Olympiad problems

1993 All-Russian Olympiad Regional Round, 10.6

Tags: inequalities

Prove the inequality $ \sqrt {2 \plus{} \sqrt [3]{3 \plus{} ... \plus{} \sqrt [{2008}]{2008}}} < 2$

1974 IMO Longlists, 33

Tags: inequalities

Let a be a real number such that $0 < a < 1$, and let $n$ be a positive integer. Define the sequence $a_0, a_1, a_2, \ldots, a_n$ an recursively by \[a_0 = a, \quad a_{k+1} = a_k +\frac 1n a_k^2 \quad \text{ for } k = 0, 1, \ldots, n - 1.\] Prove that there exists a real number $A$, depending on $a$ but independent of $n$, such that \[0 < n(A - a_n) < A^3.\]

1996 Romania Team Selection Test, 8

Tags: number theory , inequalities

Let $ p_1,p_2,\ldots,p_k $ be the distinct prime divisors of $ n $ and let $ a_n=\frac {1}{p_1}+\frac {1}{p_2}+\cdots+\frac {1}{p_k} $ for $ n\geq 2 $. Show that for every positive integer $ N\geq 2 $ the following inequality holds: $ \sum_{k=2}^{N} a_2a_3 \cdots a_k <1 $ [i]Laurentiu Panaitopol[/i]

2020 Jozsef Wildt International Math Competition, W48

Tags: geometric inequalities , inequalities , geometry

Let $ABC$ be a triangle such that $$S^2=2R^2+8Rr+3r^2$$ Then prove that $\frac Rr=2$ or $\frac Rr\ge\sqrt2+1$. [i]Proposed by Marian Cucoanoeş and Marius Drăgan[/i]

1991 APMO, 3

Tags: inequalities , n-variable inequality

Let $a_1$, $a_2$, $\cdots$, $a_n$, $b_1$, $b_2$, $\cdots$, $b_n$ be positive real numbers such that $a_1 + a_2 + \cdots + a_n = b_1 + b_2 + \cdots + b_n$. Show that \[ \frac{a_1^2}{a_1 + b_1} + \frac{a_2^2}{a_2 + b_2} + \cdots + \frac{a_n^2}{a_n + b_n} \geq \frac{a_1 + a_2 + \cdots + a_n}{2} \]

2002 China Team Selection Test, 2

Tags: induction , algebra , function , inequalities

For any two rational numbers $ p$ and $ q$ in the interval $ (0,1)$ and function $ f$, there is always $ \displaystyle f \left( \frac{p\plus{}q}{2} \right) \leq \frac{f(p) \plus{} f(q)}{2}$. Then prove that for any rational numbers $ \lambda, x_1, x_2 \in (0,1)$, there is always: \[ f( \lambda x_1 \plus{} (1\minus{}\lambda) x_2 ) \leq \lambda f(x_i) \plus{} (1\minus{}\lambda) f(x_2)\]

2004 IMC, 5

Tags: integration , logarithm , inequalities , calculus

Prove that \[ \int^1_0 \int^1_0 \frac { dx \ dy }{ \frac 1x + |\log y| -1 } \leq 1 . \]

1994 China Team Selection Test, 1

Tags: domain , linear algebra , function , logarithm , inequalities , algebra , matrix

Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.

1966 IMO Longlists, 32

Tags: geometry , triangle inequality , inequalities , similar triangles

The side lengths $a,$ $b,$ $c$ of a triangle $ABC$ form an arithmetical progression (such that $b-a=c-b$). The side lengths $a_{1},$ $b_{1},$ $c_{1}$ of a triangle $A_{1}B_{1}C_{1}$ also form an arithmetical progression (with $b_{1}-a_{1}=c_{1}-b_{1}$). [Hereby, $a=BC,$ $b=CA,$ $c=AB, $ $a_{1}=B_{1}C_{1},$ $b_{1}=C_{1}A_{1},$ $c_{1}=A_{1}B_{1}.$] Moreover, we know that $\measuredangle CAB=\measuredangle C_{1}A_{1}B_{1}.$ Show that triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.

2014 Contests, 2

Tags: inequalities

Let $a$ ,$b$ and $c$ be distinct real numbers. $a)$ Determine value of $ \frac{1+ab }{a-b} \cdot \frac{1+bc }{b-c} + \frac{1+bc }{b-c} \cdot \frac{1+ca }{c-a} + \frac{1+ca }{c-a} \cdot \frac{1+ab}{a-b} $ $b)$ Determine value of $ \frac{1-ab }{a-b} \cdot \frac{1-bc }{b-c} + \frac{1-bc }{b-c} \cdot \frac{1-ca }{c-a} + \frac{1-ca }{c-a} \cdot \frac{1-ab}{a-b} $ $c)$ Prove the following ineqaulity $ \frac{1+a^2b^2 }{(a-b)^2} + \frac{1+b^2c^2 }{(b-c)^2} + \frac{1+c^2a^2 }{(c-a)^2} \geq \frac{3}{2} $ When does eqaulity holds?

