Found problems: 6530
2011 Morocco National Olympiad, 2
Let $\alpha , \beta ,\gamma$ be the angles of a triangle $ABC$ of perimeter $ 2p $ and $R$ is the radius of its circumscribed circle.
$(a)$ Prove that
\[\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right).\]
$(b)$ When do we have equality?
2016 Estonia Team Selection Test, 8
Let $x, y$ and $z$ be positive real numbers such that $x + y + z = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ . Prove that $xy + yz + zx \ge 3$.
1991 Putnam, B6
Let $a$ and $b$ be positive numbers. Find the largest number $c$, in terms of $a$ and $b$, such that for all $x$ with $0<|x|\le c$ and for all $\alpha$ with $0<\alpha<1$, we have:
$$a^\alpha b^{1-\alpha}\le\frac{a\sinh\alpha x}{\sinh x}+\frac{b\sinh x(1-\alpha)}{\sinh x}.$$
2006 Romania National Olympiad, 3
We have in the plane the system of points $A_1,A_2,\ldots,A_n$ and $B_1,B_2,\ldots,B_n$, which have different centers of mass. Prove that there is a point $P$ such that \[ PA_1 + PA_2 + \ldots+ PA_n = PB_1 + PB_2 + \ldots + PB_n . \]
2005 Iran MO (3rd Round), 3
Find all $\alpha>0$ and $\beta>0$ that for each $(x_1,\dots,x_n)$ and $(y_1,\dots,y_n)\in\mathbb {R^+}^n$ that:\[(\sum x_i^\alpha)(\sum y_i^\beta)\geq\sum x_iy_i\]
1967 Dutch Mathematical Olympiad, 4
The following applies: $$a, b, c, d > 0 , a + b < c + d$$
Prove that $$ac + bd > ab.$$
2007 ITAMO, 6
a) For each $n \ge 2$, find the maximum constant $c_{n}$ such that
$\frac 1{a_{1}+1}+\frac 1{a_{2}+1}+\ldots+\frac 1{a_{n}+1}\ge c_{n}$
for all positive reals $a_{1},a_{2},\ldots,a_{n}$ such that $a_{1}a_{2}\cdots a_{n}= 1$.
b) For each $n \ge 2$, find the maximum constant $d_{n}$ such that
$\frac 1{2a_{1}+1}+\frac 1{2a_{2}+1}+\ldots+\frac 1{2a_{n}+1}\ge d_{n}$
for all positive reals $a_{1},a_{2},\ldots,a_{n}$ such that $a_{1}a_{2}\cdots a_{n}= 1$.
2017 Auckland Mathematical Olympiad, 2
The sum of the three nonnegative real numbers $ x_1, x_2, x_3$ is not greater than $\frac12$.
Prove that $(1 - x_1)(1 - x_2)(1 - x_3) \ge \frac12$
2023 Quang Nam Province Math Contest (Grade 11), Problem 5
a) Given an acute triangle $ABC(AB>AC).$ The circle $(O)$ with diameter $BC$ intersects $AB,AC$ at $F,E$, respectively. Let $H$ be the intersection point of $BE,CF,$ the line $AH$ intersects the line $BC$ at $D,$ the line $EF$ intersects the line $BC$ at $K.$ The line passing through $D$ and parallel to $EF$ intersects $AB,AC$ at $M,N,$ respectively.
Prove that: $M,O,N,K$ are on the same circle.
b) Given $\triangle ABC, \angle BAC=\angle BCA=30^{\circ}.$ $D,E,F$ are moving points on the side $AB,BC,CA$ such that: $\angle BFD=\angle BFE=60^{\circ}.$ Let $p,p_1$ be the perimeter of $\triangle ABC,\triangle DEF,$ respectively. Prove that: $p\le 2p_1.$
2004 India National Olympiad, 4
$ABC$ is a triangle, with sides $a$, $b$, $c$ , circumradius $R$, and exradii $r_a$, $r_b$, $r_c$If $2R\leq r_a$, show that $a > b$, $a > c$, $2R > r_b$, and $2R > r_c$.
1993 Rioplatense Mathematical Olympiad, Level 3, 4
$x$ and $y$ are real numbers such that $6 -x$, $3 + y^2$, $11 + x$, $14 - y^2$ are greater than zero.
Find the maximum of the function $$f(x,y) = \sqrt{(6 -x)(3 + y^2)} + \sqrt{(11 + x)(14 - y^2)}.$$
2003 Switzerland Team Selection Test, 6
Let $ a,b,c $ be positive real numbers satisfying $ a+b+c=2 $. Prove the inequality \[ \frac{1}{1+ab}+\frac{1}{1+bc}+\frac{1}{1+ca} \ge \frac{27}{13} \]
2019 Peru EGMO TST, 5
Define the sequence sequence $a_0,a_1, a_2,....,a_{2018}, a_{2019}$ of real numbers as follows:
$\bullet$ $a_0 = 1$.
$\bullet$ $a_{n + 1} = a_n - \frac{a_n^2}{2019}$ for $n = 0, 1, ...,2018$.
Prove that $a_{2019} < \frac12 <a_{2018}$.
