For what minimal natural $a$ the polynomial $ax^2 + bx + c$ with the integer $c$ and $b$ has two different positive roots both less than one.

Let $ a\in (1,\infty) $ and a countinuous function $ f:[0,\infty)\longrightarrow\mathbb{R} $ having the property: $$ \lim_{x\to \infty} xf(x)\in\mathbb{R} . $$ [b]a)[/b] Show that the integral $ \int_1^{\infty} \frac{f(x)}{x}dx $ and the limit $ \lim_{t\to\infty} t\int_{1}^a f\left( x^t \right) dx $ both exist, are finite and equal. [b]b)[/b] Calculate $ \lim_{t\to \infty} t\int_1^a \frac{dx}{1+x^t} . $

In the convex quadrilateral $ABCD$, it is given that $\angle BAD = \angle DCB = 90^\circ$, $AB = 7$, $CD = 11$, and that $BC$ and $AD$ are integers greater than 11. Determine the values of $BC$ and $AD$.

You have a 2 meter long string. You choose a point along the string uniformly at random and make a cut. You discard the shorter section. If you still have 0.5 meters or more of string, you repeat. You stop once you have less than 0.5 meters of string. On average, how many cuts will you make before stopping?

Let C be the number of ways to arrange the letters of the word CATALYSIS, T be the number of ways to arrange the letters of the word TRANSPORT, S be the number of ways to arrange the letters of the word STRUCTURE, and M be the number of ways to arrange the letters of the word MOTION. What is $\frac{C - T + S}{M}$ ?

Show that the only solutions of the equation $x^{3}-3xy^2 -y^3 =1$ are given by $(x,y)=(1,0),(0,-1),(-1,1),(1,-3),(-3,2),(2,1)$.

Find all continuous functions $f: \mathbb{R}\to\mathbb{R}$ such that for all real $x$ we have \[f(x)=f\left(x^{2}+\frac{x}{3}+\frac{1}{9}\right). \]

We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integers $m$ and $n$, it can be represented in the form \[f(x_1,\cdots , x_k )=\max_{i=1,\cdots , m} \min_{j=1,\cdots , n}P_{i,j}(x_1,\cdots , x_k),\] where $P_{i,j}$ are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial.

The numbers one through eight are written, in that order, on a chalkboard. A mysterious higher power in possession of both an eraser and a piece of chalk chooses three distinct numbers $x$, $y$, and $z$ on the board, and does the following. First, $x$ is erased and replaced with $y$, after which $y$ is erased and replaced with $z$, and finally $z$ is erased and replaced with $x$. The higher power repeats this process some finite number of times. For example, if $(x,y,z)=(2,4,5)$ is chosen, followed by $(x,y,z)=(1,4,3)$, the board would change in the following manner: \[12345678 \rightarrow 14352678 \rightarrow 43152678\] Compute the number of possible final orderings of the eight numbers.

Let $\triangle ABC$ be a triangle with incentre $I$. Points $E$ and $F$ are the tangency points of the incircle and the sides $AC$ and $AB$, respectively. Suppose that the lines $BI$ and $CI$ intersect the line $EF$ at $Y$ and $Z$, respectively. Let $M$ denote the midpoint of the segment $BC$, and $N$ denote the midpoint of the segment $YZ$. Prove that $AI \parallel MN$.

Evaluate $ \int_0^{\sqrt{2}\minus{}1} \frac{1\plus{}x^2}{1\minus{}x^2}\ln \left(\frac{1\plus{}x}{1\minus{}x}\right)\ dx$.

Find all real $\alpha$ s.t. \[ [ \sqrt{n + \alpha} + \sqrt{n} ] = [ \sqrt{4n+1} ] \] holds for all natural numbers $n$

Let $a$ be the largest root of the equation $x^3 - 3x^2 + 1 = 0$. Find the first $200$ decimal digits for the number $a^{2000}$.

