Let $a,b,c$ be the sides of a triangle, let $m_a,m_b,m_c$ be the corresponding medians, and let $D$ be the diameter of the circumcircle of the triangle. Prove that $\frac{a^2+b^2}{m_c}+\frac{a^2+c^2}{m_b}+\frac{b^2+c^2}{m_a} \leq 6D$.

Let $ a,b,c$ be positive real numbers. Prove that \[ \frac{a^3}{c(a^2 + bc)} + \frac{b^3}{a(b^2 + ca)} + \frac{c^3}{b(c^2 +ab )} \ge \frac{3}{2} . \]

Let $r$ be a rational number in the interval $[-1,1]$ and let $\theta = \cos^{-1} r$. Call a subset $S$ of the plane [i]good[/i] if $S$ is unchanged upon rotation by $\theta$ around any point of $S$ (in both clockwise and counterclockwise directions). Determine all values of $r$ satisfying the following property: The midpoint of any two points in a good set also lies in the set.

Let $ABC$ be an acute angle triangle. Suppose that $D,E,F$ are the feet of perpendicluar lines from $A,B,C$ to $BC,CA,AB$. Let $P,Q,R$ be the feet of perpendicular lines from $A,B,C$ to $EF,FD,DE$. Prove that \[ 2(PQ+QR+RP)\geq DE+EF+FD \]

Let $ABC$ be an acute triangle, and $H$ its orthocenter. The distance from $H$ to rays $BC$, $CA$, and $AB$ is denoted by $d_a$, $d_b$, and $d_c$, respectively. Let $R$ be the radius of circumcenter of $\triangle ABC$ and $r$ be the radius of incenter of $\triangle ABC$. Prove the following inequality: $$d_a+d_b+d_c \le \frac{3R^2}{4r}$$.

(i) Solve the simultaneous inequalities, $x<\frac{1}{4x}$ and $x<0$; i.e. find a single inequality equivalent to the two simultaneous inequalities. (ii) What is the greatest integer that satisfies both inequalities $4x+13 < 0$ and $x^{2}+3x > 16$. (iii) Give a rational number between $11/24$ and $6/13$. (iv) Express 100000 as a product of two integers neither of which is an integral multiple of 10. (v) Without the use of logarithm tables evaluate \[\frac{1}{\log_{2}36}+\frac{1}{\log_{3}36}.\]

Let $a,b,c$ be real numbers with sum equal to zero. Prove that \[ab^3+bc^3+ca^3\leqslant 0.\]

If $0<a<b$ then $$\int \limits_a^b \frac{\left (x^2-\left (\frac{a+b}{2} \right )^2\right )\ln \frac{x}{a} \ln \frac{x}{b}}{(x^2+a^2)(x^2+b^2)} dx > 0$$

Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that \[0<|xy-zw|<C\alpha ^{-4}\] where $\alpha =\frac{|X|}{n}$.

The point $A_1$ on the perimeter of a convex quadrilateral $ABCD$ is such that the line $AA_1$ divides the quadrilateral into two parts of equal area. The points $B_1$, $C_1$, $D_1$ are defined similarly. Prove that the area of the quadrilateral $A_1B_1C_1D_1$ is greater than a quarter of the area of $ABCD$. [i]L. Emelyanov [/i]

Let $x$ and $y$ be positive real numbers such that $x + y = 1$. Show that $$\left( \frac{x+1}{x} \right)^2 + \left( \frac{y+1}{y} \right)^2 \geq 18.$$

Let $a, b$ be distinct real numbers and $k,m$ be positive integers $k + m = n \ge 3, k \le 2m, m \le 2k$. Consider sequences $x_1,\dots , x_n$ with the following properties: (i) $k$ terms $x_i$, including $x_1$, are equal to $a$; (ii) $m$ terms $x_i$, including $x_n$, are equal to $b$; (iii) no three consecutive terms are equal. Find all possible values of $x_nx_1x_2 + x_1x_2x_3 + \cdots + x_{n-1}x_nx_1$.