2005 Romania National Olympiad, 4

Tags: geometry , n-variable inequality , inequalities

Let $x_1,x_2,\ldots,x_n$ be positive reals. Prove that \[ \frac 1{1+x_1} + \frac 1{1+x_1+x_2} + \cdots + \frac 1{1+x_1+\cdots + x_n} < \sqrt { \frac 1{x_1} + \frac 1{x_2} + \cdots + \frac 1{x_n}} . \] [i]Bogdan Enescu[/i]

2015 Costa Rica - Final Round, 5

Tags: inequalities , algebra

Let $a,b \in R^+$ with $ab = 1$, prove that $$\frac{1}{a^3 + 3b}+\frac{1}{b^3 + 3a}\le \frac12.$$

2011 Postal Coaching, 4

Tags: inequalities

For all $a, b, c > 0$ and $abc = 1$, prove that \[\frac{1}{a(a+1)+ab(ab+1)}+\frac{1}{b(b+1)+bc(bc+1)}+\frac{1}{c(c+1)+ca(ca+1)}\ge\frac{3}{4}\]

2015 Iran MO (3rd round), 1

Tags: inequalities

$x,y,z$ are three real numbers inequal to zero satisfying $x+y+z=xyz$. Prove that $$ \sum (\frac{x^2-1}{x})^2 \geq 4$$ [i]Proposed by Amin Fathpour[/i]

2007 Bulgaria Team Selection Test, 1

Tags: inequalities , trigonometry , geometry , circumcircle

Let $ABC$ is a triangle with $\angle BAC=\frac{\pi}{6}$ and the circumradius equal to 1. If $X$ is a point inside or in its boundary let $m(X)=\min(AX,BX,CX).$ Find all the angles of this triangle if $\max(m(X))=\frac{\sqrt{3}}{3}.$

2002 India Regional Mathematical Olympiad, 6

Tags: inequalities , limit

Prove that for any natural number $n > 1$, \[ \frac{1}{2} < \frac{1}{n^2+1} + \frac{2}{n^2 +2} + \ldots + \frac{n}{n^2 + n} < \frac{1}{2} + \frac{1}{2n}. \]

1976 IMO Longlists, 46

Tags: symmetry , search , inequalities

Let $ a,b,c,d$ be nonnegative real numbers. Prove that \[ a^4\plus{}b^4\plus{}c^4\plus{}d^4\plus{}2abcd \ge a^2b^2\plus{}a^2c^2\plus{}a^2d^2\plus{}b^2c^2\plus{}b^2d^2\plus{}c^2d^2.\]

2017 ELMO Shortlist, 2

Tags: algebra , inequalities

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$: (i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$ (ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$ [i]Proposed by Ashwin Sah[/i]

2001 Moldova National Olympiad, Problem 2

Tags: inequality , optimization , inequalities

If $n\in\mathbb N$ and $a_1,a_2,\ldots,a_n$ are arbitrary numbers in the interval $[0,1]$, find the maximum possible value of the smallest among the numbers $a_1-a_1a_2,a_2-a_2a_3,\ldots,a_n-a_na_1$.

1991 Bundeswettbewerb Mathematik, 1

Tags: algebra , inequalities , product

Given $1991$ distinct positive real numbers, the product of any ten of these numbers is always greater than $1$. Prove that the product of all $1991$ numbers is also greater than $1$.

2019 Nigeria Senior MO Round 2, 5

Tags: inequalities

Let $a$, $b$, and $c$ be real numbers such that $abc=1$. prove that $\frac{1+a+ab}{1+b+ab}$ +$\frac{1+b+bc}{1+c+bc}$ + $\frac{1+c+ac}{1+a+ac}$ $>=3$

JOM 2015 Shortlist, A9

Tags: inequalities

Let $2n$ positive reals $a_1, a_2, \cdots, a_n, b_1, b_2, \cdots, b_n$ satisfy $a_{i+1}\ge 2a_i$ and $b_{i+1} \le b_i$ for $1\le i\le n-1$. Find the least constant $C$ that satisfy: \[\displaystyle \sum^{n}_{i=1}{\frac{a_i}{b_i}} \ge \displaystyle \frac{C(a_1+a_2+\cdots+a_n)}{b_1+b_2+\cdots+b_n}\] and determine all equality case with that constant $C$.

2023 Francophone Mathematical Olympiad, 1

Tags: polynomial , algebra , inequalities

Let $P(X) = a_n X^n + a_{n-1} X^{n-1} + \cdots + a_1 X + a_0$ be a polynomial with real coefficients such that $0 \leqslant a_i \leqslant a_0$ for $i = 1, 2, \ldots, n$. Prove that, if $P(X)^2 = b_{2n} X^{2n} + b_{2n-1} X^{2n-1} + \cdots + b_{n+1} X^{n+1} + \cdots + b_1 X + b_0$, then $4 b_{n+1} \leqslant P(1)^2$.

2004 India IMO Training Camp, 3

Tags: inequalities

For $a,b,c$ positive reals find the minimum value of \[ \frac{a^2+b^2}{c^2+ab}+\frac{b^2+c^2}{a^2+bc}+\frac{c^2+a^2}{b^2+ca}. \]

2014 JBMO Shortlist, 2

Tags: algebra , positive real , three variable inequality , inequalities

Let $a, b, c$ be positive real numbers such that $abc = \dfrac {1} {8}$. Prove the inequality:$$a ^ 2 + b ^ 2 + c ^ 2 + a ^ 2b ^ 2 + b ^ 2c ^ 2 + c ^ 2a ^ 2 \geq \dfrac {15} {16}$$ When the equality holds?