Found problems: 310
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
Let $a, b, c$ be positive reals such that $a + b + c = 3$. Prove:
\[ \frac{a^2}{a + \sqrt[3]{bc}} + \frac{b^2}{b + \sqrt[3]{ca}} + \frac{c^2}{c + \sqrt[3]{ab}} \geq \frac{3}{2} \]
And determine when equality holds.
Prove that if $x$, $y$, and $z$ are non-negative numbers and $x^2+y^2+z^2=1$, then the following inequality is true:
$\frac{x}{1-x^2}+\frac{y}{1-y^2}+\frac{z}{1-z^2 }\geq \frac{3\sqrt{3}}{2}$
Suppose $x,y,z\in \mathbb R^+$. Prove that \[\frac {x}{\sqrt{yz+4xy+4xz}}+\frac {y}{\sqrt{zx+4yz+4yx}}+\frac {z}{\sqrt{xy+4zx+4zy}}\geq 1\].
Of all triangles with given perimeter, find the triangle with the maximum area. Justify your answer
Let $M$ be a point in the interior of triangle $ABC$. Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$. Define $B'$ on $CA$ and $C'$ on $AB$ similarly. Define
\[
p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}.
\]
Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?
A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane such that
[list]
[*] no three points in $P$ lie on a line and
[*] no two points in $P$ lie on a line through the origin.
[/list]
A triangle with vertices in $P$ is [i]fat[/i] if $O$ is strictly inside the triangle. Find the maximum number of fat triangles.
Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum \[ {BC\over PD}+{CA\over PE}+{AB\over PF}. \]
Let $a_i > 0$ for $i=1,2,\dots,n$ and suppose $a_1 + a_2 + \dots + a_n = 1$. Prove that for any positive integer $k$,
\[ \left( a_1^k + \frac{1}{a_1^k} \right) \left( a_2^k + \frac{1}{a_2^k} \right) \dots \left( a_n^k + \frac{1}{a_n^k} \right) \ge \left( n^k + \frac{1}{n^k} \right)^n. \]
A polynomial is [i]good[/i] if it has integer coefficients, it is monic, all its roots are distinct, and there exists a disk with radius $0.99$ on the complex plane that contains all the roots. Prove that there is no [i]good[/i] polynomial for a sufficient large degree.
[i]Proposed by Rodrigo Sanches Angelo (rsa365), Brazil.[/i]
Let $ABCD$ be a convex quadrilateral and $K,L,M,N$ be points on $[AB],[BC],[CD],[DA]$, respectively. Show that,
\[
\sqrt[3]{s_{1}}+\sqrt[3]{s_{2}}+\sqrt[3]{s_{3}}+\sqrt[3]{s_{4}}\leq 2\sqrt[3]{s}
\]
where $s_1=\text{Area}(AKN)$, $s_2=\text{Area}(BKL)$, $s_3=\text{Area}(CLM)$, $s_4=\text{Area}(DMN)$ and $s=\text{Area}(ABCD)$.
[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
\]
[i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
\]
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $A_1,B_1$ and $C_1$ be respectively the midpoints of the arcs $BAC,CBA$ and $ACB$ of $\Gamma$. Show that the inradius of triangle $A_1B_1C_1$ is not less than the inradius of triangle $ABC$.
Let $ABC$ be a triangle with $\angle A = 60^\circ$, and $AA', BB', CC'$ be its internal angle bisectors. Prove that $\angle B'A'C' \le 60^\circ$.
Let $a,b,c$ be positive real numbers satisfying $a+b+c=3$.Prove that $\sum_{sym} \frac{a^{2}}{(b+c)^{3}}\geq \frac{3}{8}$
Let $a,b,c$ be positive real numbers such that $ab+bc+ca\le 3abc$. Prove that
\[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})\]
Suppose that $a, b, c$ are positive real numbers and $a + b + c = 3$. Prove that
$$\frac{a+b}{c+2} + \frac{b+c}{a+2} + \frac{c+a}{b+2} \geq 2$$
and determine when equality holds.
Let $f(x)=(x + a)(x + b)$ where $a,b>0$. For any reals $x_1,x_2,\ldots ,x_n\geqslant 0$ satisfying $x_1+x_2+\ldots +x_n =1$, find the maximum of $F=\sum\limits_{1 \leqslant i < j \leqslant n} {\min \left\{ {f({x_i}),f({x_j})} \right\}} $.
If $ a,b,c,d,e$ are positive numbers bounded by $ p$ and $ q$, i.e, if they lie in $ [p,q], 0 < p$, prove that
\[ (a \plus{} b \plus{} c \plus{} d \plus{} e)\left(\frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c} \plus{} \frac {1}{d} \plus{} \frac {1}{e}\right) \le 25 \plus{} 6\left(\sqrt {\frac {p}{q}} \minus{} \sqrt {\frac {q}{p}}\right)^2\]
and determine when there is equality.
Let $a,b,c$ be sides of a triangle. Prove that
\[ 2 < \frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b} - \frac{a^3+b^3+c^3}{abc}\leq 3 \]
Let $n\geq 2$ be an integer, and let $0 < a_1 < a_2 < \cdots < a_{2n+1}$ be real numbers. Prove the inequality
\[ \sqrt[n]{a_1} - \sqrt[n]{a_2} + \sqrt[n]{a_3} - \cdots + \sqrt[n]{a_{2n+1}} < \sqrt[n]{a_1 - a_2 + a_3 - \cdots + a_{2n+1}}. \]
[i]Bogdan Enescu, Romania[/i]
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc=1$.
Prove that
$ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq\frac{1}{2+a}+\frac{1}{2+b}+\frac{1}{2+c}$.
Given a convex polygon with $n$ sides and perimeter $S$, which has an incircle $\omega$ with radius $R$. A regular polygon with $n$ sides, whose vertices lie on $\omega$, has a perimeter $s$. Determine whether the following inequality holds:
\[ S \ge \frac{2sRn}{\sqrt{4n^2R^2-s^2}}. \]
There are some checkers in $n\cdot n$ size chess board.Known that for all numbers $1\le i,j\le n$ if checkwork in the intersection of $i$ th row and $j$ th column is empty,so the number of checkers that are in this row and column is at least $n$.Prove that there are at least $\frac{n^2}{2}$ checkers in chess board.
The sequence $(a_n)$ is defined by $a_1=1,a_2=\frac{1}{2}$,$$n(n+1)
a_{n+1}a_{n}+na_{n}a_{n-1}=(n+1)^2a_{n+1}a_{n-1}(n\ge 2).$$
Prove that $$\frac{2}{n+1}<\sqrt[n]{a_n}<\frac{1}{\sqrt{n}}(n\ge 3).$$