Olympiad problems

2012 Ukraine Team Selection Test, 12

We shall call the triplet of numbers $a, b, c$ of the interval $[-1,1]$ [i]qualitative [/i] if these numbers satisfy the inequality $1 + 2abc\ge a^2 + b^2 + c^2$. Prove that when the triples $a, b, c$, and $x, y, z$ are qualitative, then $ax, by, cz$ is also qualitative.

2006 South East Mathematical Olympiad, 2

Tags: inequalities

Find the minimum value of real number $m$, such that inequality \[m(a^3+b^3+c^3) \ge 6(a^2+b^2+c^2)+1\] holds for all positive real numbers $a,b,c$ where $a+b+c=1$.

2011 Tournament of Towns, 5

Tags: inequalities , algebra

Given that $0 < a, b, c, d < 1$ and $abcd = (1 - a)(1 - b)(1 - c)(1 - d)$, prove that $(a + b + c + d) -(a + c)(b + d) \ge 1$

2023 Indonesia TST, A

Tags: algebra , inequalities

Let $a,b,c$ positive real numbers and $a+b+c = 1$. Prove that \[a^2 + b^2 + c^2 + \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \ge 2(ab + bc + ac)\]

MathLinks Contest 1st, 2

Tags: complex number , inequalities , algebra

Let $m$ be the greatest number such that for any set of complex numbers having the sum of all modulus of all the elements $1$, there exists a subset having the modulus of the sum of the elements in the subset greater than $m$. Prove that $$\frac14 \le m \le \frac12.$$ (Optional Task for 3p) Find a smaller value for the RHS.

2021 Latvia Baltic Way TST, P2

Tags: inequalities

Determine all functions $f: \mathbb{R} \backslash \{0 \} \rightarrow \mathbb{R}$ such that, for all nonzero $x$: $$ f(\frac{1}{x}) \ge 1 -f(x) \ge x^2f(x) $$

2008 Pre-Preparation Course Examination, 4

Tags: inequalities , probability and stats

Sarah and Darah play the following game. Sarah puts $ n$ coins numbered with $ 1,\dots,n$ on a table (Each coin is in HEAD or TAIL position.) At each step Darah gives a coin to Sarah and she (Sarah) let him (Dara) to change the position of all coins with number multiple of a desired number $ k$. At the end, all of the coins that are in TAIL position will be given to Sarah and all of the coins with HEAD position will be given to Darah. Prove that Sarah can put the coins in a position at the beginning of the game such that she gains at least $ \Omega(n)$ coins. [hide="Hint:"]Chernov inequality![/hide]

2024 Austrian MO National Competition, 1

Tags: algebra , inequalities

Determine the smallest real constant $C$ such that the inequality \[(X+Y)^2(X^2+Y^2+C)+(1-XY)^2 \ge 0\] holds for all real numbers $X$ and $Y$. For which values of $X$ and $Y$ does equality hold for this smallest constant $C$? [i](Walther Janous)[/i]

2003 Romania Team Selection Test, 10

Tags: number theory , modular arithmetic , pigeonhole principle , inequalities , relatively prime

Let $\mathcal{P}$ be the set of all primes, and let $M$ be a subset of $\mathcal{P}$, having at least three elements, and such that for any proper subset $A$ of $M$ all of the prime factors of the number $ -1+\prod_{p\in A}p$ are found in $M$. Prove that $M= \mathcal{P}$. [i]Valentin Vornicu[/i]

2008 AIME Problems, 10

Tags: geometry , trapezoid , inequalities , trigonometry , quadratic

Let $ ABCD$ be an isosceles trapezoid with $ \overline{AD}\parallel{}\overline{BC}$ whose angle at the longer base $ \overline{AD}$ is $ \dfrac{\pi}{3}$. The diagonals have length $ 10\sqrt {21}$, and point $ E$ is at distances $ 10\sqrt {7}$ and $ 30\sqrt {7}$ from vertices $ A$ and $ D$, respectively. Let $ F$ be the foot of the altitude from $ C$ to $ \overline{AD}$. The distance $ EF$ can be expressed in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

2010 Belarus Team Selection Test, 8.2

Tags: inequalities , algebra

Prove that for positive real numbers $a, b, c$ such that $abc=1$, the following inequality holds: $$\frac{a}{b(a+b)}+\frac{b}{c(b+c)}+\frac{c}{a(c+a)} \ge \frac32$$ (I. Voronovich)

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 $$1<\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{d}{cd+d+1}+\frac{a}{da+a+1}<2.$$

1996 China Team Selection Test, 2

Tags: real root , inequalities , algebra , sums and products

Let $\alpha_1, \alpha_2, \dots, \alpha_n$, and $\beta_1, \beta_2, \ldots, \beta_n$, where $n \geq 4$, be 2 sets of real numbers such that \[\sum_{i=1}^{n} \alpha_i^2 < 1 \qquad \text{and} \qquad \sum_{i=1}^{n} \beta_i^2 < 1.\] Define \begin{align*} A^2 &= 1 - \sum_{i=1}^{n} \alpha_i^2,\\ B^2 &= 1 - \sum_{i=1}^{n} \beta_i^2,\\ W &= \frac{1}{2} (1 - \sum_{i=1}^{n} \alpha_i \beta_i)^2. \end{align*} Find all real numbers $\lambda$ such that the polynomial \[x^n + \lambda (x^{n-1} + \cdots + x^3 + Wx^2 + ABx + 1) = 0,\] only has real roots.

