Olympiad problems

2020 Baltic Way, 2

Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $$\frac{1}{a\sqrt{c^2 + 1}} + \frac{1}{b\sqrt{a^2 + 1}} + \frac{1}{c\sqrt{b^2+1}} > 2.$$

2006 Bosnia and Herzegovina Team Selection Test, 3

Tags: fractional part , Inequality , number theory , positive integer , inequalities

Prove that for every positive integer $n$ holds inequality $\{n\sqrt{7}\}>\frac{3\sqrt{7}}{14n}$, where $\{x\}$ is fractional part of $x$.

1989 Bundeswettbewerb Mathematik, 2

Tags: Inequality , algebra , real numbers

Find all pairs $(a,b)$ of real numbers such that $$|\sqrt{1-x^2 }-ax-b| \leq \frac{\sqrt{2} -1}{2}$$ holds for all $x\in [0,1]$.

2021 China Team Selection Test, 4

Tags: Inequality , algebra , TST

Suppose $x_1,x_2,...,x_{60}\in [-1,1]$ , find the maximum of $$ \sum_{i=1}^{60}x_i^2(x_{i+1}-x_{i-1}),$$ where $x_{i+60}=x_i$.

2014 China Western Mathematical Olympiad, 1

Tags: inequalities proposed , algebra , China , BPSQ , Inequality

Let $x,y$ be positive real numbers .Find the minimum of $x+y+\frac{|x-1|}{y}+\frac{|y-1|}{x}$.

2024 Belarus Team Selection Test, 1.3

Tags: Inequality , algebra , TST

Prove that for any real numbers $a,b,c,d \geq \frac{1}{3}$ the following inequality holds: $$\sqrt{\frac{a^6}{b^4+c^3}+\frac{b^6}{c^4+d^3}+\frac{c^6}{d^4+a^3}+\frac{d^6}{a^4+b^3}}\geq \frac{a+b+c+d}{4}$$ [i]D. Zmiaikou[/i]

2021 Olympic Revenge, 1

Tags: Inequality , algebra , inequalities , olympic revenge

Let $a$, $b$, $c$, $k$ be positive reals such that $ab+bc+ca \leq 1$ and $0 < k \leq \frac{9}{2}$. Prove that: \[\sqrt[3]{ \frac{k}{a} + (9-3k)b} + \sqrt[3]{\frac{k}{b} + (9-3k)c} + \sqrt[3]{\frac{k}{c} + (9-3k)a } \leq \frac{1}{abc}.\] [i]Proposed by Zhang Yanzong and Song Qing[/i]

2019 IFYM, Sozopol, 6

Tags: algebra , Inequality , complex numbers

Prove that for $\forall$ $z\in \mathbb{C}$ the following inequality is true: $|z|^2+2|z-1|\geq 1$, where $"="$ is reached when $z=1$.

1983 IMO Shortlist, 21

Tags: algebra , Summation , series summation , approximation , Inequality , IMO Shortlist

Find the greatest integer less than or equal to $\sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1}.$

2020 Regional Olympiad of Mexico Southeast, 5

Tags: geometry , circumcircle , Inequality , Tangents , inequalities

Let $ABC$ an acute triangle with $\angle BAC\geq 60^\circ$ and $\Gamma$ it´s circumcircule. Let $P$ the intersection of the tangents to $\Gamma$ from $B$ and $C$. Let $\Omega$ the circumcircle of the triangle $BPC$. The bisector of $\angle BAC$ intersect $\Gamma$ again in $E$ and $\Omega$ in $D$, in the way that $E$ is between $A$ and $D$. Prove that $\frac{AE}{ED}\leq 2$ and determine when equality holds.

1998 IMO Shortlist, 3

Tags: rearrangement inequality , IMO Shortlist , algebra , Inequality

Let $x,y$ and $z$ be positive real numbers such that $xyz=1$. Prove that \[ \frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)} \geq \frac{3}{4}. \]

2022 JBMO Shortlist, A3

Tags: Inequality , algebra , JBMO Shortlist

Let $a, b,$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove the following inequality $$a \sqrt[3]{\frac{b}{a}} + b \sqrt[3]{\frac{c}{b}} + c \sqrt[3]{\frac{a}{c}} \le ab + bc + ca + \frac{2}{3}.$$ Proposed by [i]Anastasija Trajanova, Macedonia[/i]

