Olympiad problems

2010 Contests, 2

Tags: combinatorics , inequalities

There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.

1952 Moscow Mathematical Olympiad, 214

Tags: algebra , inequalities , absolute value , moscow

Prove that if $|x| < 1$ and $|y| < 1$, then $\left|\frac{x - y}{1 -xy}\right|< 1$.

1998 Denmark MO - Mohr Contest, 4

Tags: algebra , inequalities

Let $a$ and $b$ be positive real numbers with $a + b =1$. Show that $$\left(a+\frac{1}{a}\right)^2 + \left(b+\frac{1}{b}\right)^2 \ge \frac{25}{2}.$$

Kvant 2024, M2797

Tags: algebra , inequalities , inequality

For real numbers $0 \leq a_1 \leq a_2 \leq ... \leq a_n$ and $0 \leq b_1 \leq b_2 \leq ... \leq b_n$ prove that \[ \left( \frac{a_1}{1 \cdot 2}+\frac{a_2}{2 \cdot 3}+...+\frac{a_n}{n(n+1)} \right) \times \left( \frac{b_1}{1 \cdot 2}+\frac{b_2}{2 \cdot 3}+...+\frac{b_n}{n(n+1)} \right) \leq \frac{a_1b_1}{1 \cdot 2}+\frac{a_2b_2}{2 \cdot 3}+...+\frac{a_nb_n}{n(n+1)}.\] [i]Proposed by A. Antropov[/i]

1986 Traian Lălescu, 2.3

Tags: function , inequalities , calculus , derivative , ftc , rolle theorem , integral

Let $ f:[0,2]\longrightarrow \mathbb{R} $ a differentiable function having a continuous derivative and satisfying $ f(0)=f(2)=1 $ and $ |f’|\le 1. $ Show that $$ \left| \int_0^2 f(t) dt\right| >1. $$

1988 India National Olympiad, 4

Tags: function , inequalities

If $ a$ and $ b$ are positive and $ a \plus{} b \equal{} 1$, prove that \[ \left(a\plus{}\frac{1}{a}\right)^2\plus{}\left(b\plus{}\frac{1}{b}\right)^2 \geq \frac{25}{2}\]

1974 Swedish Mathematical Competition, 2

Tags: algebra , inequalities

Show that \[ 1 - \frac{1}{k} \leq n\left(\sqrt[n]{k}-1\right) \leq k - 1 \] for all positive integers $n$ and positive reals $k$.

2010 Contests, 3

Tags: trigonometry , circumcircle , geometric transformation , triangle inequality , inequalities , reflection , geometry

Let $h_a, h_b, h_c$ be the lengths of the altitudes of a triangle $ABC$ from $A, B, C$ respectively. Let $P$ be any point inside the triangle. Show that \[\frac{PA}{h_b+h_c} + \frac{PB}{h_a+h_c} + \frac{PC}{h_a+h_b} \ge 1.\]

1998 All-Russian Olympiad, 4

Tags: inequalities , euler , combinatorics

A connected graph has $1998$ points and each point has degree $3$. If $200$ points, no two of them joined by an edge, are deleted, show that the result is a connected graph.

2011 Mathcenter Contest + Longlist, 8 sl12

Tags: algebra , inequalities

Let $a,b,c\in\mathbb{R^+}$. Prove that $$\frac{a^{11}}{b^5c^5}+\frac{b^{11}}{ c^5a^5}+\frac{c^{11}}{a^5b^5}\ge a+b+c$$ [i](Real Matrik)[/i]

1971 Polish MO Finals, 5

Tags: inequalities , number theory , algebra

Find the largest integer $A$ such that, for any permutation of the natural numbers not exceeding $100$, the sum of some ten successive numbers is at least $A$.

1967 Swedish Mathematical Competition, 5

Tags: inequalities , algebra

$a_1, a_2, a_3, ...$ are positive reals such that $a_n^2 \ge a_1 + a_2 +... + a_{n-1}$. Show that for some $C > 0$ we have $a_n \ge C n$ for all $n$.

