Olympiad problems

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Found problems: 93

2009 Iran Team Selection Test, 3

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

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}$

2017 Federal Competition For Advanced Students, P2, 4

Tags: maximum , three variable inequality , rational , algebra , inequalities

(a) Determine the maximum $M$ of $x+y +z$ where $x, y$ and $z$ are positive real numbers with $16xyz = (x + y)^2(x + z)^2$. (b) Prove the existence of infinitely many triples $(x, y, z)$ of positive rational numbers that satisfy $16xyz = (x + y)^2(x + z)^2$ and $x + y + z = M$. Proposed by Karl Czakler

2018 Moldova Team Selection Test, 6

Tags: three variable inequality , inequalities , algebra

Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Show that $$\frac{a}{1+b^2}+\frac{b}{1+c^2}+\frac{c}{1+a^2}\geq \frac{3}{2}.$$

1984 IMO Shortlist, 5

Tags: inequality , three variable inequality , maximization , calculus

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

2015 Caucasus Mathematical Olympiad, 4

Tags: three variable inequality , algebra , inequalities

The sum of the numbers $a,b$ and $c$ is zero, and their product is negative. Prove that the number $\frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}$ is positive.

2014 Indonesia MO Shortlist, A3

Tags: three variable inequality , algebra , inequalities

Prove for each positive real number $x, y, z$, $$\frac{x^2y}{x+2y}+\frac{y^2z}{y+2z}+\frac{z^2x}{z+2x}<\frac{(x+y+z)^2}{8}$$

2015 Greece JBMO TST, 1

Tags: three variable inequality , algebra , inequalities

If $x,y,z>0$, prove that $(3x+y)(3y+z)(3z+x) \ge 64xyz$. When we have equality;

2007 Nicolae Coculescu, 1

Tags: three variable inequality , geometry , floor , algebra , inequalities

Calculate $ \left\lfloor \frac{(a^2+b^2+c^2)(a+b+c)}{a^3+b^3+c^3} \right\rfloor , $ where $ a,b,c $ are the lengths of the side of a triangle. [i]Costel Anghel[/i]

2016 Balkan MO Shortlist, A4

Tags: three variable inequality , minimum , inequalities , algebra

The positive real numbers $a, b, c$ satisfy the equality $a + b + c = 1$. For every natural number $n$ find the minimal possible value of the expression $$E=\frac{a^{-n}+b}{1-a}+\frac{b^{-n}+c}{1-b}+\frac{c^{-n}+a}{1-c}$$

2018 China Northern MO, 2

Tags: three variable inequality , inequalities

Let $a$,$b$,$c$ be nonnegative reals such that $$a^2+b^2+c^2+ab+\frac{2}{3}ac+\frac{4}{3}bc=1$$ Find the maximum and minimum value of $a+b+c$.

1997 USAMO, 5

Tags: symmetry , three variable inequality , inequalities

Prove that, for all positive real numbers $ a$, $ b$, $ c$, the inequality \[ \frac {1}{a^3 \plus{} b^3 \plus{} abc} \plus{} \frac {1}{b^3 \plus{} c^3 \plus{} abc} \plus{} \frac {1}{c^3 \plus{} a^3 \plus{} abc} \leq \frac {1}{abc} \] holds.

2014 JBMO Shortlist, 7

Tags: three variable inequality , inequality

$a,b,c\in\mathbb{R^+}$ and $a^2+b^2+c^2=48$. Prove that \[a^2\sqrt{2b^3+16}+b^2\sqrt{2c^3+16}+c^2\sqrt{2a^3+16}\le24^2\]

1987 IMO Shortlist, 6

Tags: rearrangement inequality , three variable inequality , inequality , triangle inequality

Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then \[\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N \] [i]Proposed by Greece.[/i]

1987 IMO Longlists, 33

Tags: inequality , three variable inequality , rearrangement inequality , triangle inequality

Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then \[\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N \] [i]Proposed by Greece.[/i]

2018 Azerbaijan Senior NMO, 5

Tags: inequalities , three variable inequality

Prove that if $x$, $y$, $z$ are positive real numbers and $xyz = 1$ then \[\frac{x^3}{x^2+y}+\frac{y^3}{y^2+z}+\frac{z^3}{z^2+x}\geq \dfrac {3} {2}.\] [i]A. Golovanov[/i]

2006 USA Team Selection Test, 3

Tags: function , quadratic , three variable inequality , inequalities

Find the least real number $k$ with the following property: if the real numbers $x$, $y$, and $z$ are not all positive, then \[k(x^{2}-x+1)(y^{2}-y+1)(z^{2}-z+1)\geq (xyz)^{2}-xyz+1.\]

2004 USAMO, 5

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

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

2005 Greece JBMO TST, 2

Tags: three variable inequality , algebra , inequalities

Prove that for each $x,y,z \in R$ it holds that $$\frac{x^2-y^2}{2x^2+1} +\frac{y^2-z^2}{2y^2+1}+\frac{z^2-x^2}{2z^2+1}\le 0$$

2015 Indonesia MO Shortlist, A5

Tags: three variable inequality , algebra , inequalities

Let $a,b,c$ be positive real numbers. Prove that $\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3$

2006 Junior Tuymaada Olympiad, 4

Tags: three variable inequality , algebra , inequalities

The sum of non-negative numbers $ x $, $ y $ and $ z $ is $3$. Prove the inequality $$ {1 \over x ^ 2 + y + z} + {1 \over x + y ^ 2 + z} + {1 \over x + y + z ^ 2} \leq 1. $$

2014 Federal Competition For Advanced Students, P2, 5

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

Show that the inequality $(x^2 + y^2z^2) (y^2 + x^2z^2) (z^2 + x^2y^2) \ge 8xy^2z^3$ is valid for all integers $x, y$ and $z$.When does equality apply?

2015 Indonesia MO, 7

Tags: three variable inequality , algebra , inequalities

Let $a,b,c$ be positive real numbers. Prove that $\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3$

2011 All-Russian Olympiad Regional Round, 9.4

Tags: three variable inequality , calculus , inequalities

$x$, $y$ and $z$ are positive real numbers. Prove the inequality \[\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x}{y}+\frac{y}{z}+\frac{z}{x}.\] (Authors: A. Khrabrov, B. Trushin)

2014 ELMO Shortlist, 1

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

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]

1999 Greece JBMO TST, 2

Tags: three variable inequality , inequalities , algebra

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) $a(1+b)+b(1+c)+c(1+a)\ge 6\sqrt{abc}$