Olympiad problems

2019 IFYM, Sozopol, 6

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

2021 Spain Mathematical Olympiad, 4

Tags: inequality , algebra , inequalities

Let $a,b,c,d$ real numbers such that: $$ a+b+c+d=0 \text{ and } a^2+b^2+c^2+d^2 = 12 $$ Find the minimum and maximum possible values for $abcd$, and determine for which values of $a,b,c,d$ the minimum and maximum are attained.

2014 IFYM, Sozopol, 4

Tags: algebra , inequality , inequalities

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

2000 Bosnia and Herzegovina Team Selection Test, 4

Tags: inequality , inequalities , algebra

Prove that for all positive real $a$, $b$ and $c$ holds: $$ \frac{bc}{a^2+2bc}+\frac{ac}{b^2+2ac}+\frac{ab}{c^2+2ab} \leq 1 \leq \frac{a^2}{a^2+2bc}+\frac{b^2}{b^2+2ac}+\frac{c^2}{c^2+2ab}$$

2016 Nigerian Senior MO Round 2, Problem 6

Tags: algebra , inequality , inequalities

Given that $a, b, c, d \in \mathbb{R}$, prove that $(ab+cd)^2 \leq (a^2+c^2)(b^2+d^2)$.

1969 IMO Shortlist, 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 $1.$

2019 Moldova Team Selection Test, 6

Tags: inequality , inequalities

Let $a,b,c \ge 0$ such that $a+b+c=1$ and $s \ge 5$. Prove that $s(a^2+b^2+c^2) \le 3(s-3)(a^3+b^3+c^3)+1$

2010 BAMO, 5

Tags: inequality , cyclic quadrilateral , algebra , inequalities

Let $a$, $b$, $c$, $d$ be positive real numbers such that $abcd=1$. Prove that $1/[(1/2 +a+ab+abc)^{1/2}]+ 1/[(1/2+b+bc+bcd)^{1/2}] + 1/[(1/2+c+cd+cda)^{1/2}] + 1/[1(1/2+d+da+dab)^{1/2}]$ is greater than or equal to $2^{1/2}$.

2014 Contests, 1

Tags: algebra , inequality

Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds. [i]UK - David Monk[/i]

2014 JBMO Shortlist, 3

Tags: inequality , algebra

For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$

2012 Polish MO Finals, 6

Tags: algebra , inequality , inequalities

Show that for any positive real numbers $a, b, c$ true is inequality: $\left(\frac{a - b}{c}\right)^2 + \left(\frac{b - c}{a}\right)^2 + \left(\frac{c - a}{b}\right)^2 \ge 2\sqrt{2}\left(\frac{a - b}{c} + \frac{b - c}{a} + \frac{c - a}{b} \right)$.

2002 Rioplatense Mathematical Olympiad, Level 3, 4

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

Let $a, b$ and $c$ be positive real numbers. Show that $\frac{a+b}{c^2}+ \frac{c+a}{b^2}+ \frac{b+c}{a^2}\ge \frac{9}{a+b+c}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

2021 Science ON all problems, 3

Tags: inequality , algebra , identities

Real numbers $a,b,c$ with $0\le a,b,c\le 1$ satisfy the condition $$a+b+c=1+\sqrt{2(1-a)(1-b)(1-c)}.$$ Prove that $$\sqrt{1-a^2}+\sqrt{1-b^2}+\sqrt{1-c^2}\le \frac{3\sqrt 3}{2}.$$ [i] (Nora Gavrea)[/i]

2010 Kosovo National Mathematical Olympiad, 5

Tags: inequality , algebra , inequalities

Let $x,y$ be positive real numbers such that $x+y=1$. Prove that $\left(1+\frac {1}{x}\right)\left(1+\frac {1}{y}\right)\geq 9$.

