Three quadratic polynomials $f_1(x) = x^2+2a_1x+b_1$, $f_2(x) = x^2+2a_2x+b_2$, $f_3(x) = x^2 + 2a_3x + b_3$ are such that $a_1a_2a_3 = b_1b_2b_3 > 1$. Prove that at least one polynomial has two distinct roots.

Seven skiers with numbers 1,2, . . . ,7 left the start one by one and walked the distance - each at their own constant speed. It turned out that each skier participated in overtaking exactly twice. (In each overtaking, exactly two skiers participate - the one who overtakes and the one who is overtaken.) At the end of the race, a ranking must be drawn up consisting of numbers skiers in finishing order. Prove that you are in the race with the described properties, no more than two different ranking can be obtained. sourceRussia Regional Olympiad 2010 9.2

Is it possible for some natural number $k$ to divide all natural numbers from $1$ to $k$ into two groups and write down the numbers in each group in a row in some order so that you get two the same numbers? original wording beacuse it doesn't make much senseМожно ли при каком-то натуральном k разбить все натуральные числа от 1 до k на две группы и выписать числа в каждой группе подряд в некотором порядке так, чтобы получились два одинаковых числа?

In triangle $ABC$, $\angle A =60^o$. Let $BB_1$ and $CC_1$ be angle bisectors of this triangle. Prove that the point symmetrical to vertex $A$ with respect to line $B_1C_1$ lies on side $BC$.

Dunno wrote down $11$ natural numbers in a circle. For every two adjacent numbers, he calculated their difference. As a result among the differences found there were four units, four twos and three threes. Prove that Dunno made a mistake somewhere an error.

Let points $A$, $B$, $C$ lie on a circle, and line $b$ be the tangent to the circle at point $B$. Perpendiculars $PA_1$ and $PC_1$ are dropped from a point $P$ on line $b$ onto lines $AB$ and $BC$ respectively. Points $A_1$ and $C_1$ lie inside line segments $AB$ and $BC$ respectively. Prove that $A_1C_1$ is perpendicular to $AC$.

In a company of seven people, any six can sit at a round table so that every two neighbors turn out to be acquaintances. Prove that the whole company can be seated at a round table so that every two neighbors turn out to be acquaintances.

For every positive integer $n$, let $S_n$ be the sum of the first $n$ prime numbers: $S_1 = 2, S_2 = 2 + 3 = 5, S_3 = 2 + 3 + 5 = 10$, etc. Can both $S_n$ and $S_{n+1}$ be perfect squares?

Nine skiers left the start line in turn and covered the distance, each at their own constant speed. Could it turn out that each skier participated in exactly four overtakes? (In each overtaking, exactly two skiers participate - the one who is overtaking, and the one who is being overtaken.)

In triangle $ABC$, the angle bisectors $AD$, $BE$ and $CF$ are drawn, intersecting at point $I$. The perpendicular bisector of the segment $AD$ intersects lines $BE$ and $CF$ at points $M$ and $N$, respectively. Prove that points $A$, $I$, $M$ and $ N$ lie on the same circle.

We call a natural number $b$ lucky if for any natural number $a$ such that $a^5$ is divisible by $b^2$, the number $a^2$ is divisible by $b$. Find the number of lucky natural numbers less than $2010$.

Non-zero numbers $a, b, c$ are such that $ax^2+bx+c > cx$ for any $x$. Prove that $cx^2-bx + a > cx-b$ for any $x$.

The tangent lines to the circle $\omega$ at points $B$ and $D$ intersect at point $P$. The line passing through $P$ cuts out from circle chord $AC$. Through an arbitrary point on the segment $AC$ a straight line parallel to $BD$ is drawn. Prove that it divides the lengths of polygonal $ABC$ and $ADC$ in the same ratio. last sentence was in Russian:Докажите, что она делит длины ломаных ABC и ADC в одинаковых отношениях.

Are there three pairwise distinct non-zero integers whose sum is zero and whose sum of thirteenth powers is the square of some natural number?

Let's call it a staircase of height $n$, a figure consisting from all square cells $n\times n$ lying no higher diagonals (the figure shows a staircase of height $4$ ). In how many different ways can a staircase of height $n$ can be divided into several rectangles whose sides go along the grid lines, but the areas are different in pairs?

Each leg of a right triangle is increased by one. Could its hypotenuse increase by more than $\sqrt2$?

In a row of $2009$ weights, the weight of each weight is an integer grams and does not exceed $1$ kg. The weights of any two adjacent weights differ by exactly $1$ g, and the total weight of all weights in grams is an even number. Prove that weights can be separated into two piles, the sums of the weights in which are equal.

Quadrangle $ABCD$ is inscribed in a circle with diameter $AC$. Points $K$ and $M$ are projections of vertices $A$ and $C$, respectively, onto line $BD$. A line parallel to $BC$ is drawn through point $K$ and intersecting $AC$ at point $P$. Prove that angle $KPM$ is a right angle.

We call a triple of natural numbers $(a, b, c)$ square if they form an arithmetic progression (in exactly this order), the number $b$ is coprime to each of the numbers $a$ and $c$, and the number $abc$ is a perfect square. Prove that for any given a square triple, there is another square triple that has at least one common number with it.

The angles of the triangle $\alpha, \beta, \gamma$ satisfy the inequalities $$\sin \alpha > \cos \beta, \sin \beta > \cos \gamma, \sin \gamma > \cos \alpha. $$Prove that the trαiangle is acute-angled.

At the base of the quadrangular pyramid $SABCD$ lies the parallelogram $ABCD$. Prove that for any point $O$ inside the pyramid, the sum of the volumes of the tetrahedra $OSAB$ and $OSCD$ is equal to the sum of the volumes of the tetrahedra $OSBC$ and $OSDA$ .

Integers $a,b,c$ are such that the values of the trinomials $bx^2+cx+a$ and $cx^2+ax+b$ at $x=1234$ coincide. Can the first trinomial at $x = 1$ take the value $2009$?

The numbers $1, 2,. . . , 10000, $ were placed in the cells of a $100 \times 100$ square, each once; in this case, numbers differing by $1$ are written in cells adjacent to each side. After that we calculated distances between the centers of every two cells whose numbers differ by exactly $5000$. Let $S$ be the minimum of these distances What is the largest value $S$ can take?