2004 Romania National Olympiad, 3
Let $f : \left[ 0,1 \right] \to \mathbb R$ be an integrable function such that \[ \int_0^1 f(x) \, dx = \int_0^1 x f(x) \, dx = 1 . \] Prove that \[ \int_0^1 f^2 (x) \, dx \geq 4 . \]
[i]Ion Rasa[/i]
2017 Taiwan TST Round 1, 1
Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$
[i]Proposed by Tigran Margaryan, Armenia[/i]
2011 Mathcenter Contest + Longlist, 3 sl3
We will call the sequence of positive real numbers. $a_1,a_2,\dots ,a_n$ of [i]length [/i] $n$ when $$a_1\geq \frac{a_1+a_2}{2}\geq \dots \geq \frac{a_1+a_2+\cdots +a_n}{n}.$$
Let $x_1,x_2,\dots ,x_n$ and $y_1,y_2,\dots ,y_n$ be sequences of length $n.$ Prove that
$$\sum_{i = 1}^{n}x_iy_i\geq\frac{1}{n}\left(\sum_{i = 1}^{n}x_i\right)\left(\sum_{i = 1}^{n}y_i\right).$$
[i](tatari/nightmare)[/i]
1996 Iran MO (2nd round), 4
Let $n$ blue points $A_i$ and $n$ red points $B_i \ (i = 1, 2, \ldots , n)$ be situated on a line. Prove that
\[\sum_{i,j} A_i B_j \geq \sum_{i<j} A_iA_j + \sum_{i<j} B_iB_j.\]
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 $\sum_{j=1}^n |a_j| \leq 3$.
2020 Latvia Baltic Way TST, 3
Prove that for all positive integers $n$ the following inequality holds:
$$ \frac{1}{1^2 +2020}+\frac{1}{2^2+2020} + \ldots + \frac{1}{n^2+2020} < \frac{1}{22} $$
2013 Kosovo National Mathematical Olympiad, 5
Let $P$ be a point inside or outside (but not on) of a triangle $ABC$. Prove that $PA +PB +PC$ is greater than half of the perimeter of the triangle
2021 Olympic Revenge, 1
Let $a$, $b$, $c$, $k$ be positive reals such that $ab+bc+ca \leq 1$ and $0 < k \leq \frac{9}{2}$. Prove that:
\[\sqrt[3]{ \frac{k}{a} + (9-3k)b} + \sqrt[3]{\frac{k}{b} + (9-3k)c} + \sqrt[3]{\frac{k}{c} + (9-3k)a } \leq \frac{1}{abc}.\]
[i]Proposed by Zhang Yanzong and Song Qing[/i]
2020 Baltic Way, 3
A real sequence $(a_n)_{n=0}^\infty$ is defined recursively by $a_0 = 2$ and the recursion formula
$$
a_{n} =
\begin{dcases}
a_{n-1}^2 & \text{if $a_{n-1}<\sqrt3$} \\
\frac{a_{n-1}^2}{3} & \text{if $a_{n-1}\geq\sqrt 3$.}
\end{dcases}
$$
Another real sequence $(b_n)_{n=1}^\infty$ is defined in terms of the first by the formula
$$
b_{n} =
\begin{dcases}
0 & \text{if $a_{n-1}<\sqrt3$} \\
\frac{1}{2^{n}} & \text{if $a_{n-1}\geq\sqrt 3$,}
\end{dcases}
$$
valid for each $n\geq 1$. Prove that
$$
b_1 + b_2 + \cdots + b_{2020} < \frac23.
$$
2020-IMOC, A5
Let $0<c<1$ be a given real number. Determine the least constant $K$ such that the following holds: For all positive real $M$ that is greater than $1$, there exists a strictly increasing sequence $x_0, x_1, \ldots, x_n$ (of arbitrary length) such that $x_0=1, x_n\geq M$ and
\[\sum_{i=0}^{n-1}\frac{\left(x_{i+1}-x_i\right)^c}{x_i^{c+1}}\leq K.\]
(From 2020 IMOCSL A5. I think this problem is particularly beautiful so I want to make a separate thread for it :D )
2008 Irish Math Olympiad, 5
Suppose that $ x, y$ and $ z$ are positive real numbers such that $ xyz \ge 1$.
(a) Prove that
$ 27 \le (1 \plus{} x \plus{} y)^2 \plus{} (1\plus{} x \plus{} z)^2 \plus{} (1 \plus{} y \plus{} z)^2$,
with equality if and only if $ x \equal{} y \equal{} z \equal{} 1$.
(b) Prove that
$ (1 \plus{} x \plus{} y)^2 \plus{} (1\plus{} x \plus{} z)^2 \plus{} (1 \plus{} y \plus{} z)^2$ $ \le 3(x \plus{} y \plus{} z)^2$,
with equality if and only if $ x \equal{} y \equal{} z \equal{} 1$.
2002 JBMO ShortLists, 13
Let $ A_1,A_2,...,A_{2002}$ be arbitrary points in the plane. Prove that for every circle of radius $ 1$ and for every rectangle inscribed in this circle, there exist $3$ vertices $ M,N,P$ of the rectangle such that
$ MA_1 + MA_2 + \cdots + MA_{2002} + $ $NA_1 + NA_2 + \cdots + NA_{2002} + $ $PA_1 + PA_2 + \cdots + PA_{2002}\ge 6006$.