Found problems: 6530
1990 India National Olympiad, 5
Let $ a$, $ b$, $ c$ denote the sides of a triangle. Show that the quantity
\[ \frac{a}{b\plus{}c}\plus{}\frac{b}{c\plus{}a}\plus{}\frac{c}{a\plus{}b}\]
must lie between the limits $ 3/2$ and 2. Can equality hold at either limits?
2017 NMTC Junior, 4
a) $a,b,c,d$ are positive reals such that $abcd=1$. Prove that \[\sum_{cyc} \frac{1+ab}{1+a}\geq 4.\]
(b)In a scalene triangle $ABC$, $\angle BAC =120^{\circ}$. The bisectors of angles $A,B,C$ meets the opposite sides in $P,Q,R$ respectively. Prove that the circle on $QR$ as diameter passes through the point $P$.
1970 Czech and Slovak Olympiad III A, 3
Let $p>0$ be a given parameter. Determine all real $x$ such that \[\frac{1}{\,x+\sqrt{p-x^2\,}\,}+\frac{1}{\,x-\sqrt{p-x^2\,}\,}\ge\frac{1}{\,p\,}.\]
2012 Kyrgyzstan National Olympiad, 2
Given positive real numbers $ {a_1},{a_2},...,{a_n} $ with $ {a_1}+{a_2}+...+{a_n}= 1 $. Prove that
$ \left({\frac{1}{{a_1^2}}-1}\right)\left({\frac{1}{{a_2^2}}-1}\right)...\left({\frac{1}{{a_n^2}}-1}\right)\geqslant{({n^2}-1)^n} $.
2013 Argentina Cono Sur TST, 3
$1390$ ants are placed near a line, such that the distance between their heads and the line is less than $1\text{cm}$ and the distance between the heads of two ants is always larger than $2\text{cm}$. Show that there is at least one pair of ants such that the distance between their heads is at least $10$ meters (consider the head of an ant as point).
2021 Centroamerican and Caribbean Math Olympiad, 5
Let $n \geq 3$ be an integer and $a_1,a_2,...,a_n$ be positive real numbers such that $m$ is the smallest and $M$ is the largest of these numbers. It is known that for any distinct integers $1 \leq i,j,k \leq n$, if $a_i \leq a_j \leq a_k$ then $a_ia_k \leq a_j^2$. Show that
\[ a_1a_2 \cdots a_n \geq m^2M^{n-2} \]
and determine when equality holds
Kvant 2021, M2642
The nonzero numbers $x{}$ and $y{}$ satisfy the inequalities $x^{2n}-y^{2n}>x$ and $y^{2n}-x^{2n}>y$ for some natural number $n{}$. Can the product $xy$ be a negative number?
[i]Proposed by N. Agakhanov[/i]
1982 Czech and Slovak Olympiad III A, 5
Given is a sequence of real numbers $\{a_n\}^{\infty}_{n=1}$ such that $a_n \ne a_m$ for $n\ne m,$ given is a natural number $k$. Construct an injective map $P:\{1,2,\ldots,20k\}\to\mathbb Z^+$ such that the following inequalities hold:
$$a_{p(1)}<a_{p(2)}<...<a_{p(10)}$$
$$ a_{p(10)}>a_{p(11)}>...>a_{p(20)}$$
$$a_{p(20)}<a_{p(21)}<...<a_{p(30)}$$
$$...$$
$$a_{p(20k-10)}>a_{p(20k-9)}>...>a_{p(20k)}$$
$$a_{p(10)}>a_{p(30)}>...>a_{p((20k-10))} $$
$$a_{p(1)}<a_{p(20)}<...<a_{p(20k)},$$
2011 CentroAmerican, 5
If $x$, $y$, $z$ are positive numbers satisfying
\[x+\frac{y}{z}=y+\frac{z}{x}=z+\frac{x}{y}=2.\]
Find all the possible values of $x+y+z$.
2014 Czech and Slovak Olympiad III A, 6
For arbitrary non-negative numbers $a$ and $b$ prove inequality
$\frac{a}{\sqrt{b^2+1}}+\frac{b}{\sqrt{a^2+1}}\ge\frac{a+b}{\sqrt{ab+1}}$,
and find, where equality occurs.
(Day 2, 6th problem
authors: Tomáš Jurík, Jaromír Šimša)
2018 China National Olympiad, 6
China Mathematical Olympiad 2018 Q6
Given the positive integer $n ,k$ $(n>k)$ and $ a_1,a_2,\cdots ,a_n\in (k-1,k)$ ,if positive number $x_1,x_2,\cdots ,x_n$ satisfying:For any set $\mathbb{I} \subseteq \{1,2,\cdots,n\}$ ,$|\mathbb{I} |=k$,have $\sum_{i\in \mathbb{I} }x_i\le \sum_{i\in \mathbb{I} }a_i$ , find the maximum value of $x_1x_2\cdots x_n.$
2017 Harvard-MIT Mathematics Tournament, 7
Determine the largest real number $c$ such that for any $2017$ real numbers $x_1, x_2, \dots, x_{2017}$, the inequality $$\sum_{i=1}^{2016}x_i(x_i+x_{i+1})\ge c\cdot x^2_{2017}$$ holds.
