Found problems: 583
2024 Junior Balkan Team Selection Tests - Moldova, 10
Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that
$$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$
2016 India Regional Mathematical Olympiad, 4
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Determine, with certainty, the largest possible value of the expression $$ \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b}$$
2023 JBMO Shortlist, A3
Prove that for all non-negative real numbers $x,y,z$, not all equal to $0$, the following inequality holds
$\displaystyle \dfrac{2x^2-x+y+z}{x+y^2+z^2}+\dfrac{2y^2+x-y+z}{x^2+y+z^2}+\dfrac{2z^2+x+y-z}{x^2+y^2+z}\geq 3.$
Determine all the triples $(x,y,z)$ for which the equality holds.
[i]Milan Mitreski, Serbia[/i]
1975 IMO Shortlist, 2
We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$
2003 Junior Tuymaada Olympiad, 5
Prove that for any real $ x $ and $ y $ the inequality $x^2 \sqrt {1+2y^2} + y^2 \sqrt {1+2x^2} \geq xy (x+y+\sqrt{2})$ .
2000 Moldova National Olympiad, Problem 6
Let $(a_n)_{n\ge0}$ be a sequence of positive numbers that satisfy the relations $a_{i-1}a_{i+1}\le a_i^2$ for all $i\in\mathbb N$. For any integer $n>1$, prove the inequality
$$\frac{a_0+\ldots+a_n}{n+1}\cdot\frac{a_1+\ldots+a_{n-1}}{n-1}\ge\frac{a_0+\ldots+a_{n-1}}n\cdot\frac{a_1+\ldots+a_n}n.$$
2021 Spain Mathematical Olympiad, 4
Let $a,b,c,d$ real numbers such that:
$$
a+b+c+d=0 \text{ and } a^2+b^2+c^2+d^2 = 12
$$
Find the minimum and maximum possible values for $abcd$, and determine for which values of $a,b,c,d$ the minimum and maximum are attained.
2022 Azerbaijan JBMO TST, A2
For positive real numbers $a,b,c$, $\frac{1}{a}+\frac{1}{b} + \frac{1}{c} \ge \frac{3}{abc}$ is true. Prove that:
$$ \frac{a^2+b^2}{a^2+b^2+1}+\frac{b^2+c^2}{b^2+c^2+1}+\frac{c^2+a^2}{c^2+a^2+1} \ge 2$$
2023 Vietnam National Olympiad, 7
Let $\triangle{ABC}$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Incircle $(I)$ of the $\triangle{ABC}$ is tangent to the sides $BC,CA,AB$ at $M,N,P$ respectively. Denote $\Omega_A$ to be the circle passing through point $A$, external tangent to $(I)$ at $A'$ and cut again $AB,AC$ at $A_b,A_c$ respectively. The circles $\Omega_B,\Omega_C$ and points $B',B_a,B_c,C',C_a,C_b$ are defined similarly.
$a)$ Prove $B_cC_b+C_aA_c+A_bB_a \ge NP+PM+MN$.
$b)$ Suppose $A',B',C'$ lie on $AM,BN,CP$ respectively. Denote $K$ as the circumcenter of the triangle formed by lines $A_bA_c,B_cB_a,C_aC_b.$ Prove $OH//IK$.
2002 Croatia National Olympiad, Problem 2
Prove that for any positive number $a,b,c$ and any nonnegative integer $p$
$$a^{p+2}+b^{p+2}+c^{p+2}\ge a^pbc+b^pca+c^pab.$$
2018 China National Olympiad, 3
Let $q$ be a positive integer which is not a perfect cube. Prove that there exists a positive constant $C$ such that for all natural numbers $n$, one has
$$\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}}$$
where $\{ x \}$ denotes the fractional part of $x$.
2024 Junior Macedonian Mathematical Olympiad, 1
Let $a, b$, and $c$ be positive real numbers. Prove that
\[\frac{a^4 + 3}{b} + \frac{b^4 + 3}{c} + \frac{c^4 + 3}{a} \ge 12.\]
When does equality hold?
