Found problems: 6530
2014 JBMO Shortlist, 9
Let $n$ a positive integer and let $x_1, \ldots, x_n, y_1, \ldots, y_n$ real positive numbers such that $x_1+\ldots+x_n=y_1+\ldots+y_n=1$. Prove that:
$$|x_1-y_1|+\ldots+|x_n-y_n|\leq 2-\underset{1\leq i\leq n}{min} \;\dfrac{x_i}{y_i}-\underset{1\leq i\leq n}{min} \;\dfrac{y_i}{x_i}$$
2014 Cezar Ivănescu, 2
[b]a)[/b] Let be two nonegative integers $ n\ge 1,k, $ and $ n $ real numbers $ a,b,\ldots ,c. $ Prove that
$$ (1/a+1/b+\cdots 1/c)\left( a^{1+k} +b^{1+k}+\cdots c^{1+k} \right)\ge n\left(a^k+b^k+\cdots +c^k\right) . $$
[b]b)[/b] If $ 1\le d\le e\le f\le g\le h\le i\le 1000 $ are six real numbers, determine the minimum value the expression
$$ d/e+f/g+h/i $$
can take.
2016 Hanoi Open Mathematics Competitions, 13
Find all triples $(a,b,c)$ of real numbers such that $|2a + b| \ge 4$ and $|ax^2 + bx + c| \le 1$ $ \forall x \in [-1, 1]$.
2005 Indonesia MO, 6
Find all triples $ (x,y,z)$ of integers which satisfy
$ x(y \plus{} z) \equal{} y^2 \plus{} z^2 \minus{} 2$
$ y(z \plus{} x) \equal{} z^2 \plus{} x^2 \minus{} 2$
$ z(x \plus{} y) \equal{} x^2 \plus{} y^2 \minus{} 2$.
2000 Turkey Team Selection Test, 3
Suppose $f:\mathbb{R} \to \mathbb{R}$ is a function such that
\[|f(x+y)-f(x)-f(y)|\le 1\ \ \ \text{for all} \ \ x, y \in\mathbb R.\]
Prove that there is a function $g:\mathbb{R}\to\mathbb{R}$ such that $|f(x)-g(x)|\le 1$ and $g(x+y)=g(x)+g(y)$ for all $x,y \in\mathbb R.$
2014 USAMO, 6
Prove that there is a constant $c>0$ with the following property: If $a, b, n$ are positive integers such that $\gcd(a+i, b+j)>1$ for all $i, j\in\{0, 1, \ldots n\}$, then\[\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.\]
1943 Eotvos Mathematical Competition, 3
Let $a < b < c < d$ be real numbers and $(x,y, z,t)$ be any permutation of $a$,$b$, $c$ and $d$. What are the maximum and minimum values of the expression $$(x - y)^2 + (y- z)^2 + (z - t)^2 + (t - x)^2?$$
1986 Dutch Mathematical Olympiad, 3
The following apply: $a,b,c,d \ge 0$ and $abcd=1$
Prove that $$ a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd \ge 10$$
2011 Federal Competition For Advanced Students, Part 2, 2
Let $k$ and $n$ be positive integers.
Show that if $x_j$ ($1\leqslant j\leqslant n$) are real numbers with $\sum_{j=1}^n\frac{1}{x_j^{2^k}+k}=\frac{1}{k}$, then
\[\sum_{j=1}^n\frac{1}{x_j^{2^{k+1}}+k+2}\leqslant\frac{1}{k+1}\mbox{.}\]
2014 Saudi Arabia IMO TST, 1
Let $a_1,\dots,a_n$ be a non increasing sequence of positive real numbers. Prove that \[\sqrt{a_1^2+a_2^2+\cdots+a_n^2}\le a_1+\frac{a_2}{\sqrt{2}+1}+\cdots+\frac{a_n}{\sqrt{n}+\sqrt{n-1}}.\] When does equality hold?
1970 IMO Longlists, 5
Prove that $\sqrt[n]{\sum_{i=1}^{n}{\frac{i}{n+1}}}\ge 1$ for $2 \le n \in \mathbb{N}$.
2023 Mexican Girls' Contest, 7
Suppose $a$ and $b$ are real numbers such that $0 < a < b < 1$. Let
$$x= \frac{1}{\sqrt{b}} - \frac{1}{\sqrt{b+a}},\hspace{1cm}
y= \frac{1}{b-a} - \frac{1}{b}\hspace{0.5cm}\textrm{and}\hspace{0.5cm}
z= \frac{1}{\sqrt{b-a}} - \frac{1}{\sqrt{b}}.$$
Show that $x$, $y$, $z$ are always ordered from smallest to largest in the same way, regardless of the choice of $a$ and $b$. Find this order among $x$, $y$, $z$.
