Let $ f,g:[0,1]\longrightarrow{R} $ be two continuous functions such that $ f(x)g(x)\ge 4x^2, $ for all $ x\in [0,1] . $ Prove that $$ \left| \int_0^1 f(x)dx \right| \ge 1\text{ or } \left| \int_0^1 g(x)dx \right| \ge 1. $$

Let $ a$, $ a'$, $ b$, and $ b'$ be real numbers with $ a$ and $ a'$ nonzero. The solution to $ ax \plus{} b \equal{} 0$ is less than the solution to $ a'x \plus{} b' \equal{} 0$ if and only if $ \textbf{(A)}\ a'b < ab' \qquad \textbf{(B)}\ ab' < a'b \qquad \textbf{(C)}\ ab < a'b' \qquad \textbf{(D)}\ \frac {b}{a} < \frac {b'}{a'}$ $ \textbf{(E)}\ \frac {b'}{a'} < \frac {b}{a}$

Let $a,b,c$ be positive real numbers. Prove that \[ \frac{2a}{a^{2}+bc}+\frac{2b}{b^{2}+ca}+\frac{2c}{c^{2}+ab}\leq\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab} \]

Let $ABC$ be a triangle with inradius $r$, circumradius $R$, and with sides $a=BC,b=CA,c=AB$. Prove that \[\frac{R}{2r} \ge \left(\frac{64a^2b^2c^2}{(4a^2-(b-c)^2)(4b^2-(c-a)^2)(4c^2-(a-b)^2)}\right)^2.\]

Let $f$ be a linear function such that $f(0) = -5$ and $f(f(0)) = -15$. Find the values of $ k \in R$ for which the solutions of the inequality $f(x) \cdot f(k - x) > 0$, lie in an interval of[u][/u] length $2$.

Prove that for all real numbers $a,b,c$ the inequality $(a^2+b^2-c^2)(b^2+c^2-a^2)(c^2+a^2-b^2) \leq (a+b-c)^2(b+c-a)^2(c+a-b)^2$ holds.

The numerator of a fraction is $ 6x \plus{} 1$, then denominator is $ 7 \minus{} 4x$, and $ x$ can have any value between $ \minus{}2$ and $ 2$, both included. The values of $ x$ for which the numerator is greater than the denominator are: $ \textbf{(A)}\ \frac{3}{5} < x \le 2\qquad \textbf{(B)}\ \frac{3}{5} \le x \le 2\qquad \textbf{(C)}\ 0 < x \le 2\qquad \\ \textbf{(D)}\ 0 \le x \le 2\qquad \textbf{(E)}\ \minus{}2 \le x \le 2$

Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that \[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0. \]

Let $H$ be the orthocenter of the acute triangle $ABC$. Let $BB'$ and $CC'$ be altitudes of the triangle ($B^{\prime} \in AC$, $C^{\prime} \in AB$). A variable line $\ell$ passing through $H$ intersects the segments $[BC']$ and $[CB']$ in $M$ and $N$. The perpendicular lines of $\ell$ from $M$ and $N$ intersect $BB'$ and $CC'$ in $P$ and $Q$. Determine the locus of the midpoint of the segment $[ PQ]$. [i]Gheorghe Szolosy[/i]

The non-negative numbers $x,y,z$ satisfy the relation $x + y+ z = 3$. Find the smallest possible numerical value and the largest possible numerical value for the expression $$E(x,y, z) = \sqrt{x(y + 3)} + \sqrt{y(z + 3)} + \sqrt{z(x + 3)} .$$

Find all real numbers $ a,b,c,d$ such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\ ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\]

(a) Prove that if $n$ is a integer such that $n \geq 4011^2$ then there exists an integer $l$ such that \[ n < l^2 < (1 + \frac{1}{{2005}})n . \] (b) Find the smallest positive integer $M$ for which whenever an integer $n$ is such that $n \geq M$ then there exists an integer $l$ such that \[ n < l^2 < (1 + \frac{1}{{2005}})n . \]

Let $M=\{3,4,5,6,7,8,9,10,11,12,13,15,17,19,21,23\}$. Prove that for any $a_i>0,i=\overline{1,n},n\in M$ the inequality holds: $$\frac{a_1^2}{(a_2+a_3)^4}+\frac{a_2^2}{(a_3+a_4)^4}+\ldots+\frac{a_{n-1}^2}{(a_n+a_1)^4}+\frac{a_n^2}{(a_1+a_2)^4}\ge\frac{n^3}{16s^2},$$ where $s=\sum_{i=1}^na_i$. [i]Proposed by Marius Olteanu[/i]