2000 AMC 10, 22

Tags: diophantine , inequality

One morning each member of Angela's family drank an $ 8$-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family? $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 7$

2004 China Western Mathematical Olympiad, 4

Tags: cauchy inequality , inequality , three variable inequality , inequalities

Suppose that $ a$, $ b$, $ c$ are positive real numbers, prove that \[ 1 < \frac {a}{\sqrt {a^{2} \plus{} b^{2}}} \plus{} \frac {b}{\sqrt {b^{2} \plus{} c^{2}}} \plus{} \frac {c}{\sqrt {c^{2} \plus{} a^{2}}}\leq\frac {3\sqrt {2}}{2} \]

2020 IMO Shortlist, A1

Tags: algebra , inequality , inequalities

[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

2016 PAMO, 4

Tags: inequality , algebra

Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove that $\frac{1}{(x+1)^2+y^2+1}$ $+$ $\frac{1}{(y+1)^2+z^2+1}$ $+$ $\frac{1}{(z+1)^2+x^2+1}$ $\leq$ ${\frac{1}{2}}$.

2021 Greece JBMO TST, 1

Tags: algebra , inequality

If positive reals $x,y$ are such that $2(x+y)=1+xy$, find the minimum value of expression $$A=x+\frac{1}{x}+y+\frac{1}{y}$$

2018 Korea National Olympiad, 4

Tags: algebra , inequality , inequalities , sequence

Find all real values of $K$ which satisfies the following. Let there be a sequence of real numbers $\{a_n\}$ which satisfies the following for all positive integers $n$. (i). $0 < a_n < n^K$. (ii). $a_1 + a_2 + \cdots + a_n < \sqrt{n}$. Then, there exists a positive integer $N$ such that for all integers $n>N$, $$a^{2018}_1 + a^{2018}_2 + \cdots +a^{2018}_n < \frac{n}{2018}$$

2021 Junior Macedonian Mathematical Olympiad, Problem 4

Tags: algebra , inequality

Let $a$, $b$, $c$ be positive real numbers such that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2} = \frac{27}{4}.$ Show that: $$\frac{a^3+b^2}{a^2+b^2} + \frac{b^3+c^2}{b^2+c^2} + \frac{c^3+a^2}{c^2+a^2} \geq \frac{5}{2}.$$ [i]Authored by Nikola Velov[/i]

1979 IMO Shortlist, 13

Tags: trigonometry , inequality , geometric inequality , trigonometric identities , inequalities

Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$

2024 Turkey EGMO TST, 4

Tags: algebra , sequence , inequality , induction

Let $(a_n)_{n=1}^{\infty}$ be a strictly increasing sequence such that inequality $$a_n(a_n-2a_{n-1})+a_{n-1}(a_{n-1}-2a_{n-2})\geq 0$$ holds for all $n \geq 3$. Prove that for all $n\geq2$ the inequality $$a_n \geq a_{n-1}+a_{n-2}+\dots+a_1$$ holds as well.

2023 European Mathematical Cup, 4

Tags: algebra , inequality

We say that a $2023$-tuple of nonnegative integers $(a_1,\hdots,a_{2023})$ is [i]sweet[/i] if the following conditions hold: [list] [*] $a_1+\hdots+a_{2023}=2023$ [*] $\frac{a_1}{2}+\frac{a_2}{2^2}+\hdots+\frac{a_{2023}}{2^{2023}}\le 1$ [/list] Determine the greatest positive integer $L$ so that \[a_1+2a_2+\hdots+2023a_{2023}\ge L\] holds for every sweet $2023$-tuple $(a_1,\hdots,a_{2023})$ [i]Ivan Novak[/i]

2017 Bulgaria EGMO TST, 3

Tags: tangent line , inequality , algebra

Let $a$, $b$, $c$ and $d$ be positive real numbers with $a+b+c+d = 4$. Prove that $\frac{a}{b^2 + 1} + \frac{b}{c^2+1} + \frac{c}{d^2+1} + \frac{d}{a^2+1} \geq 2$.