2022 Kosovo National Mathematical Olympiad, 2
Show that for any positive real numbers $a$ and $b$ the following inequality hold,
$$\frac{a(a+1)}{b+1}+\frac{b(b+1)}{a+1}\geq a+b.$$
1991 Kurschak Competition, 1
Let $n$ be a positive integer, and $a,b\ge 1$, $c>0$ arbitrary real numbers. Prove that
\[\frac{(ab+c)^n-c}{(b+c)^n-c}\le a^n.\]
1932 Eotvos Mathematical Competition, 3
Let $\alpha$, $\beta$ and $\gamma$ be the interior angles of an acute triangle. Prove that if $\alpha < \beta < \gamma$ then $$\sin 2\alpha >\ sin 2 \beta > \sin 2\gamma.$$
2023 Brazil EGMO Team Selection Test, 3
Let $a_1, a_2, \ldots , a_n$ be positive real numbers such that $a_1 + a_2 + \cdots + a_n = 1$. Prove that $$\dfrac{a_1}{\sqrt{1-a_1}}+\cdots+\dfrac{a_n}{\sqrt{1-a_n}} \geq \dfrac{1}{\sqrt{n-1}}(\sqrt{a_1}+\cdots+\sqrt{a_n}).$$
2019 Belarus Team Selection Test, 7.3
Given a positive integer $n$, determine the maximal constant $C_n$ satisfying the following condition: for any partition of the set $\{1,2,\ldots,2n \}$ into two $n$-element subsets $A$ and $B$, there exist labellings $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ of $A$ and $B$, respectively, such that
$$
(a_1-b_1)^2+(a_2-b_2)^2+\ldots+(a_n-b_n)^2\ge C_n.
$$
[i](B. Serankou, M. Karpuk)[/i]
2001 IMO Shortlist, 6
Prove that for all positive real numbers $a,b,c$, \[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]
2021 Nigerian Senior MO Round 2, 4
let $x_1$, $x_2$ .... $x_6$ be non-negative reals such that $x_1+x_2+x_3+x_4+x_5+x_6=1$ and $x_1x_3x_5$ + $x_2x_4x_6$ $\geq$ $\frac{1}{540}$. Let $p$ and $q$ be relatively prime integers such that $\frac{p}{q}$ is the maximum value of $x_1x_2x_3+x_2x_3x_4+x_3x_4x_5+x_4x_5x_6+x_5x_6x_1+x_6x_1x_2$. Find $p+q$
1996 Canada National Olympiad, 2
Find all real solutions to the following system of equations. Carefully justify your answer.
\[ \left\{ \begin{array}{c} \displaystyle\frac{4x^2}{1+4x^2} = y \\ \\ \displaystyle\frac{4y^2}{1+4y^2} = z \\ \\ \displaystyle\frac{4z^2}{1+4z^2} = x \end{array} \right. \]
2010 Contests, 3
Let $ABCD$ be a convex quadrilateral. $AC$ and $BD$ meet at $P$, with $\angle APD=60^{\circ}$. Let $E,F,G$, and $H$ be the midpoints of $AB,BC,CD$ and $DA$ respectively. Find the greatest positive real number $k$ for which
\[EG+3HF\ge kd+(1-k)s \]
where $s$ is the semi-perimeter of the quadrilateral $ABCD$ and $d$ is the sum of the lengths of its diagonals. When does the equality hold?
2013 Princeton University Math Competition, 16
Is $\cos 1^\circ$ rational? Prove.
2018 Balkan MO Shortlist, A6
Let $ x_1, x_2, \cdots, x_n$ be positive real numbers . Prove that:
$$\sum_ {i = 1}^n x_i ^2\geq \frac {1} {n + 1} \left (\sum_ {i = 1}^n x_i \right)^2+\frac{12(\sum_ {i = 1}^n i x_i)^2}{n (n + 1) (n + 2) (3n + 1)}. $$
2009 China Team Selection Test, 2
Given an integer $ n\ge 2$, find the maximal constant $ \lambda (n)$ having the following property: if a sequence of real numbers $ a_{0},a_{1},a_{2},\cdots,a_{n}$ satisfies $ 0 \equal{} a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n},$ and $ a_{i}\ge\frac {1}{2}(a_{i \plus{} 1} \plus{} a_{i \minus{} 1}),i \equal{} 1,2,\cdots,n \minus{} 1,$ then $ (\sum_{i \equal{} 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i \equal{} 1}^n{a_{i}^2}.$
2024 India IMOTC, 2
Let $x_1, x_2 \dots, x_{2024}$ be non-negative real numbers such that $x_1 \le x_2\cdots \le x_{2024}$, and $x_1^3 + x_2^3 + \dots + x_{2024}^3 = 2024$. Prove that
\[\sum_{1 \le i < j \le 2024} (-1)^{i+j} x_i^2 x_j \ge -1012.\]
[i]Proposed by Shantanu Nene[/i]