2018 Junior Regional Olympiad - FBH, 4

Tags: inequalities , positive real

Let $a$, $b$ and $c$ be positive real numbers such that $a \geq b \geq c$. Prove the inequality: $\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \leq \frac{b}{a}+\frac{c}{b}+\frac{a}{c}$

2015 Balkan MO Shortlist, A3

Tags: inequalities , geometric inequality , median

Let a$,b,c$ be sidelengths of a triangle and $m_a,m_b,m_c$ the medians at the corresponding sides. Prove that $$m_a\left(\frac{b}{a}-1\right)\left(\frac{c}{a}-1\right)+ m_b\left(\frac{a}{b}-1\right)\left(\frac{c}{b}-1\right) +m_c\left(\frac{a}{c}-1\right)\left(\frac{b}{c}-1\right)\geq 0.$$ (FYROM)

2005 Junior Balkan Team Selection Tests - Moldova, 3

Tags: inequalities

Let $a_1,a_2,...a_n$ be positive numbers. And let $s=a_1+a_2+...+a_n$,and $p=a_1*a_2*...*a_n$.Prove that $2^{n}*\sqrt{p} \leq 1+\frac{s}{1!}+\frac{s^2}{2!}+...+\frac{s^n}{n!}$

2014 Harvard-MIT Mathematics Tournament, 7

Tags: inequalities

Find the largest real number $c$ such that \[\sum_{i=1}^{101}x_i^2\geq cM^2\] whenever $x_1,\ldots,x_{101}$ are real numbers such that $x_1+\cdots+x_{101}=0$ and $M$ is the median of $x_1,\ldots,x_{101}$.

2003 Switzerland Team Selection Test, 6

Tags: inequalities

Let $ a,b,c $ be positive real numbers satisfying $ a+b+c=2 $. Prove the inequality \[ \frac{1}{1+ab}+\frac{1}{1+bc}+\frac{1}{1+ca} \ge \frac{27}{13} \]

1985 Iran MO (2nd round), 1

Tags: circumcircle , inequalities , geometry

Inscribe in the triangle $ABC$ a triangle with minimum perimeter.

2006 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: factorial , induction , inequalities , number theory

(a) For each integer $k\ge 3$, find a positive integer $n$ that can be represented as the sum of exactly $k$ mutually distinct positive divisors of $n$. (b) Suppose that $n$ can be expressed as the sum of exactly $k$ mutually distinct positive divisors of $n$ for some $k\ge 3$. Let $p$ be the smallest prime divisor of $n$. Show that \[\frac1p+\frac1{p+1}+\cdots+\frac{1}{p+k-1}\ge1.\]

1993 Poland - Second Round, 1

Tags: inequalities

If $ x,y,u,v$ are positiv real numbers, prove the inequality : \[ \frac {xu \plus{} xv \plus{} yu \plus{} yv}{x \plus{} y \plus{} u \plus{} v} \geq \frac {xy}{x \plus{} y} \plus{} \frac {uv}{u \plus{} v} \]

1991 IberoAmerican, 2

Tags: inequalities , geometric transformation , rotation , parallelogram , geometry

A square is divided in four parts by two perpendicular lines, in such a way that three of these parts have areas equal to 1. Show that the square has area equal to 4.

1995 Austrian-Polish Competition, 9

Tags: algebra , inequalities

Prove that for all positive integers $n,m$ and all real numbers $x, y > 0$ the following inequality holds: \[(n - 1)(m- 1)(x^{n+m} + y^{n+m}) + (n + m - 1)(x^ny^m + x^my^n)\\ \\ \ge nm(x^{n+m-1}y + xy^{n+m-1}).\]

2021 Tuymaada Olympiad, 1

Tags: inequalities

Polynomials $F$ and $G$ satisfy: $$F(F(x))>G(F(x))>G(G(x))$$ for all real $x$.Prove that $F(x)>G(x)$ for all real $x$.

2002 Poland - Second Round, 3

Tags: search , geometry , inequalities , combinatorics

A positive integer $ n$ is given. In an association consisting of $ n$ members work $ 6$ commissions. Each commission contains at least $ \large \frac{n}{4}$ persons. Prove that there exist two commissions containing at least $ \large \frac{n}{30}$ persons in common.