Inside the triangle $ABC$, the point $P$ is chosen. Let $ a, b, c $ be the lengths of the sides $ BC $, $ CA $, $ AB $, respectively, and $ x, y, z $ the distances of the point $ P $ from the vertices $ B, C, A $. Prove that if $$ x^2 + xy + y^2 = a^2 $$ $$y^2 + yz + z^2 = b^2 $$ $$z^2 + zx + x^2 = c^2$$ this $$ a^2 + ab + b^2 > c^2.$$

Let be given the sequence $(x_n)$ defined by $x_1 = 1$ and $x_{n+1} = 3x_n + \lfloor x_n \sqrt5 \rfloor$ for all $n = 1,2,3,...,$ where $\lfloor x \rfloor$ denotes the greatest integer that does not exceed $x$. Prove that for any positive integer $n$ we have $$x_nx_{n+2} - x^2_{n+1} = 4^{n-1}$$ Trần Nam Dũng

Consider all products by $2, 4, 6, ..., 2000$ of the elements of the set $A =\left\{\frac12, \frac13, \frac14,...,\frac{1}{2000},\frac{1}{2001}\right\}$ . Find the sum of all these products.

A sequence $a_n$ is defined such that $a_0 =\frac{1 + \sqrt3}{2}$ and $a_{n+1} =\sqrt{a_n}$ for $n \ge 0$. Evaluate $$\prod_{k=0}^{\infty} 1 - a_k + a^2_k$$

Let \( \mathbb R \) be the set of real numbers. Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that, for all real numbers \( x \) and \( y \), the following equation holds:$$\big (x^2-y^2\big )f\big (xy\big )=xf\big (x^2y\big )-yf\big (xy^2\big ).$$

Call a polynomial $P$ [i]prime-covering[/i] if for every prime $p$, there exists an integer $n$ for which $p$ divides $P(n)$. Determine the number of ordered triples of integers $(a,b,c)$, with $1\leq a < b < c \leq 25$, for which $P(x)=(x^2-a)(x^2-b)(x^2-c)$ is prime-covering.

Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$ holds for all integers $n>N.$ [i]Proposed by Carl Schildkraut[/i]

Let $f(n)$ and $g(n)$ be polynomials of degree $2014$ such that $f(n)+(-1)^ng(n)=2^n$ for $n=1,2,\ldots,4030$. Find the coefficient of $x^{2014}$ in $g(x)$.

Find the largest possible integer $k$, such that the following statement is true: Let $2009$ arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain \[ \left. \begin{array}{rcl} & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }\\ & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }\\ \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }\\ \end{array}\right.\] Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_j$, $r_j$, $w_j$. [i]Proposed by Michal Rolinek, Czech Republic[/i]

Let $m<n$ be two positive integers and $x_m<x_{m+1}<\cdots<x_n$ be a sequence of rational numbers. Suppose that $kx_k$ is an integer for all integers $k$ which $m\leq k\leq n$. Prove that $$x_n-x_m\geq \frac{1}{m}-\frac{1}{n}.$$

Let $m \neq -1$ be a real number. Consider the quadratic equation $$ (m + 1)x^2 + 4mx + m - 3 =0. $$ Which of the following must be true? $\quad\rm(I)$ Both roots of this equation must be real. $\quad\rm(II)$ If both roots are real, then one of the roots must be less than $-1.$ $\quad\rm(III)$ If both roots are real, then one of the roots must be larger than $1.$ $$ \mathrm a. ~ \text{Only} ~(\mathrm I)\rm \qquad \mathrm b. ~(I)~and~(II)\qquad \mathrm c. ~Only~(III) \qquad \mathrm d. ~Both~(I)~and~(III) \qquad \mathrm e. ~(I), (II),~and~(III) $$

Compute $$\sum^{\infty}_{k=1} \frac{(-1)^k}{(2k - 1)(2k + 1)}$$

For positive real numbers $x$ and $y$, define their special mean to be average of their arithmetic and geometric means. Find the total number of pairs of integers $(x, y)$, with $x \le y$, from the set of numbers $\{1,2,...,2016\}$, such that the special mean of $x$ and $y$ is a perfect square.

Find the greatest positive integer $m$ with the following property: For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference.

Given that $f(x)=(3+2x)^3(4-x)^4$ on the interval $\frac{-3}{2}<x<4$. Find the a. Maximum value of $f(x)$ b. The value of $x$ that gives the maximum in (a).

Let $p(x)$ and $q(x)$ be two cubic polynomials such that $p(0)=-24$, $q(0)=30$, and \[p(q(x))=q(p(x))\] for all real numbers $x$. Find the ordered pair $(p(3),q(6))$.

Given an integer $n > 2$, consider all sequences $x_1,x_2,...,x_n$ of nonnegative real numbers such that $$x_1+ 2x_2 + ... + nx_n = 1.$$ Find the maximum value and the minimum value of $x_1^2+x_2^2+...+x_n^2$ and determine all the sequences $x_1,x_2,...,x_n$ for which these values are obtained.

Given several nonnegative integers (repetitions allowed), the allowed operation is to choose a positive integer $a$ and replace each number $b$ greater than or equal to $a$ by $b-a$ (the numbers $a$ , if any, are replaced by $0$). Initially, the integers from $1$ are written on the blackboard until $2013$ inclusive. After a few operations the numbers on the board have a sum equal to $10$. Determine what the numbers that remained on the board could be. Find all the possibilities.

Let $\alpha,\beta,\gamma$ measures of angles of opposite to the sides of triangle with measures $a,b,c$ respectively.Prove that $$2(cos^2\alpha+cos^2\beta+cos^2\gamma)\geq \frac{a^2}{b^2+c^2}+\frac{b^2}{a^2+c^2}+\frac{c^2}{a^2+b^2}$$

Find all the solutions of the system $$\begin{cases} y^2 = x^3 - 3x^2 + 2x \\ x^2 = y^3 - 3y^2 + 2y \end{cases}$$

$$F(x)=x^{6}+15x^{5}+85x^{4}+225x^{3}+274x^{2}+120x+1$$

Three natural numbers greater than or equal to $2$ are written, not necessarily different, and from them a sequence is constructed using the following procedure: in each step, if the penultimate number written is $a$, the penultimate one is $b$ and the last one is $c$, it is written $x$ such that $$x\cdot c=a+b+186.$$Determine all the possible values of the three numbers initially written so that when the process continues indefinitely all the written numbers are natural numbers greater than or equal to $2$.

In January Kolya and Vasya have been assessed at school $20$ times and each has been given $20$ marks (each being an integer no greater than $5$ , with both Kolya and Vasya receiving at least twos on each occasion). Kolya has been given as many fives as Vasya fours, as many fours as Vasya threes, as many threes as Vasya twos and as many twos as Vasya fives. If each has the same average mark , determine how many twos were given to Kolya. (S . Fomin, Leningrad)

$n$ is called [i]diamond 2005[/i] if $n=\overline{...ab999...99999cd...}$, e.g. $2005 \times 9$. Let $\{a_n\}:a_n< C\cdot n,\{a_n\}$ is increasing. Prove that $\{a_n\}$ contain infinite [i]diamond 2005[/i]. Compare with [url=http://www.mathlinks.ro/Forum/topic-15091.html]this problem.[/url]

If $a_1,a_2,\ldots,a_n$ are positive real numbers, prove the inequality $$\dfrac1{\dfrac1{1+a_1}+\dfrac1{1+a_2}+\ldots+\dfrac1{1+a_n}}-\dfrac1{\dfrac1{a_1}+\dfrac1{a_2}+\ldots+\dfrac1{a_n}}\ge\frac1n.$$