Olympiad problems

2022 Bundeswettbewerb Mathematik, 1

Tags: algebra , am-gm

Find all quadrupels $(a, b, c, d)$ of positive real numbers that satisfy the following two equations: \begin{align*} ab + cd &= 8,\\ abcd &= 8 + a + b + c + d. \end{align*}

2023 India Regional Mathematical Olympiad, 5

Tags: inequalities , am-gm

Let $n>k>1$ be positive integers. Determine all positive real numbers $a_1, a_2, \ldots, a_n$ which satisfy $$ \sum_{i=1}^n \sqrt{\frac{k a_i^k}{(k-1) a_i^k+1}}=\sum_{i=1}^n a_i=n . $$

2023 UMD Math Competition Part II, 5

Tags: inequalities , am-gm

Let $0 \le a_1 \le a_2 \le \dots \le a_n \le 1$ be $n$ real numbers with $n \ge 2$. Assume $a_1 + a_2 + \dots + a_n \ge n-1$. Prove that \[ a_2a_3\dots a_n \ge \left( 1 - \frac 1n \right)^{n-1} \]

2019 NMTC Junior, 7

Tags: inequalities , am-gm

The perimeter of $\triangle ABC$ is $2$ and it's sides are $BC=a, CA=b,AB=c$. Prove that $$abc+\frac{1}{27}\ge ab+bc+ca-1\ge abc. $$

2021 Hong Kong TST, 1

Tags: am-gm , equation , algebra

Find all real triples $(a,b,c)$ satisfying \[(2^{2a}+1)(2^{2b}+2)(2^{2c}+8)=2^{a+b+c+5}.\]

2019 India Regional Mathematical Olympiad, 3

Tags: am-gm , inequality , inequalities

Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that $$\frac{a}{a^2+b^3+c^3}+\frac{b}{b^2+a^3+c^3}+\frac{c}{c^2+a^3+b^3}\leq\frac{1}{5abc}$$

2022 Bulgaria JBMO TST, 2

Tags: inequalities , am-gm , titu's lemma , cauchy schwarz

Let $a$, $b$ and $c$ be positive real numbers with $abc = 1$. Determine the minimum possible value of $$ \left(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\right) \cdot \left(\frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a}\right) $$ as well as all triples $(a,b,c)$ which attain the minimum.

2019 Centers of Excellency of Suceava, 1

Tags: inequalities , am-gm

Prove that $ \binom{m+n}{\min (m,n)}\le \sqrt{\binom{2m}{m}\cdot \binom{2n}{n}} , $ for nonnegative $ m,n. $ [i]Gheorghe Stoica[/i]