Prove that ${d((n^2 +1)}^2)$ does not become monotonic from any given point onwards.

Find the smallest possible value of the expression $$\frac{(a+b) (b + c)}{a + 2b+c}$$ where $a, b, c$ are arbitrary numbers from the interval $[1,2]$.

Let $n \geq 4$ be a natural and let $x_1,\ldots,x_n$ be non-negative reals such that $x_1 + \cdots + x_n = 1$. Determine the maximum value of $x_1x_2x_3 + x_2x_3x_4 + \cdots + x_nx_1x_2$.

Let $ f(x,y,z)$ be a nonnegative harmonic function in the unit ball of $ \mathbb{R}^3$ for which the inequality $ f(x_0,0,0) \leq \varepsilon^2$ holds for some $ 0\leq x_0 \leq 1$ and $ 0<\varepsilon<(1\minus{}x_0)^2$. Prove that $ f(x,y,z) \leq \varepsilon$ in the ball with center at the origin an radius $ (1\minus{}3\varepsilon^{1/4}).$ [i]P. Turan[/i]

Let $a,b,c,d$ be real positive numbers such that $a^2+b^2+c^2+d^2 = 1$. Prove that $(1-a)(1-b)(1-c)(1-d) \geq abcd$.

Let $ \alpha(n)$ be the number of digits equal to one in the binary representation of a positive integer $ n.$ Prove that: (a) the inequality $ \alpha(n) (n^2 ) \leq \frac{1}{2} \alpha(n)(\alpha(n) + 1)$ holds; (b) the above inequality is an equality for infinitely many positive integers, and (c) there exists a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i }$ goes to zero as $ i$ goes to $ \infty.$ [i]Alternative problem:[/i] Prove that there exists a sequence a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i )}$ (d) $ \infty;$ (e) an arbitrary real number $ \gamma \in (0,1)$; (f) an arbitrary real number $ \gamma \geq 0$; as $ i$ goes to $ \infty.$

Let $x,y,z$ be real numbers such that $(x-z)(y-z)=x+y+z-3.$ Prove that $x^2+y^2+z^2\ge3.$

Let $D$ and $E$ be points on sides $AB$ and $AC$ respectively of a triangle $ABC$ such that $DE$ is parallel to $BC$ and tangent to the incircle of $ABC$. Prove that \[DE\le\frac{1}{8}(AB+BC+CA) \]

Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ be a monic polynomial of degree $4$. It is known that all the roots of $P$ are real, distinct and belong to the interval $[-1, 1]$. (a) Prove that $P(x) > -4$ for all real $x$. (b) Find the highest value of the real constant $k$ such that $P(x) > k$ for every real $x$ and for every polynomial $P(x)$ satisfying the given conditions.

For three points $A,B,C$ in the plane, we define $m(ABC)$ to be the smallest length of the three heights of the triangle $ABC$, where in the case $A$, $B$, $C$ are collinear, we set $m(ABC) = 0$. Let $A$, $B$, $C$ be given points in the plane. Prove that for any point $X$ in the plane, \[ m(ABC) \leq m(ABX) + m(AXC) + m(XBC). \]

Let $ABC$ be an acute-angled triangle with circumcenter $O$ and let $M$ be a point on $AB$. The circumcircle of $AMO$ intersects $AC$ a second time on $K$ and the circumcircle of $BOM$ intersects $BC$ a second time on $N$. Prove that $\left[MNK\right] \geq \frac{\left[ABC\right]}{4}$ and determine the equality case.

For $ a,b,c\geq 0$ with $ a\plus{}b\plus{}c\equal{}1$, prove that $ \sqrt{a\plus{}\frac{(b\minus{}c)^2}{4}}\plus{}\sqrt{b}\plus{}\sqrt{c}\leq \sqrt{3}$

Find the maximum possible value of $9\sqrt{x} + 8\sqrt{y} + 5\sqrt{z}$ where $x, y,$ and $z$ are positive real numbers satisfying $9x + 4y + z = 128$.