If $x$ and $y$ are nonnegative real numbers such that $x+y=1$, determine minimal and maximal value of $$A=x\sqrt{1+y}+y\sqrt{1+x}$$

Let $n \geq 2, n \in \mathbb{N},$ find the least positive real number $\lambda$ such that for arbitrary $a_i \in \mathbb{R}$ with $i = 1, 2, \ldots, n$ and $b_i \in \left[0, \frac{1}{2}\right]$ with $i = 1, 2, \ldots, n$, the following holds: \[\sum^n_{i=1} a_i = \sum^n_{i=1} b_i = 1 \Rightarrow \prod^n_{i=1} a_i \leq \lambda \sum^n_{i=1} a_i b_i.\]

Let O be the circumcenter of a triangle ABC. Points M and N are choosen on the sides AB and BC respectively so that the angle AOC is two times greater than angle MON. Prove that the perimeter of triangle MBN is not less than the lenght of side AC

Let $a$ be some positive number. Find the number of integer solutions $x$ of inequality $2 < xa < 3$ given that inequality $1 < xa < 2$ has exactly $3$ integer solutions. Consider all possible cases. [i](4 points)[/i]

Let $ a,b,c\in (0,1)$ and $ x,y,z\in (0, \plus{} \infty)$ be six real numbers such that \[ a^x \equal{} bc , \quad b^y \equal{} ca , \quad c^z \equal{} ab .\] Prove that \[ \frac 1{2 \plus{} x} \plus{} \frac 1{2 \plus{} y} \plus{} \frac 1{2 \plus{} z} \leq \frac 34 .\] [i]Cezar Lupu[/i]

In a triangle $ABC$, let $AB$ be the shortest side. Points $X$ and $Y$ are given on the circumcircle of $\triangle ABC$ such that $CX=AX+BX$ and $CY=AY+BY$. Prove that $\measuredangle XCY<60^{o}$.

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove the inequality: $\frac{1}{\sqrt{(a+2b)(b+2a)}}+\frac{1}{\sqrt{(b+2c)(c+2b)}}+\frac{1}{\sqrt{(c+2a)(a+2c)}} \geq 3$

Let $a,b, c$ denote the real numbers such that $1 \le a, b, c\le 2$. Consider $T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}$. Determine the largest possible value of $T$.

Suppose $a,b,c,A,B,C$ are real numbers with $a\ne0$ and $A\ne0$ such that for all $x$, $$\left|ax^2+bx+c\right|\le\left|Ax^2+Bx+C\right|.$$Prove that $$\left|b^2-4ac\right|\le\left|B^2-4AC\right|.$$

Let $p>1,\frac1p+\frac1q=1$ and $r>1$. If $u(x,y),v(x,y)>0$, and $f(x,y),g(x,y)$ are continuous functions on $[a,b]\times[c,d]$, then prove $$\left(\frac{\left(\int^b_a\int^d_c(f(x,y)+g(x,y))^rdxdy\right)^{1/r}}{(u(x,y)+v(x,y))^{1/q}}\right)^p\le\left(\frac{\left(\int^b_a\int^d_cf(x,y)^rdxdy\right)^{1/r}}{u(x,y)^{1/q}}\right)^p+\left(\frac{\left(\int^b_a\int^d_cg(x,y)^rdxdy\right)^{1/r}}{v(x,y)^{1/q}}\right)^p,$$ with equality if and only if either $$\left(\lVert f(x,y)\rVert^r_r,\lVert g(x,y)\rVert^r_r\right)=\alpha\left(\lVert u(x,y)\rVert^r_r,\lVert v(x,y)\rVert^r_r\right)$$ for some $\alpha>0$ or $\lVert f(x,y)\rVert^r_r=\lVert g(x,y)\rVert^r_r=0$. [i]Proposed by Chang-Jian Zhao[/i]

Find the minimum possible value of \[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}\] given that $a$, $b$, $c$, $d$ are nonnegative real numbers such that $a+b+c+d=4$. [i]Proposed by Titu Andreescu[/i]

Given a finite nonempty set $X$ with real values, let $f(X) = \frac{1}{|X|} \displaystyle\sum\limits_{a\in X} a$, where $\left\lvert X \right\rvert$ denotes the cardinality of $X$. For ordered pairs of sets $(A,B)$ such that $A\cup B = \{1, 2, \dots , 100\}$ and $A\cap B = \emptyset$ where $1\leq |A| \leq 98$, select some $p\in B$, and let $A_{p} = A\cup \{p\}$ and $B_{p} = B - \{p\}.$ Over all such $(A,B)$ and $p\in B$ determine the maximum possible value of $(f(A_{p})-f(A))(f(B_{p})-f(B)).$