Found problems: 310
Let $n \geq 2$ be a positive integer and $a_{1},\ldots , a_{n}$ be positive real numbers such that $a_{1}+...+a_{n}= 1$. Prove that:
\[\frac{a_{1}}{1+a_{2}+\cdots +a_{n}}+\cdots +\frac{a_{n}}{1+a_{1}+a_{2}+\cdots +a_{n-1}}\geq \frac{n}{2n-1}\]
Prove that for all positive real numbers $a_1,a_2,\cdots a_n$ \[\frac{1}{\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots +\frac{1}{1+a_n}}-\frac{1}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots +\frac{1}{a_n}}\geq \frac{1}{n}\] When does the inequality hold?
Let $ABC$ be an acute triangle with $\omega,S$, and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\omega_{A}$ is tangent internally to $S$ at $A$ and tangent externally to $\omega$. Circle $S_{A}$ is tangent internally to $S$ at $A$ and tangent internally to $\omega$. Let $P_{A}$ and $Q_{A}$ denote the centers of $\omega_{A}$ and $S_{A}$, respectively. Define points $P_{B}, Q_{B}, P_{C}, Q_{C}$ analogously. Prove that
\[8P_{A}Q_{A}\cdot P_{B}Q_{B}\cdot P_{C}Q_{C}\leq R^{3}\; , \]
with equality if and only if triangle $ABC$ is equilateral.
If $a, b, c, d$ are positive numbers satisfying $a^3 + b^3 +3ab = c + d = 1,$ prove that \[\left(a+\frac{1}{a}\right)^3+\left(b+\frac{1}{b}\right)^3+\left(c+\frac{1}{c}\right)^3+\left(d+\frac{1}{d}\right)^3\geq 40.\]
Which configurations of five (not necessarily distinct) points $p_1 ,\ldots, p_5$ on the circle $x^2 +y^2 =1$ maximize the sum of the ten distances
$$\sum_{i<j} d(p_i, p_j)?$$
If the inradius of a triangle is half of its circumradius, prove that the triangle is equilateral.
In a group of mathematicians, every mathematician has some friends (the relation of friend is reciprocal). Prove that there exists a mathematician, such that the average of the number of friends of all his friends is no less than the average of the number of friends of all these mathematicians.
Let $\Delta ABC$ be a triangle. The internal bisectors of angles $\angle CAB$ and $\angle ABC$ intersect segments $BC$, respectively $AC$ in $D$, respectively $E$. Prove that \[DE\leq (3-2\sqrt{2})(AB+BC+CA).\]
Let $a, b, c$ be positive real numbers such that
$$a^2 + b^2 + c^2 = \frac{1}{4}.$$
Prove that
$$\frac{1}{\sqrt{b^2 + c^2}} + \frac{1}{\sqrt{c^2 + a^2}} + \frac{1}{\sqrt{a^2 + b^2}} \le \frac{\sqrt{2}}{(a + b)(b + c)(c + a)}.$$
[i]Proposed by Petar Filipovski, Macedonia[/i]
Let $a,b,c$ be positive real numbers satisfying $a+b+c=3$.Prove that $\sum_{sym} \frac{a^{2}}{(b+c)^{3}}\geq \frac{3}{8}$
Find $n$ points $p_1, p_2, \ldots, p_n$ on the circumference of a unit circle, such that $\sum_{1\leq i< j \leq n} p_i p_j$ is maximal.
Prove the inequality
[b]a.)[/b] $
\left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left(
a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) , $
where $k\geq 1$ is a natural number and $a_{1},$ $a_{2},$ $...,$ $a_{k}$ are arbitrary real numbers.
[b]b.)[/b] Using the inequality (1), show that if the real numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ satisfy the inequality
\[
a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left(
a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) },
\]
then all of these numbers $a_{1},$ $a_{2},$ $\ldots,$ $a_{n}$ are non-negative.
Let $ABC$ be a triangle inscribed in a circle and let
\[ l_a = \frac{m_a}{M_a} \ , \ \ l_b = \frac{m_b}{M_b} \ , \ \ l_c = \frac{m_c}{M_c} \ , \]
where $m_a$,$m_b$, $m_c$ are the lengths of the angle bisectors (internal to the triangle) and $M_a$, $M_b$, $M_c$ are the lengths of the angle bisectors extended until they meet the circle. Prove that
\[ \frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \geq 3 \]
and that equality holds iff $ABC$ is an equilateral triangle.
We have an equilateral triangle with circumradius $1$. We extend its sides. Determine the point $P$ inside the triangle such that the total lengths of the sides (extended), which lies inside the circle with center $P$ and radius $1$, is maximum.
(The total distance of the point P from the sides of an equilateral triangle is fixed )
[i]Proposed by Erfan Salavati[/i]
$x,y,z$ are three real numbers inequal to zero satisfying $x+y+z=xyz$.
Prove that
$$ \sum (\frac{x^2-1}{x})^2 \geq 4$$
[i]Proposed by Amin Fathpour[/i]
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:
$$d_a+d_b+d_c \le \frac{3R^2}{4r}$$.
Given real numbers $x,y,z$ such that $x+y+z=0$, show that
\[\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0\]
When does equality hold?
Let $I \subset \mathbb{R}$ be an open interval and $f:I \to \mathbb{R}$ a twice differentiable function such that $f(x)f''(x)=0,$ for any $x \in I.$ Prove that $f''(x)=0,$ for any $x \in I.$
Set $T$ consists of $66$ points in plane, and $P$ consists of $16$ lines in plane. Pair $(A,l)$ is [i]good[/i] if $A \in T$, $l \in P$ and $A \in l$. Prove that maximum number of good pairs is no greater than $159$, and prove that there exits configuration with exactly $159$ good pairs.
Let $(a_n)^{+\infty}_{n=1}$ be a sequence defined recursively as follows: $a_1=1$ and $$a_{n+1}=1 + \sum\limits_{k=1}^{n}ka_k$$
For every $n > 1$, prove that $\sqrt[n]{a_n} < \frac {n+1}{2}$.
Let $ ABC$ be a triangle with circumradius $ R$, perimeter $ P$ and area $ K$. Determine the maximum value of: $ \frac{KP}{R^3}$.
Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$.
Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$
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]
Graph $G$ is defined in 3d space. It has $e$ edges and every vertex are connected if the distance between them is $1.$ Given that there exists the Hamilton cycle, prove that for $e>1,$ we have $$\min d(v)\le 1+2\left(\frac{e}{2}\right)^{0.4}.$$
Let $a, b, c$ be positive reals so that $a^2+b^2+c^2=1$. Find the minimum value of $S=1/a^2+1/b^2+1/c^2-2(a^3+b^3+c^3)/abc$