Let $a$, $b$, $c$ be positive real numbers and let $m$, $n$ $(m \geq n)$ be positive integers. Prove that$$\frac{a^{n-1}b^{n-1}c^{m-n-1}}{a^{m+n}+b^{m+n}+a^nb^nc^{m-n}}+\frac{b^{n-1}bc^{n-1}a^{m-n-1}}{b^{m+n}+c^{m+n}+b^nc^na^{m-n}}+\frac{c^{n-1}a^{n-1}b^{m-n-1}}{c^{m+n}+a^{m+n}+c^na^nb^{m-n}}\leq \frac{1}{abc}$$

Let $ a_1\geq \cdots \geq a_n \geq a_{n \plus{} 1} \equal{} 0$ be real numbers. Show that \[ \sqrt {\sum_{k \equal{} 1}^n a_k} \leq \sum_{k \equal{} 1}^n \sqrt k (\sqrt {a_k} \minus{} \sqrt {a_{k \plus{} 1}}). \] [i]Proposed by Romania[/i]

Let $n$ be an integer greater than $2$. Prove that the $n$th power of the length of the hypotenuse of a right triangle is greater than the sum of the $n$th powers of the lengths of the legs.

Find all polynomials $p$ with real coefficients such that for all real $x$ , $xp(x)p(1-x)+x^3 +100 \ge 0$.

Let $x,y,z$ be positive real numbers that satisfy at least one of the inequalities, $2xy>1$, $yz>1$. Find the least possible value of $$xy^3z^2+\frac{4z}{x}-8yz-\frac{4}{yz}$$ .

For each real number $a$ and integer $n \ge 1$ prove the inequality $a^n +\frac{1}{a^n} -2 \ge n^2 \left(a +\frac{1}{a} -2\right)$ and find the cases of equality.

Let $ABC$ be an acute-angled triangle whose side lengths satisfy the inequalities $AB < AC < BC$. If point $I$ is the center of the inscribed circle of triangle $ABC$ and point $O$ is the center of the circumscribed circle, prove that line $IO$ intersects segments $AB$ and $BC$.

11.6 Construct for each vertex of the quadrilateral of area $S$ a symmetric point wrt to the diagonal, which doesn't contain this vertex. Let $S'$ be an area of the obtained quadrilateral. Prove that $\frac{S'}{S}<3$. ([i]L. Emel'yanov[/i])

To be considered the following complex and distinct $a,b,c,d$. Prove that the following affirmations are equivalent: i)For every $z\in \mathbb{C}$ the inequality takes place :$\left| z-a \right|+\left| z-b \right|\ge \left| z-c \right|+\left| z-d \right|$; ii)There is $t\in \left( 0,1 \right)$ so that $c=ta+\left( 1-t \right)b$ si $d=\left( 1-t \right)a+tb$

Let $a,b, c > 0$ and $abc = 1$. Prove that $\frac{a - 1}{c}+\frac{c - 1}{b}+\frac{b - 1}{a} \ge 0$

For a set $S$ of $n$ points, let $d_1 > d_2 >... > d_k > ...$ be the distances between the points. A function $f_k: S \to N$ is called a [i]coloring function[/i] if, for any pair $M,N$ of points in $S$ with $MN \le d_k$ , it takes the value $f_k(M)+ f_k(N)$ at some point. Prove that for each $m \in N$ there are positive integers $n,k$ and a set $S$ of $n$ points such that every coloring function $f_k$ of $S$ satisfies $| f_k(S)| \le m$

Prove that $$x^2 +\frac{8}{xy}+ y^2 \ge 8$$ for all positive real numbers $x$ and $y$.

Two masses are connected with spring constant $k$. The masses have magnitudes $m$ and $M$. The center-of-mass of the system is fixed. If $ k = \text {100 N/m} $ and $m=\dfrac{1}{2}M=\text{1 kg}$, let the ground state energy of the system be $E$. If $E$ can be expressed in the form $ a \times 10^p $ eV (electron-volts), find the ordered pair $(a,p)$, where $ 0 < a < 10 $, and it is rounded to the nearest positive integer and $p$ is an integer. For example, $ 4.2 \times 10^7 $ should be expressed as $(4,7)$. [i](Trung Phan, 10 points)[/i]

Solve in $(\mathbb{R}_{+}^{*})^{4}$ the following system : $\left\{\begin{matrix} x+y+z+t=4\\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}=5-\frac{1}{xyzt} \end{matrix}\right.$