2014 India IMO Training Camp, 2
For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$.Suppose that the following statements hold:
$x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$
$v_{1},v_{2},v_{3}$ satisfy triangle inequality
$v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality.
Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.
2022 Serbia JBMO TST, 1
Prove that for all positive real numbers $a$, $b$ the following inequality holds:
\begin{align*}
\sqrt{\frac{a^2+b^2}{2}}+\frac{2ab}{a+b}\ge \frac{a+b}{2}+ \sqrt{ab}
\end{align*}
When does equality hold?
1999 Bosnia and Herzegovina Team Selection Test, 2
Prove the inequality $$\frac{a^2}{b+c-a}+\frac{b^2}{a+c-b}+\frac{c^2}{a+b-c} \geq 3\sqrt{3}R$$ in triangle $ABC$ where $a$, $b$ and $c$ are sides of triangle and $R$ radius of circumcircle of $ABC$
1976 Bulgaria National Olympiad, Problem 4
Let $0<x_1\le x_2\le\ldots\le x_n$. Prove that
$$\frac{x_1}{x_2}+\frac{x_2}{x_3}+\ldots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}\ge\frac{x_2}{x_1}+\frac{x_3}{x_2}+\ldots+\frac{x_n}{x_{n-1}}+\frac{x_1}{x_n}$$
[i]I. Tonov[/i]
2009 Turkey MO (2nd round), 2
Show that
\[ \frac{(b+c)(a^4-b^2c^2)}{ab+2bc+ca}+\frac{(c+a)(b^4-c^2a^2)}{bc+2ca+ab}+\frac{(a+b)(c^4-a^2b^2)}{ca+2ab+bc} \geq 0 \]
for all positive real numbers $a, \: b , \: c.$
1968 IMO Shortlist, 5
Let $h_n$ be the apothem (distance from the center to one of the sides) of a regular $n$-gon ($n \geq 3$) inscribed in a circle of radius $r$. Prove the inequality
\[(n + 1)h_n+1 - nh_n > r.\]
Also prove that if $r$ on the right side is replaced with a greater number, the inequality will not remain true for all $n \geq 3.$
2009 China Girls Math Olympiad, 5
Let $ x,y,z$ be real numbers greater than or equal to $ 1.$ Prove that
\[ \prod(x^{2} \minus{} 2x \plus{} 2)\le (xyz)^{2} \minus{} 2xyz \plus{} 2.\]
2003 China Team Selection Test, 1
Let $g(x)= \sum_{k=1}^{n} a_k \cos{kx}$, $a_1,a_2, \cdots, a_n, x \in R$. If $g(x) \geq -1$ holds for every $x \in R$, prove that $\sum_{k=1}^{n}a_k \leq n$.
2011 AMC 10, 25
Let $T_1$ be a triangle with sides $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n=\triangle ABC$ and $D,E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB,BC$ and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD,BE,$ and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $(T_n)$?
$ \textbf{(A)}\ \frac{1509}{8} \qquad
\textbf{(B)}\ \frac{1509}{32} \qquad
\textbf{(C)}\ \frac{1509}{64} \qquad
\textbf{(D)}\ \frac{1509}{128} \qquad
\textbf{(E)}\ \frac{1509}{256} $
2010 Contests, 4
Given $n$ positive real numbers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n \ge 0$ and $x_1^2+x_2^2+\cdots+x_n^2=1$, prove that
\[\frac{x_1}{\sqrt{1}}+\frac{x_2}{\sqrt{2}}+\cdots+\frac{x_n}{\sqrt{n}}\ge 1.\]
1991 IMO Shortlist, 27
Determine the maximum value of the sum
\[ \sum_{i < j} x_ix_j (x_i \plus{} x_j)
\]
over all $ n \minus{}$tuples $ (x_1, \ldots, x_n),$ satisfying $ x_i \geq 0$ and $ \sum^n_{i \equal{} 1} x_i \equal{} 1.$
2018 Macedonia JBMO TST, 3
Let $x$, $y$, and $z$ be positive real numbers such that $x + y + z = 1$. Prove that
$\frac{(x+y)^3}{z} + \frac{(y+z)^3}{x} + \frac{(z+x)^3}{y} + 9xyz \ge 9(xy + yz + zx)$.
When does equality hold?
2024 Dutch IMO TST, 3
Let $a,b,c$ be real numbers such that $0 \le a \le b \le c$ and $a+b+c=1$. Show that
\[ab\sqrt{b-a}+bc\sqrt{c-b}+ac\sqrt{c-a}<\frac{1}{4}.\]