Olympiad problems

II Soros Olympiad 1995 - 96 (Russia), 10.1

Tags: inequalities , algebra

Find all values of $a$ for which the inequality $$a^2x^2 + y^2 + z^2 \ge ayz+xy+xz$$ holds for all $x$, $y$ and $z$.

2020 Jozsef Wildt International Math Competition, W12

Tags: inequalities

If $m,n,p,q\in\mathbb N,m,n,p,q\ge4$ then prove that: $$4^n(4^n+1)+4^m(4^m+1)+4^p(4^p+1)+4^q(4^q+1)\ge4mnpq(mnpq+1)$$ [i]Proposed by Daniel Sitaru[/i]

2010 Victor Vâlcovici, 1

Tags: inequalities , trigonometry

[b]a)[/b] Let be real numbers $ s,t\ge 0 $ and $ a,b\ge 1. $ Show that for any real $ x, $ it holds: $$ a^{s\sin x+t\cos x}b^{s\cos x+t\sin x}\le 10^{(s+t)\sqrt{\text{tg}^2 a+\text{tg}^2 b}} $$ [b]b)[/b] For $ a,b>0 $ is the above inequality still true? [i]Ilie Diaconu[/i]

2018 JBMO Shortlist, G6

Tags: chord , geometric inequality , inequalities , geometry

Let $XY$ be a chord of a circle $\Omega$, with center $O$, which is not a diameter. Let $P, Q$ be two distinct points inside the segment $XY$, where $Q$ lies between $P$ and $X$. Let $\ell$ the perpendicular line drawn from $P$ to the diameter which passes through $Q$. Let $M$ be the intersection point of $\ell$ and $\Omega$, which is closer to $P$. Prove that $$ MP \cdot XY \ge 2 \cdot QX \cdot PY$$

2024 Bulgarian Spring Mathematical Competition, 10.1

Tags: inequalities , algebra

The reals $x, y$ satisfy $x(x-6)\leq y(4-y)+7$. Find the minimal and maximal values of the expression $x+2y$.

2009 USAMTS Problems, 4

Tags: pigeonhole principle , inequalities

Let $S$ be a set of $10$ distinct positive real numbers. Show that there exist $x,y \in S$ such that \[0 < x - y < \frac{(1 + x)(1 + y)}{9}.\]

1975 Kurschak Competition, 3

Tags: recurrence relation , sequence , inequalities , algebra

Let $$x_0 = 5\,\, ,\, \,\,x_{n+1} = x_n +\frac{1}{x_n}.$$ Prove that $45 < x_{1000} < 45.1$.

2020 IMO Shortlist, A3

Tags: 4-variable inequality , inequality , inequalities , algebra

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

2008 Tournament Of Towns, 3

Tags: sum , root , inequalities , polynomial , algebra

A polynomial $x^n + a_1x^{n-1} + a_2x^{n-2} +... + a_{n-2}x^2 + a_{n-1}x + a_n$ has $n$ distinct real roots $x_1, x_2,...,x_n$, where $n > 1$. The polynomial $nx^{n-1}+ (n - 1)a_1x^{n-2} + (n - 2)a_2x^{n-3} + ...+ 2a_{n-2}x + a_{n-1}$ has roots $y_1, y_2,..., y_{n_1}$. Prove that $\frac{x^2_1+ x^2_2+ ...+ x^2_n}{n}>\frac{y^2_1 + y^2_2 + ...+ y^2_{n-1}}{n - 1}$

2020 SJMO, 5

Tags: inequalities , algebra

A nondegenerate triangle with perimeter $1$ has side lengths $a, b,$ and $c$. Prove that \[\left|\frac{a - b}{c + ab}\right| + \left|\frac{b - c}{a + bc}\right| + \left|\frac{c - a}{b + ac}\right| < 2.\] [i]Proposed by Andrew Wen[/i]

1990 IMO Longlists, 12

Tags: combinatorial inequality , inequalities , permutation , algebra

For any permutation $p$ of set $\{1, 2, \ldots, n\}$, define $d(p) = |p(1) - 1| + |p(2) - 2| + \ldots + |p(n) - n|$. Denoted by $i(p)$ the number of integer pairs $(i, j)$ in permutation $p$ such that $1 \leqq < j \leq n$ and $p(i) > p(j)$. Find all the real numbers $c$, such that the inequality $i(p) \leq c \cdot d(p)$ holds for any positive integer $n$ and any permutation $p.$