2007 Middle European Mathematical Olympiad, 1

Tags: inequalities

Let $ a,b,c,d$ be positive real numbers with $ a\plus{}b\plus{}c\plus{}d \equal{} 4$. Prove that \[ a^{2}bc\plus{}b^{2}cd\plus{}c^{2}da\plus{}d^{2}ab\leq 4.\]

STEMS 2021 Math Cat A, Q2

Tags: function , inequalities

Suppose $f: \mathbb{R}^{+} \mapsto \mathbb{R}^{+}$ is a function such that $\frac{f(x)}{x}$ is increasing on $\mathbb{R}^{+}$. For $a,b,c>0$, prove that $$2\left (\frac{f(a)+f(b)}{a+b} + \frac{f(b)+f(c)}{b+c}+ \frac{f(c)+f(a)}{c+a} \right) \geq 3\left(\frac{f(a)+f(b)+f(c)}{a+b+c}\right) + \frac{f(a)}{a}+ \frac{f(b)}{b}+ \frac{f(c)}{c}$$

2004 Thailand Mathematical Olympiad, 18

Tags: algebra , inequalities , max , product

Find positive reals $a, b, c$ which maximizes the value of $abc$ subject to the constraint that $b(a^2 + 2) + c(a + 2) = 12$.

Kyiv City MO Seniors 2003+ geometry, 2003.11.3

Tags: inequalities , 3d geometry , geometrical inequality , tetrahedron , geometry

Let $x_1, x_2, x_3, x_4$ be the distances from an arbitrary point inside the tetrahedron to the planes of its faces, and let $h_1, h_2, h_3, h_4$ be the corresponding heights of the tetrahedron. Prove that $$\sqrt{h_1+h_2+h_3+h_4} \ge \sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}$$ (Dmitry Nomirovsky)

2018 Costa Rica - Final Round, 2

Tags: algebra , inequalities , max

Let $a, b, c$, and $d$ be real numbers. The six sums of two numbers $x$ and $y$, different from the previous four, are $117$, $510$, $411$, $252$, in no particular order. Determine the maximum possible value of $x + y$.

2009 JBMO Shortlist, 4

Tags: inequalities , symmetry

Let $ x$, $ y$, $ z$ be real numbers such that $ 0 < x,y,z < 1$ and $ xyz \equal{} (1 \minus{} x)(1 \minus{} y)(1 \minus{} z)$. Show that at least one of the numbers $ (1 \minus{} x)y,(1 \minus{} y)z,(1 \minus{} z)x$ is greater than or equal to $ \frac {1}{4}$

2011 Indonesia TST, 1

Tags: inequalities , geometric inequalities , algebra

Let $a, b, c$ be the sides of a triangle with $abc = 1$. Prove that $$\frac{\sqrt{b + c -a}}{a}+\frac{\sqrt{c + a - b}}{b}+\frac{\sqrt{a + b - c}}{c} \ge a + b + c$$

2006 IMO Shortlist, 2

Tags: integration , calculus , inequalities , sequence , summation , algebra

The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$. [i]Proposed by Mariusz Skalba, Poland[/i]

2021-IMOC, A6

Tags: inequalities , algebra , am-gm

Let $n$ be some positive integer and $a_1 , a_2 , \dots , a_n$ be real numbers. Denote $$S_0 = \sum_{i=1}^{n} a_i^2 , \hspace{1cm} S_1 = \sum_{i=1}^{n} a_ia_{i+1} , \hspace{1cm} S_2 = \sum_{i=1}^{n} a_ia_{i+2},$$ where $a_{n+1} = a_1$ and $a_{n+2} = a_2.$ 1. Show that $S_0 - S_1 \geq 0$. 2. Show that $3$ is the minimum value of $C$ such that for any $n$ and $a_1 , a_2 , \dots , a_n,$ there holds $C(S_0 - S_1) \geq S_1 - S_2$.

2008 Bundeswettbewerb Mathematik, 2

Tags: relatively prime , inequalities , number theory , greatest common divisor , least common multiple

Let the positive integers $ a,b,c$ chosen such that the quotients $ \frac{bc}{b\plus{}c},$ $ \frac{ca}{c\plus{}a}$ and $ \frac{ab}{a\plus{}b}$ are integers. Prove that $ a,b,c$ have a common divisor greater than 1.

2007 Hungary-Israel Binational, 3

Tags: derivative , trigonometry , inequalities , function , geometry , calculus

Let $ AB$ be the diameter of a given circle with radius $ 1$ unit, and let $ P$ be a given point on $ AB$. A line through $ P$ meets the circle at points $ C$ and $ D$, so a convex quadrilateral $ ABCD$ is formed. Find the maximum possible area of the quadrilateral.

2006 Finnish National High School Mathematics Competition, 2

Tags: inequalities

Show that the inequality \[3(1 + a^2 + a^4)\geq (1 + a + a^2)^2\] holds for all real numbers $a.$

2014 Cuba MO, 8

Tags: inequalities , algebra

Let $a$ and $b$ be real numbers. It is known that the graph of the parabola $y =ax^2 +b$ cuts the graph of the curve $y = x+1/x$ in exactly three points. Prove that $3ab < 1$.