Given a triangle $ABC$, the internal bisectors through $A$ and $B$ meet the opposite sides in $D$ and $E$, respectively. Prove that \[DE \le (3 - 2\sqrt2)(AB + BC + CA)\] and determine the cases of equality.

Given a positive integer $n$, determine the maximal constant $C_n$ satisfying the following condition: for any partition of the set $\{1,2,\ldots,2n \}$ into two $n$-element subsets $A$ and $B$, there exist labellings $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ of $A$ and $B$, respectively, such that $$ (a_1-b_1)^2+(a_2-b_2)^2+\ldots+(a_n-b_n)^2\ge C_n. $$ [i](B. Serankou, M. Karpuk)[/i]

Determine all integers $n\ge1$ for which there exists $n$ real numbers $x_1,\ldots,x_n$ in the closed interval $[-4,2]$ such that the following three conditions are fulfilled: - the sum of these real numbers is at least $n$. - the sum of their squares is at most $4n$. - the sum of their fourth powers is at least $34n$. [i](Proposed by Gerhard Woeginger, Austria)[/i]

Each of 2010 boxes in a line contains a single red marble, and for $ 1 \le k \le 2010$, the box in the $ kth$ position also contains $ k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $ P(n)$ be the probability that Isabella stops after drawing exactly $ n$ marbles. What is the smallest value of $ n$ for which $ P(n) < \frac {1}{2010}$? $ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 63 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 201 \qquad \textbf{(E)}\ 1005$

Let $a$, $b$ and $c$ be real numbers such that $a^2+b^2=c^2$, solve the system: \[ z^2=x^2+y^2 \] \[ (z+c)^2=(x+a)^2+(y+b)^2 \] in real numbers $x, y$ and $z$.

Prove that $$\frac{\sin^3 a}{\sin b} +\frac{\cos^3 a}{\cos b} \ge \frac{1}{\cos(a - b)}$$ for all $a$ and $b$ in the interval $(0, \pi/2)$ .

Prove that $$\sqrt{x-1}+\sqrt{2x+9}+\sqrt{19-3x}<9$$ for all real $x$ for which the left-hand side is well defined.

Find the maximal value of the following expression, if $a,b,c$ are nonnegative and $a+b+c=1$. \[ \frac{1}{a^2 -4a+9} + \frac {1}{b^2 -4b+9} + \frac{1}{c^2 -4c+9} \]

Let $ ABC$ be a triangle inscribed in a circle of radius $ R$, and let $ P$ be a point in the interior of triangle $ ABC$. Prove that \[ \frac {PA}{BC^{2}} \plus{} \frac {PB}{CA^{2}} \plus{} \frac {PC}{AB^{2}}\ge \frac {1}{R}. \] [i]Alternative formulation:[/i] If $ ABC$ is a triangle with sidelengths $ BC\equal{}a$, $ CA\equal{}b$, $ AB\equal{}c$ and circumradius $ R$, and $ P$ is a point inside the triangle $ ABC$, then prove that $ \frac {PA}{a^{2}} \plus{} \frac {PB}{b^{2}} \plus{} \frac {PC}{c^{2}}\ge \frac {1}{R}$.

Let $f : R \to R$ be a function satisfying the inequality $|f(x + y) -f(x) - f(y)| < 1$ for all reals $x, y$. Show that $\left| f\left( \frac{x}{2008 }\right) - \frac{f(x)}{2008} \right|  < 1$ for all real numbers $x$.

Let, $ a,b,c$ real positive numbers with $ abc \equal{} 1$ Prove: $ (\frac {a}{1 \plus{} ab})^2 \plus{} (\frac {b}{1 \plus{} bc})^2 \plus{} (\frac {c}{1 \plus{} ca})^2\geq \frac {3}{4}$ Thanks!

Draw a $2004 \times 2004$ array of points. What is the largest integer $n$ for which it is possible to draw a convex $n$-gon whose vertices are chosen from the points in the array?