[i]Proposed by Petar Filipovski[/i]
2020 Junior Balkan Team Selection Tests-Serbia, 3#
Given are real numbers $a_1, a_2,...,a_{101}$ from the interval $[-2,10]$ such that their sum is $0$. Prove that the sum of their squares is smaller than $2020$.
Russian TST 2016, P3
Prove that for any points $A,B,C,D$ in the plane, the following inequality holds \[\frac{AB}{DA+DB}+\frac{BC}{DB+DC}\geqslant\frac{AC}{DA+DC}.\]
1974 IMO Shortlist, 4
The sum of the squares of five real numbers $a_1, a_2, a_3, a_4, a_5$ equals $1$. Prove that the least of the numbers $(a_i - a_j)^2$, where $i, j = 1, 2, 3, 4,5$ and $i \neq j$, does not exceed $\frac{1}{10}.$
2018 Baltic Way, 3
Let $a,b,c,d$ be positive real numbers such that $abcd=1$. Prove the inequality
\[\frac{1}{\sqrt{a+2b+3c+10}}+\frac{1}{\sqrt{b+2c+3d+10}}+\frac{1}{\sqrt{c+2d+3a+10}}+\frac{1}{\sqrt{d+2a+3b+10}} \le 1.\]
2013 European Mathematical Cup, 4
Let $a,b,c$ be positive reals satisfying :
\[ \frac{a}{1+b+c}+\frac{b}{1+c+a}+\frac{c}{1+a+b}\ge \frac{ab}{1+a+b}+\frac{bc}{1+b+c}+\frac{ca}{1+c+a} \]
Then prove that :
\[ \frac{a^2+b^2+c^2}{ab+bc+ca}+a+b+c+2\ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}) \]
[i]Proposed by Dimitar Trenevski[/i]
2004 Bulgaria Team Selection Test, 2
Prove that if $a,b,c \ge 1$ and $a+b+c=9$, then $\sqrt{ab+bc+ca} \le \sqrt{a} +\sqrt{b} + \sqrt{c}$
2009 Philippine MO, 4
Let $k$ be a positive real number such that $$\frac{1}{k+a} + \frac{1}{k+b} + \frac{1}{k+c} \leq 1$$ for any positive positive real numbers $a$, $b$ and $c$ with $abc = 1$. Find the minimum value of $k$.
1989 IMO Shortlist, 16
The set $ \{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions:
[b](i)[/b] $ a_0 \equal{} a_n \equal{} 0,$
[b](ii)[/b] for $ 1 \leq k \leq n \minus{} 1,$ \[ a_k \equal{} c \plus{} \sum^{n\minus{}1}_{i\equal{}k} a_{i\minus{}k} \cdot \left(a_i \plus{} a_{i\plus{}1} \right)\]
Prove that $ c \leq \frac{1}{4n}.$
2019 Balkan MO Shortlist, A5
Let $a,b,c$ be positive real numbers, such that $(ab)^2 + (bc)^2 + (ca)^2 = 3$. Prove that
\[ (a^2 - a + 1)(b^2 - b + 1)(c^2 - c + 1) \ge 1. \]
[i]Proposed by Florin Stanescu (wer), România[/i]
2024 Indonesia Regional, 1
Given a real number $C\leqslant 2$. Prove that for every positive real number $x,y$ with $xy=1$, the following inequality holds:
\[ \sqrt{\frac{x^2+y^2}{2}} + \frac{C}{x+y} \geqslant 1 + \frac{C}{2}.\]
[i]Proposed by Fajar Yuliawan, Indonesia[/i]
2000 Moldova National Olympiad, Problem 2
Show that if real numbers $x<1<y$ satisfy the inequality
$$2\log x+\log(1-x)\ge3\log y+\log(y-1),$$then $x^3+y^3<2$.
2021 Thailand TST, 1
[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 $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
\]
2022 District Olympiad, P2
$a)$ Prove that $2x^3-3x^2+1\geq 0,~(\forall)x\geq0.$
$b)$ Let $x,y,z\geq 0$ such that $\frac{2}{1+x^3}+\frac{2}{1+y^3}+\frac{2}{1+z^3}=3.$ Prove that $\frac{1-x}{1-x+x^2}+\frac{1-y}{1-y+y^2}+\frac{1-z}{1-z+z^2}\geq 0.$