2022 Tuymaada Olympiad, 4

Tags: Inequality , inequalities

For every positive $a_1, a_2, \dots, a_6$, prove the inequality \[ \sqrt[4]{\frac{a_1}{a_2 + a_3 + a_4}} + \sqrt[4]{\frac{a_2}{a_3 + a_4 + a_5}} + \dots + \sqrt[4]{\frac{a_6}{a_1 + a_2 + a_3}} \ge 2 \]

2025 Belarusian National Olympiad, 9.6

Tags: algebra , Inequality

Numbers $a,b,c$ are lengths of sides of some triangle. Prove the inequality$$\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c} \geq \frac{a+b}{2c}+\frac{b+c}{2a}+\frac{c+a}{2b}$$ [i]M. Karpuk[/i]

2023 Junior Balkan Team Selection Tests - Moldova, 12

Tags: algebra , TST , Inequality

Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2=3. $ Prove that $$\frac{a^4+3ab^3}{a^3+2b^3}+\frac{b^4+3bc^3}{b^3+2c^3}+\frac{c^4+3ca^3}{c^3+2a^3}\leq4.$$

2021 Indonesia TST, A

Tags: Inequality , inequalities , algebra

A positive real $M$ is $strong$ if for any positive reals $a$, $b$, $c$ satisfying $$ \text{max}\left\{ \frac{a}{b+c} , \frac{b}{c+a} , \frac{c}{a+b} \right\} \geqslant M $$ then the following inequality holds: $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} > 20.$$ (a) Prove that $M=20-\frac{1}{20}$ is not $strong$. (b) Prove that $M=20-\frac{1}{21}$ is $strong$.

2018 IFYM, Sozopol, 4

Tags: algebra , Inequality , inequalities

Find all real numbers $k$ for which the inequality $(1+t)^k (1-t)^{1-k} \leq 1$ is true for every real number $t \in (-1, 1)$.

1992 Mexico National Olympiad, 5

Tags: inequalities , Inequality , algebra

$x, y, z$ are positive reals with sum $3$. Show that $$6 < \sqrt{2x+3} + \sqrt{2y+3} + \sqrt{2z+3}\le 3\sqrt5$$

2017 Vietnamese Southern Summer School contest, Problem 1

Tags: Inequality , algebra , inequalities

Let $x,y,z$ be the non-negative real numbers satisfying $xy+yz+zx\leq 1$. Prove that: $$1-xy-yz-zx\leq (6-2\sqrt{6})(1-\min\{x,y,z\}).$$

1980 IMO Longlists, 2

Tags: algebra , recurrence relation , Sequence , Inequality , IMO Shortlist

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 \[ 1 - \frac{1}{n} < a_n < 1.\]

2007 Junior Macedonian Mathematical Olympiad, 3

Tags: Junior , JMMO , 2007 , Macedonia , algebra , Inequality

Let $a$, $b$, $c$ be real numbers such that $0 < a \le b \le c$. Prove that $(a + 3b)(b + 4c)(c + 2a) \ge 60abc$. When does equality hold?

2020 Taiwan TST Round 1, 1

Tags: Inequality , inequalities , Taiwan

Let $a$, $b$, $c$, $d$ be real numbers satisfying \begin{align*} (a + c)(b + d) = \sqrt{2}(ac - 2bd - 1). \end{align*} Show that \begin{align*} (ab - 1)^2 + (bc - 1)^2 + (cd - 1)^2 + (da - 1)^2 + (ac - 1)^2 + (2bd + 1)^2 \ge 4. \end{align*}

1979 IMO Shortlist, 19

Tags: number theory , Sequences , recurrence relation , Inequality , IMO Shortlist , IMO Longlist

Consider the sequences $(a_n), (b_n)$ defined by \[a_1=3, \quad b_1=100 , \quad a_{n+1}=3^{a_n} , \quad b_{n+1}=100^{b_n} \] Find the smallest integer $m$ for which $b_m > a_{100}.$

1971 IMO Longlists, 52

Tags: Inequality , 4-variable inequality , four variables , IMO Shortlist

Prove the inequality \[ \frac{a_1+ a_3}{a_1 + a_2} + \frac{a_2 + a_4}{a_2 + a_3} + \frac{a_3 + a_1}{a_3 + a_4} + \frac{a_4 + a_2}{a_4 + a_1} \geq 4, \] where $a_i > 0, i = 1, 2, 3, 4.$

1989 IMO Longlists, 55

Tags: algebra , Sequence , recurrence relation , Inequality , IMO Shortlist

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}.$