Let $ AD$ is centroid of $ ABC$ triangle. Let $ (d)$ is the perpendicular line with $ AD$. Let $ M$ is a point on $ (d)$. Let $ E, F$ are midpoints of $ MB, MC$ respectively. The line through point $ E$ and perpendicular with $ (d)$ meet $ AB$ at $ P$. The line through point $ F$ and perpendicular with $ (d)$ meet $ AC$ at $ Q$. Let $ (d')$ is a line through point $ M$ and perpendicular with $ PQ$. Prove $ (d')$ always pass a fixed point.

Let $x,y$ be positive real numbers such that $x+y=1$. Prove that$\frac{x}{x^2+y^3}+\frac{y}{x^3+y^2}\leq2(\frac{x}{x+y^2}+\frac{y}{x^2+y})$.

$x,y\in\mathbb{R},(4x^3-3x)^2+(4y^3-3y)^2=1.\text { Find the maximum of } x+y.$

Let there be $n$ students, numbered $1$ through $n$. Let there be $n$ cards with numbers $1$ through $n$ written on them. Each student picks a card from the stack, and two students are called a pair if they pick each other's number. Let the probability that there are no pairs be $p_n$. Prove that $p_n - p_{n-1}=0$ if $n$ is odd, and prove that $p_n - p_{n-1}= \frac{1}{(-2)^kk^{1-k}}$ if $n = 2k$.

We will say that two sets of distinct numbers are $\textit{linked}$ to each other if between any two numbers of each set lies at least one number of the other set. Is it possible to fill the cells of a $100 \times 200$ rectangle with distinct numbers so that any two rows of the rectangle are linked to one another, and any two columns of the rectangle are linked to one another?

For reals $1\leq a\leq b \leq c \leq d \leq e \leq f$ prove inequality $(af + be + cd)(af + bd + ce) \leq (a + b^2 + c^3 )(d + e^2 + f^3 )$.

Let $ABC$ be a triangle such that $AC\not= BC,AB<AC$ and let $K$ be it's circumcircle. The tangent to $K$ at the point $A$ intersects the line $BC$ at the point $D$. Let $K_1$ be the circle tangent to $K$ and to the segments $(AD),(BD)$. We denote by $M$ the point where $K_1$ touches $(BD)$. Show that $AC=MC$ if and only if $AM$ is the bisector of the $\angle DAB$. [i]Neculai Roman[/i]

Prove that from $x + y = 1 \ (x, y \in \mathbb R)$ it follows that \[x^{m+1} \sum_{j=0}^n \binom{m+j}{j} y^j + y^{n+1} \sum_{i=0}^m \binom{n+i}{i} x^i = 1 \qquad (m, n = 0, 1, 2, \ldots ).\]

Let $\mathbb{R}^*$ be the set of all real numbers, except $1$. Find all functions $f:\mathbb{R}^* \rightarrow \mathbb{R}$ that satisfy the functional equation $$x+f(x)+2f\left(\frac{x+2009}{x-1}\right)=2010$$.

Do there exist postive integers $a_1<a_2<\cdots<a_{100}$ such that for $2\le k\le100$ the greatest common divisor of $a_{k-1}$ and $a_k$ is greater than the greatest common divisor of $a_k$ and $a_{k+1}$?

Show that the only solutions of te equation \[ p^{k} + 1 = q^{m} \], in positive integers $k,q,m > 1$ and prime $p$ are (i) $(p,k,q,m) = (2,3,3,2)$ (ii) $k=1 , q=2,$and $p$ is a prime of the form $2^{m} -1$, $m > 1 \in \mathbb{N}$

Let $ABC$ be an isosceles triangle with $AB = AC$. Points $D$, $E$ and $F$ are on sides $BC$, $CA $ and $AB$ respectively, such that $\angle FDE =\angle ABC$ and $FE$ is not parallel to $BC$. Prove that $BC$ is tangent to the circumcircle of the triangle $DEF$, if and only if, $D$ is the midpoint of $BC$.