Olympiad problems

1976 Spain Mathematical Olympiad, 7

Tags: inequalities , algebra

The price of a diamond is proportional to the square of its weight. Show that, breaking it into two parts, there is a depreciation of its value. When is it the maximum depreciation?

1998 Poland - Second Round, 3

Tags: inequalities

If $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ are nonnegative real numbers satisfying $ a \plus{} b \plus{} c \plus{} d \plus{} e \plus{} f \equal{} 1$ and $ ace \plus{} bdf \geq \frac {1}{108}$, then prove that \[ abc \plus{} bcd \plus{} cde \plus{} de f \plus{} efa \plus{} fab \leq \frac {1}{36} \]

2021 Kazakhstan National Olympiad, 1

Tags: inequalities

Given $a,b,c>0$ such that $$a+b+c+\frac{1}{abc}=\frac{19}{2}$$ What is the greatest value for $a$?

1922 Eotvos Mathematical Competition, 3

Tags: algebra , inequalities , number theory

Show that, if $a,b,...,n$ are distinct natural numbers, none divisible by any primes greater than $3$, then $$\frac{1}{a}+\frac{1}{b}+...+ \frac{1}{n}< 3$$

2008 China Team Selection Test, 1

Tags: cyclic quadrilateral , circumcircle , inequalities , geometry , trigonometry , ratio

Let $P$ be an arbitrary point inside triangle $ABC$, denote by $A_{1}$ (different from $P$) the second intersection of line $AP$ with the circumcircle of triangle $PBC$ and define $B_{1},C_{1}$ similarly. Prove that $\left(1 \plus{} 2\cdot\frac {PA}{PA_{1}}\right)\left(1 \plus{} 2\cdot\frac {PB}{PB_{1}}\right)\left(1 \plus{} 2\cdot\frac {PC}{PC_{1}}\right)\geq 8$.

2018 India PRMO, 23

Tags: inequalities , algebra

What is the largest positive integer $n$ such that $$\frac{a^2}{\frac{b}{29} + \frac{c}{31}}+\frac{b^2}{\frac{c}{29} + \frac{a}{31}}+\frac{c^2}{\frac{a}{29} + \frac{b}{31}} \ge n(a+b+c)$$holds for all positive real numbers $a,b,c$.

V Soros Olympiad 1998 - 99 (Russia), 11.1

Tags: inequalities , algebra , trigonometry

Find all $x$ for which the inequality holds $$9 \sin x +40 \cos x \ge 41.$$

2010 IMC, 5

Tags: inequalities

Suppose that $a,b,c$ are real numbers in the interval $[-1,1]$ such that $1 + 2abc \geq a^2+b^2+c^2$. Prove that $1+2(abc)^n \geq a^{2n} + b^{2n} + c^{2n}$ for all positive integers $n$.

2001 Balkan MO, 3

Tags: algebra , inequalities

Let $a$, $b$, $c$ be positive real numbers with $abc \leq a+b+c$. Show that \[ a^2 + b^2 + c^2 \geq \sqrt 3 abc. \] [i]Cristinel Mortici, Romania[/i]

1911 Eotvos Mathematical Competition, 1

Tags: inequalities , algebra

Show that, if the real numbers $a, b, c, A, B, C$ satisfy $$aC -2bB + cA = 0 \ \ and \ \ ac - b^2 > 0,$$ then $$AC - B^2 \le 0.$$

2001 Junior Balkan Team Selection Tests - Moldova, 6

Tags: algebra , inequalities

Let the nonnegative numbers $a_1, a_2,... a_9$, where $a_1 = a_9 = 0$ and let at least one of the numbers is nonzero. Denote the sentence $(P)$: '' For $2 \le i \le 8$ there is a number $a_i$, such that $a_{i - 1} + a_{i + 1} <ka_i $”. a) Show that the sentence $(P)$ is true for $k = 2$. b) Determine whether is the sentence $(P)$ true for $k = \frac{19}{10}$

2012 China Second Round Olympiad, 3

Tags: geometry , inequalities

Let $P_0 ,P_1 ,P_2 , ... ,P_n$ be $n+1$ points in the plane. Let $d$($d>0$) denote the minimal value of all the distances between any two points. Prove that \[|P_0P_1|\cdot |P_0P_2|\cdot ... \cdot |P_0P_n|>(\frac{d}{3})^n\sqrt{(n+1)!}.\]

2022 Serbia National Math Olympiad, P4

Tags: inequalities , phi function , number theory

Let $f(n)$ be number of numbers $x \in \{1,2,\cdots ,n\}$, $n\in\mathbb{N}$, such that $gcd(x, n)$ is either $1$ or prime. Prove $$\sum_{d|n} f(d) + \varphi(n) \geq 2n$$ For which $n$ does equality hold?