For the function $f : Z^2_{\ge0} \to Z_{\ge 0}$ it is known that $$f(0, j) = f(i, 0) = 1, \,\,\,\,\, \forall i, j \in N_0$$ $$f(i, j) = if (i, j - 1) + jf(i - 1, j),\,\,\,\,\, \forall i, j \in N$$ Prove that for every natural number $n$ the following inequality holds: $$\sum_{0\le i+j\le n+1} f(i, j) \le 2 \left(\sum^n_{k=0}\frac{1}{k!}\right)\left(\sum^n_{p=1}p!\right)+ 3$$

Determine the smallest value of $(a+5)^2+(b-2)^2+(c-9)^2$ for all real numbers $a, b, c$ satisfying $a^2+b^2+c^2-ab-bc-ca=3$.

A bagel is a loop of $2a+2b+4$ unit squares which can be obtained by cutting a concentric $a\times b$ hole out of an $(a +2)\times (b+2)$ rectangle, for some positive integers a and b. (The side of length a of the hole is parallel to the side of length $a+2$ of the rectangle.) Consider an infinite grid of unit square cells. For each even integer $n \ge 8$, a bakery of order $n$ is a finite set of cells $ S$ such that, for every $n$-cell bagel $B$ in the grid, there exists a congruent copy of $B$ all of whose cells are in $S$. (The copy can be translated and rotated.) We denote by $f(n)$ the smallest possible number of cells in a bakery of order $ n$. Find a real number $\alpha$ such that, for all sufficiently large even integers $n \ge 8$, we have $$\frac{1}{100}<\frac{f (n)}{n^ {\alpha}}<100$$ [i]Proposed by Nikolai Beluhov[/i]

The natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_n $ satisfy the condition $ 1 / a_1 + 1 / a_2 + \ldots + 1 / a_n = 1 $. Prove that all these numbers do not exceed $$ n ^ {2 ^ n} $$

Let $x,y{}$ and $z{}$ be positive real numbers satisfying $x+y+z=1$. Prove that [list=a] [*]\[1-\frac{x^2-yz}{x^2+x}=\frac{(1-y)(1-z)}{x^2+x};\] [*]\[\frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leqslant 0.\] [/list]

If $x, y, z, w$ are nonnegative real numbers satisfying \[\left\{ \begin{array}{l}y = x - 2003 \\ z = 2y - 2003 \\ w = 3z - 2003 \\ \end{array} \right. \] find the smallest possible value of $x$ and the values of $y, z, w$ corresponding to it.

Let $a_1, a_2, ..., a_n$ be more than one real numbers, such that $0\leq a_i\leq \frac{\pi}{2}$. Prove that $$\Bigg(\frac{1}{n}\sum_{i=1}^{n}\frac{1}{1+\sin a_i}\Bigg)\Bigg(1+\prod_{i=1}^{n}(\sin a_i)^{\frac{1}{n}}\Bigg)\leq1.$$

Let $a$, $b$, $c$ be non-negative real numbers such that $ab + bc + ca = 3$. Suppose that \[a^3 b + b^3 c + c^3 a + 2abc(a + b + c) = \frac{9}{2}.\] What is the maximum possible value of $ab^3 + bc^3 + ca^3$?

Let $f \colon [0,1] \to \mathbb{R} $ be a differentiable function such that its derivative is an integrable function on $[0,1]$, and $f(1)=0$. Prove that \[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]

Prove that among any $9$ distinct real numbers, there exist $4$ distinct numbers $a,b,c,d$ such that $$(ac+bd)^2\ge\frac{9}{10}(a^2+b^2)(c^2+d^2)$$

On the side $ AB$ of a cyclic quadrilateral $ ABCD$ there is a point $ X$ such that diagonal $ BD$ bisects $ CX$ and diagonal $ AC$ bisects $ DX$. What is the minimum possible value of $ AB\over CD$? [i]Proposed by S. Berlov[/i]

If $ x$, $ y$, $ z$ are nonnegative real numbers with the sum $ 1$, find the maximum value of $ S \equal{} x^2(y \plus{} z) \plus{} y^2(z \plus{} x) \plus{} z^2(x \plus{} y)$ and $ C \equal{} x^2y \plus{} y^2z \plus{} z^2x$.

Let $ a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $ a^2_1 \plus{} a^2_2 \plus{} \ldots \plus{} a^2_{100} \equal{} 1.$ Prove that \[ a^2_1 \cdot a_2 \plus{} a^2_2 \cdot a_3 \plus{} \ldots \plus{} a^2_{100} \cdot a_1 < \frac {12}{25}. \] [i]Author: Marcin Kuzma, Poland[/i]