The integers $-1$, $2$, $-3$, $4$, $-5$, $6$ are written on a blackboard. At each move, we erase two numbers $a$ and $b$, then we re-write $2a+b$ and $2b+a$. How many of the sextuples $(0,0,0,3,-9,9)$, $(0,1,1,3,6,-6)$, $(0,0,0,3,-6,9)$, $(0,1,1,-3,6,-9)$, $(0,0,2,5,5,6)$ can be gotten? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

Let $N$ be a positive integer. The sequence $x_1, x_2, \ldots$ of non-negative reals is defined by $$x_n^2=\sum_{i=1}^{n-1} \sqrt{x_ix_{n-i}}$$ for all positive integers $n>N$. Show that there exists a constant $c>0$, such that $x_n \leq \frac{n} {2}+c$ for all positive integers $n$.

$ABCD$ is a quadrilateral with $AD=BC$. If $\angle ADC$ is greater than $\angle BCD$, prove that $AC>BD$.

Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.

Let $ a,b,c$ be positive reals with $ ab \plus{} bc \plus{} ca \equal{} 3$. Prove that: \[ \frac {1}{1 \plus{} a^2(b \plus{} c)} \plus{} \frac {1}{1 \plus{} b^2(a \plus{} c)} \plus{} \frac {1}{1 \plus{} c^2(b \plus{} a)}\le \frac {1}{abc}. \]

Let $h_a, h_b, h_c$ be the lengths of the altitudes of a triangle $ABC$ from $A, B, C$ respectively. Let $P$ be any point inside the triangle. Show that \[\frac{PA}{h_b+h_c} + \frac{PB}{h_a+h_c} + \frac{PC}{h_a+h_b} \ge 1.\]

Prove that the inequality $ (x^{k}\minus{}y^{k})^{n}<(x^{n}\minus{}y^{n})^{k}$ holds forall reals $ x>y>0$ and positive integers $ n>k$.

In $\mathbb{R}^3$, for a rectangular box $\Delta$, let $10\Delta$ be the box with the same center as $\Delta$ but dilated by $10$. For example, if $\Delta$ is an $1 \times 1 \times 10$ box (hence with Lebesgue measure $10$), then $10\Delta$ is the $10 \times 10 \times 100$ box with the same center and orientation as $\Delta$. \medskip If two rectangular boxes $\Delta_1$ and $\Delta_2$ satisfy $\Delta_1 \subset 10\Delta_2$ and $\Delta_2 \subset 10 \Delta_1$, we say that they are [i]almost identical[/i]. Find the largest real number $a$ such that the following holds for some $C=C(a)>0$: For every positive integer $N$ and every collection $S$ of $1 \times 1 \times N$ boxes in $\mathbb{R}^3$, assuming that (i) $\vert S\vert=N$, (ii) every pair of boxes $(\Delta_1,\Delta_2)$ taken from $S$ are not almost identical, and (iii) the long edge of each box in $S$ forms an angle $\frac{\pi}{4}$ against the $xy$-plane. Then the volume \[\left\vert \bigcup_{\Delta \in S} \Delta\right\vert \ge CN^a.\]

Let $a, b$ and $c$ be real numbers such that $0 < a, b, c < 1$. Prove that: $$\min \ \ \{ab(1 -c)^2, bc(1 - a)^2, ca(1 - b)^2 \} \le \frac{1}{16}.$$

Let $ n$ be a positive integer. Find all $ x_1,x_2,\ldots,x_n$ that satisfy the relation: \[ \sqrt{x_1\minus{}1}\plus{}2\cdot \sqrt{x_2\minus{}4}\plus{}3\cdot \sqrt{x_3\minus{}9}\plus{}\cdots\plus{}n\cdot\sqrt{x_n\minus{}n^2}\equal{}\frac{1}{2}(x_1\plus{}x_2\plus{}x_3\plus{}\cdots\plus{}x_n).\]

a) Prove that for $x,y \ge 1$, holds $$x+y - \frac{1}{x}- \frac{1}{y} \ge 2\sqrt{xy} -\frac{2}{\sqrt{xy}}$$ b) Prove that for $a,b,c,d \ge 1$ with $abcd=16$ , holds $$a+b+c+d-\frac{1}{a}-\frac{1}{b}-\frac{1}{c}-\frac{1}{d}\ge 6$$