2010 Indonesia TST, 3

Tags: inequalities , induction , algebra

Let $ a_1,a_2,\dots$ be sequence of real numbers such that $ a_1\equal{}1$, $ a_2\equal{}\dfrac{4}{3}$, and \[ a_{n\plus{}1}\equal{}\sqrt{1\plus{}a_na_{n\minus{}1}}, \quad \forall n \ge 2.\] Prove that for all $ n \ge 2$, \[ a_n^2>a_{n\minus{}1}^2\plus{}\dfrac{1}{2}\] and \[ 1\plus{}\dfrac{1}{a_1}\plus{}\dfrac{1}{a_2}\plus{}\dots\plus{}\dfrac{1}{a_n}>2a_n.\] [i]Fajar Yuliawan, Bandung[/i]

2013 Kyiv Mathematical Festival, 1

Tags: 4-variable inequality , inequality , algebra , inequalities

For every positive $a, b, c, d$ such that $a + c\le ac$ and $b + d \le bd$ prove that $ab + cd \ge 8$.

2011 Tournament of Towns, 5

Tags: algebra , inequalities

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

2012 India Regional Mathematical Olympiad, 4

Tags: area of a triangle , circumcircle , inequalities , heron's formula , trigonometry , geometry , geometric inequality

Let $a,b,c$ be positive real numbers such that $abc(a+b+c)=3.$ Prove that we have \[(a+b)(b+c)(c+a)\geq 8.\] Also determine the case of equality.

2021 Vietnam TST, 4

Tags: inequalities , algebra

Let $a,b,c$ are non-negative numbers such that $$2(a^2+b^2+c^2)+3(ab+bc+ca)=5(a+b+c)$$ then prove that $4(a^2+b^2+c^2)+2(ab+bc+ca)+7abc\le 25$

2024 Macedonian Balkan MO TST, Problem 4

Tags: absolute value , inequalities

Let $x_1, ..., x_n$ $(n \geq 2)$ be real numbers from the interval $[1,2]$. Prove that $$|x_1-x_2|+...+|x_n-x_1| + \frac{1}{3} (|x_1-x_3|+...+|x_n-x_2|) \leq \frac{2}{3} (x_1+...+x_n)$$ and determine all cases of equality.

1993 Greece National Olympiad, 4

Tags: symmetry , matrix , inequalities , linear algebra , polynomial , algebra

How many ordered four-tuples of integers $(a,b,c,d)$ with $0 < a < b < c < d < 500$ satisfy $a + d = b + c$ and $bc - ad = 93$?

2018 China Northern MO, 2

Tags: three variable inequality , inequalities

Let $a$,$b$,$c$ be nonnegative reals such that $$a^2+b^2+c^2+ab+\frac{2}{3}ac+\frac{4}{3}bc=1$$ Find the maximum and minimum value of $a+b+c$.

1970 IMO Longlists, 25

Tags: derivative , real analysis , calculus , function , inequalities , integration

A real function $f$ is defined for $0\le x\le 1$, with its first derivative $f'$ defined for $0\le x\le 1$ and its second derivative $f''$ defined for $0<x<1$. Prove that if $f(0)=f'(0)=f'(1)=f(1)-1 =0$, then there exists a number $0<y<1$ such that $|f''(y)|\ge 4$.

1996 Austrian-Polish Competition, 4

Tags: algebra , inequalities

Real numbers $x,y,z, t$ satisfy $x + y + z +t = 0$ and $x^2+ y^2+ z^2+t^2 = 1$. Prove that $- 1 \le xy + yz + zt + tx \le 0$.

2014 Cono Sur Olympiad, 1

Tags: invariant , least common multiple , inequalities , greatest common divisor , number theory , combinatorics

Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$. Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$

2009 IMO Shortlist, 4

Tags: inequalities

Let $a$, $b$, $c$ be positive real numbers such that $ab+bc+ca\leq 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\leq \sqrt{2}\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)\] [i]Proposed by Dzianis Pirshtuk, Belarus[/i]

1962 All Russian Mathematical Olympiad, 019

Tags: inequalities , algebra

Given a quartet of positive numbers $a,b,c,d$, and is known, that $abcd=1$. Prove that $$a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+dc \ge 10$$

2016 Kosovo National Mathematical Olympiad, 5

Tags: inequalities

If $a,b,c$ are sides of right triangle with $c$ hypothenuse then show that for every positive integer $n>2$ we have $c^n>a^n+b^n$ .