1992 Swedish Mathematical Competition, 4

Tags: inequalities , number theory , diophantine

Find all positive integers $a, b, c$ such that $a < b$, $a < 4c$, and $b c^3 \le a c^3 + b$.

2014 ELMO Shortlist, 3

Tags: parameterization , inequalities

Let $a,b,c,d,e,f$ be positive real numbers. Given that $def+de+ef+fd=4$, show that \[ ((a+b)de+(b+c)ef+(c+a)fd)^2 \geq\ 12(abde+bcef+cafd). \][i]Proposed by Allen Liu[/i]

2007 JBMO Shortlist, 1

Tags: function , quadratic , algebra , inequalities

Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.

2021 Philippine MO, 7

Tags: inequalities , number theory , algebra , philippines

Let $a, b, c,$ and $d$ be real numbers such that $a \geq b \geq c \geq d$ and $$a+b+c+d = 13$$ $$a^2+b^2+c^2+d^2=43.$$ Show that $ab \geq 3 + cd$.

2019 Jozsef Wildt International Math Competition, W. 24

Tags: inequalities

If $a$, $b$, $c > 0$, prove that$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq \frac{a+b}{a+b+2c}+\frac{b+c}{2a+b+c}+\frac{c+a}{a+2b+c}$$

1997 Vietnam Team Selection Test, 2

Tags: combinatorics , inequalities

There are $ 25$ towns in a country. Find the smallest $ k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied: 1) from each town there are exactly $ k$ direct routes to $ k$ other towns; 2) if two towns are not connected by a direct route, then there is a town which has direct routes to these two towns.

2017 IMO Shortlist, A5

Tags: inequalities , algebra

An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have $$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$ Find the largest constant $K = K(n)$ such that $$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$ holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.

2005 Federal Competition For Advanced Students, Part 2, 2

Tags: inequalities

Prove that for all positive reals $a,b,c,d$, we have $\frac{a+b+c+d}{abcd}\leq \frac{1}{a^{3}}+\frac{1}{b^{3}}+\frac{1}{c^{3}}+\frac{1}{d^{3}}$

2016 Kyiv Mathematical Festival, P3

Tags: inequalities

1) Let $a,b,c\ge0$ and $ab+bc+ca=2.$ Prove that \[\frac{ab}{c+1}+\frac{bc}{a+1}+\frac{ca}{b+1}+2(a+b+c)\ge6.\] 2) Let $a,b,c\ge0$ and $ab+bc+ca=3.$ Prove that \[\frac{ab}{c+1}+\frac{bc}{a+1}+\frac{ca}{b+1}\ge\frac{3}{2}.\]

2008 Junior Balkan Team Selection Tests - Romania, 4

Tags: inequalities

Determine the maximum possible real value of the number $ k$, such that \[ (a \plus{} b \plus{} c)\left (\frac {1}{a \plus{} b} \plus{} \frac {1}{c \plus{} b} \plus{} \frac {1}{a \plus{} c} \minus{} k \right )\ge k\] for all real numbers $ a,b,c\ge 0$ with $ a \plus{} b \plus{} c \equal{} ab \plus{} bc \plus{} ca$.

1997 All-Russian Olympiad Regional Round, 11.6

Tags: algebra , inequalities , logarithm

Prove that if $1 < a < b < c$, then $$\log_a(\log_a b) + \log_b(\log_b c) + \log_c(\log_c a) > 0.$$

1988 Greece National Olympiad, 1

Tags: inequalities , algebra

Let $a>0,b>0,c>0$ and $\sqrt{1987+a}+\sqrt{1987+b}=2\sqrt{1987+c}$. Prove that $\frac{1}{2} (a+b )\ge c $.