Olympiad problems

1986 French Mathematical Olympiad, Problem 2

Points $A,B,C$, and $M$ are given in the plane. (a) Let $D$ be the point in the plane such that $DA\le CA$ and $DB\le CB$. Prove that there exists point $N$ satisfying $NA\le MA,NB\le MB$, and $ND\le MC$. (b) Let $A',B',C'$ be the points in the plane such that $A'B'\le AB,A'C'\le AC,B'C'\le BC$. Does there exist a point $M'$ which satisfies the inequalities $M'A'\le MA,M'B'\le MB,M'C'\le MC$?

IV Soros Olympiad 1997 - 98 (Russia), 11.2

Tags: inequalities , algebra , trigonometry , analytic geometry

Find the area of a figure consisting of points whose coordinates satisfy the inequality $$(y^3 - arcsin x)(x^3 + arcsin y) \ge 0.$$

2002 JBMO ShortLists, 5

Tags: vector , induction , inequalities

Let $ a,b,c$ be positive real numbers. Prove the inequality: $ \frac {a^3}{b^2} \plus{} \frac {b^3}{c^2} \plus{} \frac {c^3}{a^2}\ge \frac {a^2}{b} \plus{} \frac {b^2}{c} \plus{} \frac {c^2}{a}$

2005 Silk Road, 4

Tags: inequalities , induction , algebra

Suppose $\{a(n) \}_{n=1}^{\infty}$ is a sequence that: \[ a(n) =a(a(n-1))+a(n-a(n-1)) \ \ \ \forall \ n \geq 3\] and $a(1)=a(2)=1$. Prove that for each $n \geq 1$ , $a(2n) \leq 2a(n)$.

2014 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , algebra

Determine all real numbers $x, y, z \in (0, 1)$ that satisfy simultaneously the conditions: $(x^2 + y^2)\sqrt{1- z^2}\ge z$ $(y^2 + z^2)\sqrt{1- x^2}\ge x$ $(z^2 + x^2)\sqrt{1- y^2}\ge y$

1982 Dutch Mathematical Olympiad, 1

Tags: inequalities

Which is greater: $ 17091982!^2$ or $ 17091982^{17091982}$?

1982 Kurschak Competition, 2

Tags: number theory , inequalities , lcm , least common multiple

Prove that for any integer $k > 2$, there exist infinitely many positive integers $n$ such that the least common multiple of $n$, $n + 1$,$...$, $n + k - 1$ is greater than the least common multiple of $n + 1$,$n + 2$,$...$, $n + k$.

1999 South africa National Olympiad, 1

Tags: geometry , perimeter , inequalities , ceiling function , triangle inequality

How many non-congruent triangles with integer sides and perimeter 1999 can be constructed?

1937 Eotvos Mathematical Competition, 1

Tags: algebra , inequalities

Let $n$ be a positive integer. Prove that $a_1!a_2! ... a_n! < k!$, where $k$ is an integer which is greater than the sum of the positive integers $a_1, a_2,.., a_n$.

2017 Macedonia National Olympiad, Problem 3

Tags: inequalities

Let $x,y,z \in \mathbb{R}$ such that $xyz = 1$. Prove that: $$\left(x^4 + \frac{z^2}{y^2}\right)\left(y^4 + \frac{x^2}{z^2}\right)\left(z^4 + \frac{y^2}{x^2}\right) \ge \left(\frac{x^2}{y} + 1 \right)\left(\frac{y^2}{z} + 1 \right)\left(\frac{z^2}{x} + 1 \right).$$

1999 Romania Team Selection Test, 4

Tags: search , inequalities

Show that for all positive real numbers $x_1,x_2,\ldots,x_n$ with product 1, the following inequality holds \[ \frac 1{n-1+x_1 } +\frac 1{n-1+x_2} + \cdots + \frac 1{n-1+x_n} \leq 1. \]

2018 China Team Selection Test, 5

Tags: inequalities

Given positive integers $n, k$ such that $n\ge 4k$, find the minimal value $\lambda=\lambda(n,k)$ such that for any positive reals $a_1,a_2,\ldots,a_n$, we have \[ \sum\limits_{i=1}^{n} {\frac{{a}_{i}}{\sqrt{{a}_{i}^{2}+{a}_{{i}+{1}}^{2}+{\cdots}{{+}}{a}_{{i}{+}{k}}^{2}}}} \le \lambda\] Where $a_{n+i}=a_i,i=1,2,\ldots,k$

2006 Korea Junior Math Olympiad, 6

Tags: algebra , inequalities , four variables

For all reals $a, b, c,d $ prove the following inequality: $$\frac{a + b + c + d}{(1 + a^2)(1 + b^2)(1 + c^2)(1 + d^2)}< 1$$

