Olympiad problems

2019 Nordic, 2

Let $a, b, c $ be the side lengths of a right angled triangle with c > a, b. Show that $$3<\frac{c^3-a^3-b^3}{c(c-a)(c-b)}\leq \sqrt{2}+2.$$

Let $a,b,c$ be positive real numbers such that $a + b + c = 1$. Show that $$\left(\frac{1}{a} + \frac{1}{bc}\right)\left(\frac{1}{b} + \frac{1}{ca}\right)\left(\frac{1}{c} + \frac{1}{ab}\right) \geq 1728$$

1995 All-Russian Olympiad Regional Round, 11.5

Tags: trigonometry , algebra , inequalities

Angles $\alpha, \beta, \gamma$ satisfy the inequality $\sin \alpha +\sin \beta +\sin \gamma \ge 2$. Prove that $\cos \alpha + \cos \beta +\cos \gamma \le \sqrt5.$

2019 Romania Team Selection Test, 1

Tags: algebra , minimum , inequalities

Let be a natural number $ n\ge 3. $ Find $$ \inf_{\stackrel{ x_1,x_2,\ldots ,x_n\in\mathbb{R}_{>0}}{1=P\left( x_1,x_2,\ldots ,x_n\right)}}\sum_{i=1}^n\left( \frac{1}{x_i} -x_i \right) , $$ where $ P\left( x_1,x_2,\ldots ,x_n\right) :=\sum_{i=1}^n \frac{1}{x_i+n-1} , $ and find in which circumstances this infimum is attained.

2024 Chile TST Ibero., 4

Tags: inequalities , algebra , tituslemma

Prove that if $ a $, $ b $, and $ c $ are positive real numbers, then the following inequality holds: \[ \frac{a + 3c}{a + b} + \frac{c + 3a}{b + c} + \frac{4b}{c + a} \geq 6. \]

2020 Vietnam National Olympiad, 2

Tags: inequalities , algebra , inequation

a)Let$a,b,c\in\mathbb{R}$ and $a^2+b^2+c^2=1$.Prove that: $|a-b|+|b-c|+|c-a|\le2\sqrt{2}$ b) Let $a_1,a_2,..a_{2019}\in\mathbb{R}$ and $\sum_{i=1}^{2019}a_i^2=1$.Find the maximum of: $S=|a_1-a_2|+|a_2-a_3|+...+|a_{2019}-a_1|$

2009 Postal Coaching, 1

Tags: inequalities , geometric inequalities , right triangle

Two circles $\Gamma_a$ and $\Gamma_b$ with their centres lying on the legs $BC$ and $CA$ of a right triangle, both touching the hypotenuse $AB$, and both passing through the vertex $C$ are given. Let the radii of these circles be denoted by $\gamma_a$ and $\gamma_b$. Find the greatest real number $p$ such that the inequality $\frac{1}{\gamma_a}+\frac{1}{\gamma_b}\ge p \left(\frac{1}{a}+\frac{1}{b}\right)$ ($BC = a,CA = b$) holds for all right triangles $ABC$.

2015 China Western Mathematical Olympiad, 1

Tags: algebra , inequalities

Let the integer $n \ge 2$ , and $x_1,x_2,\cdots,x_n $ be real numbers such that $\sum_{k=1}^nx_k$ be integer . $d_k=\underset{m\in {Z}}{\min}\left|x_k-m\right| $, $1\leq k\leq n$ .Find the maximum value of $\sum_{k=1}^nd_k$.

2005 Georgia Team Selection Test, 3

Tags: derivative , calculus , inequalities , cauchy inequality

Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.

2007 Korea National Olympiad, 1

Tags: inequalities

For all positive reals $ a$, $ b$, and $ c$, what is the value of positive constant $ k$ satisfies the following inequality? $ \frac{a}{c\plus{}kb}\plus{}\frac{b}{a\plus{}kc}\plus{}\frac{c}{b\plus{}ka}\geq\frac{1}{2007}$ .

1993 IMO Shortlist, 3

Tags: inequalities , function , algebra , four variables , 4-variable inequality

Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.

2022 District Olympiad, P2

Tags: inequalities , complex number

Let $z_1,z_2$ and $z_3$ be complex numbers of modulus $1,$ such that $|z_i-z_j|\geq\sqrt{2}$ for all $i\neq j\in\{1,2,3\}.$ Prove that \[|z_1+z_2|+|z_2+z_3|+|z_3+z_2|\leq 3.\][i]Mathematical Gazette[/i]

1986 China Team Selection Test, 2

Tags: inequalities , function , perimeter , geometry , 3d geometry , tetrahedron

Given a tetrahedron $ABCD$, $E$, $F$, $G$, are on the respectively on the segments $AB$, $AC$ and $AD$. Prove that: i) area $EFG \leq$ max{area $ABC$,area $ABD$,area $ACD$,area $BCD$}. ii) The same as above replacing "area" for "perimeter".

1991 Romania Team Selection Test, 7

Tags: algebra , inequalities , sum

Let $x_1,x_2,...,x_{2n}$ be positive real numbers with the sum $1$. Prove that $$x_1^2x_2^2...x_n^2+x_2^2x_3^2...x_{n+1}^2+...+x_{2n}^2x_1^2...x_{n-1}^2 <\frac{1}{n^{2n}}$$

2024-IMOC, A5

Tags: inequalities

The non-negative numbers $ x_1, x_2, \ldots, x_5$ satisfy $ \sum_{i \equal{} 1}^5 \frac {1}{1 \plus{} x_i} \equal{} 1$. Prove that $ \sum_{i \equal{} 1}^5 \frac {x_i}{4 \plus{} x_i^2} \leq 1$.

2018 Singapore Senior Math Olympiad, 4

Tags: inequalities , algebra

Let $a,b,c,d$ be positive integers such that $a+c=20$ and $\frac{a}{b}+\frac{c}{d}<1$. Find the maximum possible value of $\frac{a}{b}+\frac{c}{d}$.

2004 USA Team Selection Test, 3

Tags: inequalities , geometry , floor function , perimeter , vector , integration , calculus

Draw a $2004 \times 2004$ array of points. What is the largest integer $n$ for which it is possible to draw a convex $n$-gon whose vertices are chosen from the points in the array?

1992 Romania Team Selection Test, 4

Tags: inequalities , algebra , sum

Let $x_1,x_2,...,x_n$ be real numbers with $1 \ge x_1 \ge x_2\ge ... \ge x_n \ge 0$ and $x_1^2 +x_2^2+...+x_n^2= 1$. If $[x_1 +x_2 +...+x_n] = m$, prove that $x_1 +x_2 +...+x_m \ge 1$.

1998 Singapore MO Open, 2

Tags: inequalities , combinatorics , function , line , lattice , collinear , combinatorial geometry

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 $(n,f(n))$.

1983 AIME Problems, 9

Tags: inequalities , calculus , trigonometry

Find the minimum value of \[\frac{9x^2 \sin^2 x + 4}{x \sin x}\] for $0 < x < \pi$.

2020-IMOC, A1

Tags: functional equation , algebra , inequalities , am-gm

$\definecolor{A}{RGB}{190,0,60}\color{A}\fbox{A1.}$ Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $$\definecolor{A}{RGB}{80,0,200}\color{A} x^4+y^4+z^4\ge f(xy)+f(yz)+f(zx)\ge xyz(x+y+z)$$holds for all $a,b,c\in\mathbb{R}$. [i]Proposed by [/i][b][color=#FFFF00]usjl[/color][/b]. [color=#B6D7A8]#1733[/color]