Olympiad problems

2005 Federal Math Competition of S&M, Problem 1

Tags: inequality , inequalities

If $x,y,z$ are positive numbers, prove that $$\frac x{\sqrt{y+z}}+\frac y{\sqrt{z+x}}+\frac z{\sqrt{x+y}}\ge\sqrt{\frac32(x+y+z)}.$$

Azerbaijan Al-Khwarizmi IJMO TST 2025, 2

Tags: inequality , algebra , inequalities

For $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2 \geq 3$,show that: $\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$.

2018 Serbia Team Selection Test, 2

Tags: inequality , algebra , inequalities

Let $n$ be a fixed positive integer and let $x_1,\ldots,x_n$ be positive real numbers. Prove that $$x_1\left(1-x_1^2\right)+x_2\left(1-(x_1+x_2)^2\right)+\cdots+x_n\left(1-(x_1+...+x_n)^2\right)<\frac{2}{3}.$$

2024 Ukraine National Mathematical Olympiad, Problem 5

Tags: algebra , inequality , inequalities

For real numbers $a, b, c, d \in [0, 1]$, find the largest possible value of the following expression: $$a^2+b^2+c^2+d^2-ab-bc-cd-da$$ [i]Proposed by Mykhailo Shtandenko[/i]

2002 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: inequality , number theory , algebra , integer , positive integer

Let $\lambda$ be a real number such that the inequality $0 <\sqrt {2002} - \frac {a} {b} <\frac {\lambda} {ab}$ holds for an infinite number of pairs $ (a, b)$ of positive integers. Prove that $\lambda \geq 5 $.

1976 IMO Shortlist, 2

Tags: inequality , convexity , sequence , recurrence relation

Let $a_0, a_1, \ldots, a_n, a_{n+1}$ be a sequence of real numbers satisfying the following conditions: \[a_0 = a_{n+1 }= 0,\]\[ |a_{k-1} - 2a_k + a_{k+1}| \leq 1 \quad (k = 1, 2,\ldots , n).\] Prove that $|a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).$

ICMC 4, 3

Tags: expectation , inequality

Let $f,g,h : \mathbb R \to \mathbb R$ be continuous functions and $X$ be a random variable such that $E(g(X)h(X))=0$ and $E(g(X)^2) \neq 0 \neq E(h(X)^2)$. Prove that $$E(f(X)^2) \geq \frac{E(f(X)g(X))^2}{E(g(X)^2)} + \frac{E(f(X)h(X))^2}{E(h(X)^2)}.$$ You may assume that all expected values exist. [i]Proposed by Cristi Calin[/i]

2014 JBMO TST - Macedonia, 1

Tags: inequality , integer

Prove that $\frac{1}{1\times2013}+\frac{1}{2\times2012}+\frac{1}{3\times2011}+...+\frac{1}{2012\times2}+\frac{1}{2013\times1}<1$

2022 Bulgarian Spring Math Competition, Problem 10.1

Tags: system of equations , inequality , algebra

If $x, y, z \in \mathbb{R}$ are solutions to the system of equations $$\begin{cases} x - y + z - 1 = 0\\ xy + 2z^2 - 6z + 1 = 0\\ \end{cases}$$ what is the greatest value of $(x - 1)^2 + (y + 1)^2$?

2012 Kosovo National Mathematical Olympiad, 2

Tags: geometry , inequality

Let $a,b,c$ be the lengths of the sides of a triangle. Prove that, $\left|\frac {a}{b}+\frac {b}{c}+\frac {c}{a}-\frac {b}{a}-\frac {c}{b}-\frac {a}{c}\right|<1$

JOM 2015 Shortlist, A1

Tags: inequality , inequalities

Let $ a, b, c $ be the side lengths of a triangle. Prove that $$ \displaystyle\sum_{cyc} \frac{(a^2 + b^2)(a + c)}{b} \ge 2(a^2 + b^2 + c^2) $$

2020 Baltic Way, 1

Tags: inequality , sequence , algebra

Let $a_0>0$ be a real number, and let $$a_n=\frac{a_{n-1}}{\sqrt{1+2020\cdot a_{n-1}^2}}, \quad \textrm{for } n=1,2,\ldots ,2020.$$ Show that $a_{2020}<\frac1{2020}$.

