Found problems: 6530
2013 Vietnam Team Selection Test, 1
The $ABCD$ is a cyclic quadrilateral with no parallel sides inscribed in circle $(O, R)$. Let $E$ be the intersection of two diagonals and the angle bisector of $AEB$ cut the lines $AB, BC, CD, DA$ at $M, N, P, Q$ respectively .
a) Prove that the circles $(AQM), (BMN), (CNP), (DPQ)$ are passing through a point. Call that point $K$.
b) Denote $min \,\{AC, BD\} = m$. Prove that $OK \le \dfrac{2R^2}{\sqrt{4R^2-m^2}}$.
1987 IMO Longlists, 14
Given $n$ real numbers $0 < t_1 \leq t_2 \leq \cdots \leq t_n < 1$, prove that
\[(1-t_n^2) \left( \frac{t_1}{(1-t_1^2)^2}+\frac{t_2}{(1-t_2^3)^2}+\cdots +\frac{t_n}{(1-t_n^{n+1})^2} \right) < 1.\]
2005 Iran Team Selection Test, 1
Suppose that $ a_1$, $ a_2$, ..., $ a_n$ are positive real numbers such that $ a_1 \leq a_2 \leq \dots \leq a_n$. Let
\[ {{a_1 \plus{} a_2 \plus{} \dots \plus{} a_n} \over n} \equal{} m; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{a_1^2 \plus{} a_2^2 \plus{} \dots \plus{} a_n^2} \over n} \equal{} 1.
\]
Suppose that, for some $ i$, we know $ a_i \leq m$. Prove that:
\[ n \minus{} i \geq n \left(m \minus{} a_i\right)^2
\]
2004 South East Mathematical Olympiad, 1
Let real numbers a, b, c satisfy $a^2+2b^2+3c^2= \frac{3}{2}$, prove that $3^{-a}+9^{-b}+27^{-c}\ge1$.
2016 IFYM, Sozopol, 3
Let $x\leq y\leq z$ be real numbers such that $x+y+z=12$, $x^2+y^2+z^2=54$. Prove that:
a) $x\leq 3$ and $z\geq 5$
b) $xy$, $yz$, $zx\in [9,25]$
1998 Singapore MO Open, 2
Let $N$ be the set of natural numbers, and let $f: N \to N$ be a function satisfying $f(x) + f(x + 2) < 2 f(x + 1)$ for any $x \in N$. Prove that there exists a straight line in the $xy$-plane which contains infinitely many points with coordinates $(n,f(n))$.
MathLinks Contest 5th, 2.3
Let $a, b, c$ be positive numbers such that $abc \le 8$. Prove that
$$\frac{1}{a^2 - a + 1} +\frac{1}{b^2 - b + 1}++\frac{1}{c^2 - c + 1} \ge 1$$
2005 China Team Selection Test, 3
Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds.
Prove that $\sum_{j=1}^n |a_j| \leq 3$.
2004 Brazil Team Selection Test, Problem 3
Set $\mathbb Q_1=\{x\in\mathbb Q\mid x\ge1\}$. Suppose that a function $f:\mathbb Q_1\to\mathbb R$ satisfies the inequality $\left|f(x+y)-f(x)-f(y)\right|<\epsilon$ for all $x,y\in\mathbb Q_1$, where $\epsilon>0$ is given. Prove that there exists a real number $q$ such that
$$\left|\frac{f(x)}x-q\right|<2\epsilon\qquad\text{for all }x\in\mathbb Q_1.$$
2007 Federal Competition For Advanced Students, Part 1, 2
For every positive integer $ n$ determine the highest value $ C(n)$, such that for every $ n$-tuple $ (a_1,a_2,\ldots,a_n)$ of pairwise distinct integers
$ (n \plus{} 1)\sum_{j \equal{} 1}^n a_j^2 \minus{} \left(\sum_{j \equal{} 1}^n a_j\right)^2\geq C(n)$
1904 Eotvos Mathematical Competition, 3
Let $A_1A_2$ and $B_1B_2$ be the diagonals of a rectangle, and let $O$ be its center. Find and construct the set of all points $P$ that satisfy simultaneously the four inequaliies:
$$A_1P > OP , \\A_2P > OP, \ \ B_1P > OP , \ \ B_2P > OP.$$
2012 Bosnia And Herzegovina - Regional Olympiad, 4
Prove the inequality: $$\frac{A+a+B+b}{A+a+B+b+c+r}+\frac{B+b+C+c}{B+b+C+c+a+r}>\frac{C+c+A+a}{C+c+A+a+b+r}$$ where $A$, $B$, $C$, $a$, $b$, $c$ and $r$ are positive real numbers
2007 Today's Calculation Of Integral, 182
Find the area of the domain of the system of inequality
\[y(y-|x^{2}-5|+4)\leq 0,\ \ y+x^{2}-2x-3\leq 0. \]
2007 Romania National Olympiad, 3
a) In a triangle $ MNP$, the lenghts of the sides are less than $ 2$. Prove that the lenght of the altitude corresponding to the side $ MN$ is less than $ \sqrt {4 \minus{} \frac {MN^2}{4}}$.
b) In a tetrahedron $ ABCD$, at least $ 5$ edges have their lenghts less than $ 2$.Prove that the volume of the tetrahedron is less than $ 1$.
