(x,y,z)$ of positive integers are there such that $xyz=510510$ and $x^2y+y^2z+z^2x = xy^2 + yz^2 + zx^2$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \text{None of above} $

Two bicyclists traveled the distance from $A$ to $B$, which is $100$ km, with speed $30$ km/h and it is known that the first started $30$ minutes before the second. $20$ minutes after the start of the first bicyclist from $A$, there is a control car started whose speed is $90$ km/h and it is known that the car is reached the first bicyclist and is driving together with him for $10$ minutes, went back to the second and was driving for $10$ minutes with him and after that the car is started again to the first bicyclist with speed $90$ km/h and etc. to the end of the distance. How many times will the car drive together with the first bicyclist? [i]K. Dochev[/i]

Given trapezoid $ ABCD$ with parallel sides $ AB$ and $ CD$, assume that there exist points $ E$ on line $ BC$ outside segment $ BC$, and $ F$ inside segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Denote by $ I$ the point of intersection of $ CD$ and $ EF$, and by $ J$ the point of intersection of $ AB$ and $ EF$. Let $ K$ be the midpoint of segment $ EF$, assume it does not lie on line $ AB$. Prove that $ I$ belongs to the circumcircle of $ ABK$ if and only if $ K$ belongs to the circumcircle of $ CDJ$. [i]Proposed by Charles Leytem, Luxembourg[/i]

Let $G$ be an abelian group with $n$ elements, and let \[\{g_1=e,g_2,\dots,g_k\}\subsetneq G\] be a (not necessarily minimal) set of distinct generators of $G.$ A special die, which randomly selects one of the elements $g_1,g_2,\dots,g_k$ with equal probability, is rolled $m$ times and the selected elements are multiplied to produce an element $g\in G.$ Prove that there exists a real number $b\in(0,1)$ such that \[\lim_{m\to\infty}\frac1{b^{2m}}\sum_{x\in G}\left(\mathrm{Prob}(g=x)-\frac1n\right)^2\] is positive and finite.

Find the largest positive integer $n$, such that there exists a finite set $A$ of $n$ reals, such that for any two distinct elements of $A$, there exists another element from $A$, so that the arithmetic mean of two of these three elements equals the third one.

If 1998 is written as a product of two positive integers whose difference is as small as possible, then the difference is $\text{(A)} \ 8 \qquad \text{(B)} \ 15 \qquad \text{(C)} \ 17 \qquad \text{(D)} \ 47 \qquad \text{(E)} \ 93$

Given a set $L$ of lines in general position in the plane (no two lines in $L$ are parallel, and no three lines are concurrent) and another line $\ell$, show that the total number of edges of all faces in the corresponding arrangement, intersected by $\ell$, is at most $6|L|$. [i]Chazelle et al., Edelsbrunner et al.[/i]

Compute the number of ordered quadruples of positive integers $(a,b,c,d)$ such that \[ a!\cdot b!\cdot c!\cdot d!=24!. \] [i]Proposed by Michael Kural[/i]

$AB$ is a diameter of a circle with center $O$. Let $C$ and $D$ be two different points on the circle on the same side of $AB$, and the lines tangent to the circle at points $C$ and $D$ meet at $E$. Segments $AD$ and $BC$ meet at $F$. Lines $EF$ and $AB$ meet at $M$. Prove that $E,C,M$ and $D$ are concyclic.

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

Show that for any positive integer $n\ge 4$, there exists a multiple of $n^3$ between $n!$ and $(n + 1)!$

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

On the sides $[AB]$ and $[BC]$ of the parallelogram $ABCD$ are constructed the equilateral triangles $ABE$ and $BCF,$ so that the points $D$ and $E$ are on the same side of the line $AB$, and $F$ and $D$ on different sides of the line $BC$. If the points $E,D$ and $F$ are collinear, then prove that $ABCD$ is rhombus.

The lengths of the sides of a triangle are prime numbers of centimeters. Prove that its area cannot be an integer number of square centimeters.

The incircle $\omega$ of a triangle $ABC$ centered at $I$ touches $BC$ at point $D$. Let $P$ be the projection of the orthocenter of $ABC$ to the median from $A$. Prove that the circle $AIP$ and $\omega$ cut off equal chords on $AD$.

Decipher the equality : \[(\overline{VER}-\overline{IA})=G^{R^E} (\overline {GRE}+\overline{ECE}) \] assuming that the number $\overline {GREECE}$ has a maximum value .Each letter corresponds to a unique digit from $0$ to $9$ and different letters correspond to different digits . It's also supposed that all the letters $G$ ,$E$ ,$V$ and $I$ are different from $0$.

In some squares of a $10 \times 10$ board, a piece is placed in such a way that the following property is satisfied: For each square that has a piece, the number of pieces placed in the same row must be greater than or equal to the number of pieces placed in the same column. How many tiles can there be on the board? Give all chances.

A polynomial $P(x)$ of some degree $d$ satisfies $P(n) = n^3 + 10n^2 - 12$ and $P'(n) = 3n^2 + 20n - 1$ for $n = -2, -1, 0, 1, 2.$ Also, $P$ has $d$ distinct (not necessarily real) roots $r_1, r_2, \ldots, r_d.$ The value of \[ \sum_{k=1}^{d}\frac{1}{4 - r_k^2} \] can be expressed as a common fraction $\tfrac{p}{q}.$ What is the value of $p + q?$

Non-zero numbers are arranged in $n \times n$ square ($n>2$). Every number is exactly $k$ times less than the sum of all the other numbers in the same cross (i.e., $2n-2$ numbers written in the same row or column with this number). Find all possible $k$. [i]Proposed by D. Rostovsky, A. Khrabrov, S. Berlov [/i]

Let $ ABC$ be an equilateral triangle, and $ P,Q,R$ three points in its interior satisfying \[ \measuredangle PCA \equal{} \measuredangle CAR \equal{} 15^{\circ},\ \measuredangle RBC \equal{} \measuredangle BCQ \equal{} 20^{\circ},\ \measuredangle QAB \equal{} \measuredangle ABP \equal{} 25^{\circ}.\] Compute the angles of triangle $ PQR$.

Find a sequence of natural numbers $a_i$ such that $a_i = \displaystyle\sum_{r=1}^{i+4} d_r$, where $d_r \neq d_s$ for $r \neq s$ and $d_r$ divides $a_i$.

Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality \[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C \left(\sum_{i}x_{i} \right)^4\] holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.

A positive integer is called [i]squarefree[/i] if its only perfect square factor is $1$. Call a set of positive integers [i]squarefreeful[/i] if each product of two of its elements is squarefree, and [i]squarefreefullest[/i] if no positive integer less than the maximum element of the set can be added while preserving the set's squarefreefulness. What is the minimum number of elements in a squarefreefullest set containing $31$?

Prove that if in the product of four consequtive natural numbers we add $1$, we get a perfect square.