2014 IFYM, Sozopol, 4

Tags: algebra , inequality , inequalities

Prove that for $\forall$ $x,y,z\in \mathbb{R}^+$ the following inequality is true: $\frac{x}{y+z}+\frac{25y}{z+x}+\frac{4z}{x+y}>2$.

2009 Korea Junior Math Olympiad, 6

Tags: inequalities , algebra , four variables , inequality

If positive reals $a,b,c,d$ satisfy $abcd = 1.$ Prove the following inequality $$1<\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{d}{cd+d+1}+\frac{a}{da+a+1}<2.$$

2018 Baltic Way, 3

Tags: inequality , algebra

Let $a,b,c,d$ be positive real numbers such that $abcd=1$. Prove the inequality \[\frac{1}{\sqrt{a+2b+3c+10}}+\frac{1}{\sqrt{b+2c+3d+10}}+\frac{1}{\sqrt{c+2d+3a+10}}+\frac{1}{\sqrt{d+2a+3b+10}} \le 1.\]

2017 Iran MO (3rd round), 3

Tags: inequality , am-gm , inequalities , algebra

Let $a,b$ and $c$ be positive real numbers. Prove that $$\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $$

2022 SEEMOUS, 3

Tags: linear algebra , matrices , hermitian , inequality

Let $\alpha \in \mathbb{C}\setminus \{0\}$ and $A \in \mathcal{M}_n(\mathbb{C})$, $A \neq O_n$, be such that $$A^2 + (A^*)^2 = \alpha A\cdot A^*,$$ where $A^* = (\bar A)^T.$ Prove that $\alpha \in \mathbb{R}$, $|\alpha| \le 2$. and $A\cdot A^* = A^*\cdot A.$

1988 IMO, 1

Tags: algebra , polynomial , vieta , inequality , root

Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

2001 All-Russian Olympiad, 2

Tags: algebra , polynomial , function , inequality

The two polynomials $(x) =x^4+ax^3+bx^2+cx+d$ and $Q(x) = x^2+px+q$ take negative values on an interval $I$ of length greater than $2$, and nonnegative values outside of $I$. Prove that there exists $x_0 \in \mathbb R$ such that $P(x_0) < Q(x_0)$.

2025 239 Open Mathematical Olympiad, 7

Tags: geometry , inequality

Point $M$ is the midpoint of side $BC$ of an acute—angled triangle $ABC$. The point $U$ is symmetric to the orthocenter $ABC$ relative to its circumcenter. The point $S$ inside triangle $ABC$ is such that $US = UM$. Prove that $SA + SB + SC + AM < AB + BC + CA$.

2021 CHKMO, 4

Tags: algebra , inequality , inequalities

Let $a,b$ and $c$ be positive real numbers satisfying $abc=1$. Prove that \[\dfrac{1}{a^3+2b^2+2b+4}+\dfrac{1}{b^3+2c^2+2c+4}+\dfrac{1}{c^3+2a^2+2a+4}\leq \dfrac13.\]

2001 Korea Junior Math Olympiad, 6

Tags: inequality , algebra

For real variables $0 \leq x, y, z, w \leq 1$, find the maximum value of $$x(1-y)+2y(1-z)+3z(1-w)+4w(1-x)$$

2015 JBMO TST-Turkey, 4

Tags: inequality

Prove that $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} \ge \dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+2(a+b+c)$$ for the all $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2+2abc \le 1$.

1980 IMO, 2

Tags: algebra , recurrence relation , sequence , inequality

Define the numbers $a_0, a_1, \ldots, a_n$ in the following way: \[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \] Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]

2000 District Olympiad (Hunedoara), 1

Tags: inequalities , inequality , algebra

[b]a)[/b] Show that $ \frac{n}{2}\ge \frac{2\sqrt{x} +3\sqrt[3]{x}+\cdots +n\sqrt[n]{x}}{n-1} -x, $ for all non-negative reals $ x $ and integers $ n\ge 2. $ [b]b)[/b] If $ x,y,z\in (0,\infty ) , $ then prove the inequality $$ \sum_{\text{cyc}} \frac{x}{(2x+y+z)^2+4} \le 3/16 $$