Olympiad problems

1999 Greece JBMO TST, 2

For $a,b,c>0$, prove that (i) $\frac{a+b+c}{2}-\frac{ab}{a+b}-\frac{bc}{b+c}-\frac{ca}{c+a}\ge 0$ (ii) $a(1+b)+b(1+c)+c(1+a)\ge 6\sqrt{abc}$

2005 IMAR Test, 1

Tags: inequality , three variable inequality

Let $a,b,c$ be positive real numbers such that $abc\geq 1$. Prove that \[ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq 1. \] [hide="Remark"]This problem derives from the well known inequality given in [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=185470#p185470]USAMO 1997, Problem 5[/url]. [/hide]

2006 Vietnam Team Selection Test, 1

Tags: inequalities , function , convex-concave inequalities , three variable inequality

Prove that for all real numbers $x,y,z \in [1,2]$ the following inequality always holds: \[ (x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\geq 6(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}). \] When does the equality occur?

1969 IMO Longlists, 69

Tags: inequality , three variable inequality , algebra

$(YUG 1)$ Suppose that positive real numbers $x_1, x_2, x_3$ satisfy $x_1x_2x_3 > 1, x_1 + x_2 + x_3 <\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}$ Prove that: $(a)$ None of $x_1, x_2, x_3$ equals $1$. $(b)$ Exactly one of these numbers is less than $1.$

2016 IMO Shortlist, A1

Tags: inequalities , three variable inequality , ineqstd

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]

2007 Korea Junior Math Olympiad, 5

Tags: algebra , inequalities , inequality , three variable inequality

For all positive real numbers $a, b,c.$ Prove the folllowing inequality$$\frac{a}{c+5b}+\frac{b}{a+5c}+\frac{c}{b+5a}\geq\frac{1}{2}.$$

2014 ELMO Shortlist, 1

Tags: inequalities , trigonometry , calculus , topology , three variable inequality

In a non-obtuse triangle $ABC$, prove that \[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]

2016 Greece JBMO TST, 1

Tags: algebra , inequalities , three variable inequality

a) Prove that, for any real $x>0$, it is true that $x^3-3x\ge -2$ . b) Prove that, for any real $x,y,z>0$, it is true that $$\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y}+2\left(\frac{y}{xz}+\frac{z}{xy}+\frac{x}{yz} \right)\ge 9$$ . When we have equality ?

1984 IMO, 1

Tags: inequality , three variable inequality , calculus , maximization

Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.

2004 USAMO, 5

Tags: linear algebra , calculus , three variable inequality , algebra , inequality , inequalities

Let $a, b, c > 0$. Prove that $(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \geq (a + b + c)^3$.

2017 Estonia Team Selection Test, 8

Tags: inequalities , three variable inequality , ineqstd

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]

2009 Iran Team Selection Test, 3

Tags: calculus , function , three variable inequality , algebra , inequality

Suppose that $ a$,$ b$,$ c$ be three positive real numbers such that $ a\plus{}b\plus{}c\equal{}3$ . Prove that : $ \frac{1}{2\plus{}a^{2}\plus{}b^{2}}\plus{}\frac{1}{2\plus{}b^{2}\plus{}c^{2}}\plus{}\frac{1}{2\plus{}c^{2}\plus{}a^{2}} \leq \frac{3}{4}$

2005 Vietnam Team Selection Test, 1

Tags: calculus , inequalities , three variable inequality

Let be given positive reals $a$, $b$, $c$. Prove that: $\frac{a^{3}}{\left(a+b\right)^{3}}+\frac{b^{3}}{\left(b+c\right)^{3}}+\frac{c^{3}}{\left(c+a\right)^{3}}\geq \frac{3}{8}$.

1967 IMO Longlists, 37

Tags: inequality , three variable inequality , muirhead , algebra , inequalities

Prove that for arbitrary positive numbers the following inequality holds \[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.\]

2009 Dutch IMO TST, 3

Tags: inequalities , 3-variable inequality , three variable inequality

Let $a, b$ and $c$ be positive reals such that $a + b + c \ge abc$. Prove that $a^2 + b^2 + c^2 \ge \sqrt3 abc$.

2011 Balkan MO, 2

Tags: inequalities , algebra , three variable inequality

Given real numbers $x,y,z$ such that $x+y+z=0$, show that \[\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0\] When does equality hold?

2015 Postal Coaching, Problem 3

Tags: inequalities , holder , three variable inequality

Let $a,b,c \in \mathbb{R^+}$ such that $abc=1$. Prove that $$\sum_{a,b,c} \sqrt{\frac{a}{a+8}} \ge 1$$

2009 Serbia National Math Olympiad, 5

Tags: inequalities , three variable inequality

Let $x$, $y$, $z$ be arbitrary positive numbers such that $xy+yz+zx=x+y+z$. Prove that $$\frac{1}{x^2+y+1} + \frac{1}{y^2+z+1} + \frac{1}{z^2+x+1} \leq 1$$. When does equality occur? [i]Proposed by Marko Radovanovic[/i]

2017 Balkan MO Shortlist, A4

Tags: three variable inequality , inequalities , algebra

Let $M = \{(a,b,c)\in R^3 :0 <a,b,c<\frac12$ with $a+b+c=1 \}$ and $f: M\to R$ given as $$f(a,b,c)=4\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)-\frac{1}{abc}$$ Find the best (real) bounds $\alpha$ and $\beta$ such that $f(M) = \{f(a,b,c): (a,b,c)\in M\}\subseteq [\alpha,\beta]$ and determine whether any of them is achievable.