2015 Thailand TSTST, 2
Let $a, b, c\in (0, 1)$ with $a + b + c = 1$. Prove that $$\frac{a^5+b^5}{a^3+b^3}+\frac{b^5+c^5}{b^3+c^3}+\frac{c^5+a^5}{c^3+a^3}\geq\frac{a}{8+b^3+c^3}+\frac{b}{8+c^3+a^3}+\frac{c}{8+a^3+b^3}.$$
1992 Mexico National Olympiad, 5
$x, y, z$ are positive reals with sum $3$. Show that $$6 < \sqrt{2x+3} + \sqrt{2y+3} + \sqrt{2z+3}\le 3\sqrt5$$
2015 Sharygin Geometry Olympiad, 1
Circles $\alpha$ and $\beta$ pass through point $C$. The tangent to $\alpha$ at this point meets $\beta$ at point $B$, and the tangent to $\beta$ at $C$ meets $\alpha$ at point $A$ so that $A$ and $B$ are distinct from $C$ and angle $ACB$ is obtuse. Line $AB$ meets $\alpha$ and $\beta$ for the second time at points $N$ and $M$ respectively. Prove that $2MN < AB$.
(D. Mukhin)
2015 Balkan MO, 1
If ${a, b}$ and $c$ are positive real numbers, prove that
\begin{align*}
a ^ 3b ^ 6 + b ^ 3c ^ 6 + c ^ 3a ^ 6 + 3a ^ 3b ^ 3c ^ 3 &\ge{ abc \left (a ^ 3b ^ 3 + b ^ 3c ^ 3 + c ^ 3a ^ 3 \right) + a ^ 2b ^ 2c ^ 2 \left (a ^ 3 + b ^ 3 + c ^ 3 \right)}.
\end{align*}
[i](Montenegro).[/i]
2005 Taiwan TST Round 3, 1
Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,A_2,\dots,A_n$ the sequence of arithmetic means defined by \[ A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. \] Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality \[
n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 \] and establish the cases of equality.
[i]Proposed by Finbarr Holland, Ireland[/i]
2015 CCA Math Bonanza, L3.4
Compute the greatest constant $K$ such that for all positive real numbers $a,b,c,d$ measuring the sides of a cyclic quadrilateral, we have
\[
\left(\frac{1}{a+b+c-d}+\frac{1}{a+b-c+d}+\frac{1}{a-b+c+d}+\frac{1}{-a+b+c+d}\right)(a+b+c+d)\geq K.
\]
[i]2015 CCA Math Bonanza Lightning Round #3.4[/i]
2014 Saudi Arabia BMO TST, 3
Let $a, b$ be two nonnegative real numbers and $n$ a positive integer. Prove that \[\left(1-2^{-n}\right)\left|a^{2^n}-b^{2^n}\right|\ge\sqrt{ab}\left|a^{2^n-1}-b^{2^n-1}\right|.\]
1995 China National Olympiad, 1
Let $a_1,a_2,\cdots ,a_n; b_1,b_2,\cdots ,b_n (n\ge 3)$ be real numbers satisfying the following conditions:
(1) $a_1+a_2+\cdots +a_n= b_1+b_2+\cdots +b_n $;
(2) $0<a_1=a_2, a_i+a_{i+1}=a_{i+2}$ ($i=1,2,\cdots ,n-2$);
(3) $0<b_1\le b_2, b_i+b_{i+1}\le b_{i+2}$ ($i=1,2,\cdots ,n-2$).
Prove that $a_{n-1}+a_n\le b_{n-1}+b_n$.
2010 Romania Team Selection Test, 1
Let $n$ be a positive integer and let $x_1, x_2, \ldots, x_n$ be positive real numbers such that $x_1x_2 \cdots x_n = 1$. Prove that \[\displaystyle\sum_{i=1}^n x_i^n (1 + x_i) \geq \dfrac{n}{2^{n-1}} \prod_{i=1}^n (1 + x_i).\]
[i]IMO Shortlist[/i]
2013 Mexico National Olympiad, 6
Let $A_1A_2 ... A_8$ be a convex octagon such that all of its sides are equal and its opposite sides are parallel. For each $i = 1, ... , 8$, define $B_i$ as the intersection between segments $A_iA_{i+4}$ and $A_{i-1}A_{i+1}$, where $A_{j+8} = A_j$ and $B_{j+8} = B_j$ for all $j$. Show some number $i$, amongst 1, 2, 3, and 4 satisfies
\[\frac{A_iA_{i+4}}{B_iB_{i+4}} \leq \frac{3}{2}\]
2005 Junior Balkan Team Selection Tests - Romania, 9
Let $ABC$ be a triangle with $BC>CA>AB$ and let $G$ be the centroid of the triangle. Prove that \[ \angle GCA+\angle GBC<\angle BAC<\angle GAC+\angle GBA . \]
[i]Dinu Serbanescu[/i]