2019 Jozsef Wildt International Math Competition, W. 45

Tags: inequalities , trigonometry , complex number

Consider the complex numbers $a_1, a_2,\cdots , a_n$, $n \geq 2$. Which have the following properties: [list] [*] $|a_i|=1$ $\forall$ $i=1,2,\cdots , n$ [*] $\sum \limits_{k=1}^n arg(a_k)\leq \pi$ [/list] Show that the inequality$$\left(n^2\cot \left(\frac{\pi}{2n}\right)\right)^{-1}\left |\sum \limits_{k=0}^n(-1)^k\left[3n^2-(8k+5)n+4k(k+1)\sigma_k\right]\right |\geq \sqrt{\left(1+\frac{1}{n}\right)^2\cot^2 \left(\frac{\pi}{2n}\right)}+16\left |\sum \limits_{k=0}^n(-1)^k\sigma_k\right |$$where $\sigma_0=1$, $\sigma_k=\sum \limits_{1\leq i_1\leq i_2\leq \cdots \leq i_k\leq n}a_{i_1}a_{i_2}\cdots a_{i_k}$, $\forall$ $k=1,2,\cdots , n$

2005 Germany Team Selection Test, 2

Tags: inequalities , n-variable inequality

Let $ n$ be a positive integer such that $ n\geq 3$. Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2n$ positive real numbers satisfying the equations \[ a_1 \plus{} a_2 \plus{} ... \plus{} a_n \equal{} 1, \quad \text{and} \quad b_1^2 \plus{} b_2^2 \plus{} ... \plus{} b_n^2 \equal{} 1.\] Prove the inequality \[a_1\left(b_1 \plus{} a_2\right) \plus{} a_2\left(b_2 \plus{} a_3\right) \plus{} ... \plus{} a_{n \minus{} 1}\left(b_{n \minus{} 1} \plus{} a_n\right) \plus{} a_n\left(b_n \plus{} a_1\right) < 1.\]

2003 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: inequalities

Let $x$, $y$, and $z$ be positive real numbers satisfying $x^2+y^2+z^2=1$. Prove that \[x^2yz+xy^2z+xyz^2\le\frac{1}{3}.\]

2016 Silk Road, 1

Tags: algebra , inequalities

Let $a,b$ and $c$ be real numbers such that $| (a-b) (b-c) (c-a) | = 1$. Find the smallest value of the expression $| a | + | b | + | c |$. (K.Satylhanov )

1929 Eotvos Mathematical Competition, 2

Tags: binomial coefficient , binomial , algebra , inequalities

Let $k \le n$ be positive integers and $x$ be a real number with $0 \le x < 1/n$. Prove that $${n \choose 0} - {n \choose 1} x +{n \choose 2} x^2 - ... + (-1)^k {n \choose k} x^k > 0$$

2015 Junior Balkan Team Selection Tests - Romania, 3

Tags: inequalities

Prove that if $a,b,c>0$ and $a+b+c=1,$ then $$\frac{bc+a+1}{a^2+1}+\frac{ca+b+1}{b^2+1}+\frac{ab+c+1}{c^2+1}\leq \frac{39}{10}$$

2017 China Team Selection Test, 1

Tags: inequalities

Let $n \geq 4$ be a natural and let $x_1,\ldots,x_n$ be non-negative reals such that $x_1 + \cdots + x_n = 1$. Determine the maximum value of $x_1x_2x_3 + x_2x_3x_4 + \cdots + x_nx_1x_2$.

1990 Czech and Slovak Olympiad III A, 4

Tags: inequalities , algebra , max , parameter

Determine the largest $k\ge0$ such that the inequality \[\left(\sum_{j=1}^n x_j\right)^2\left(\sum_{j=1}^n x_jx_{j+1}\right)\ge k\sum_{j=1}^n x_j^2x_{j+1}^2\] holds for every $n\ge2$ and any $n$-tuple $x_1,\ldots,x_n$ of non-negative numbers (given that $x_{n+1}=x_1$)

Russian TST 2021, P3

Tags: inequalities , n-variable inequality

Let $n$ and $k$ be positive integers. Prove that for $a_1, \dots, a_n \in [1,2^k]$ one has \[ \sum_{i = 1}^n \frac{a_i}{\sqrt{a_1^2 + \dots + a_i^2}} \le 4 \sqrt{kn}. \]

2013 Federal Competition For Advanced Students, Part 2, 1

Tags: floor function , ceiling function , inequalities , algebra

For each pair $(a,b)$ of positive integers, determine all non-negative integers $n$ such that \[b+\left\lfloor{\frac{n}{a}}\right\rfloor=\left\lceil{\frac{n+b}{a}}\right\rceil.\]

2018 Macedonia JBMO TST, 3

Tags: algebra , inequality , inequalities

Let $x$, $y$, and $z$ be positive real numbers such that $x + y + z = 1$. Prove that $\frac{(x+y)^3}{z} + \frac{(y+z)^3}{x} + \frac{(z+x)^3}{y} + 9xyz \ge 9(xy + yz + zx)$. When does equality hold?

2001 India IMO Training Camp, 3

Tags: inequalities , incenter , geometry

In a triangle $ABC$ with incircle $\omega$ and incenter $I$ , the segments $AI$ , $BI$ , $CI$ cut $\omega$ at $D$ , $E$ , $F$ , respectively. Rays $AI$ , $BI$ , $CI$ meet the sides $BC$ , $CA$ , $AB$ at $L$ , $M$ , $N$ respectively. Prove that: \[AL+BM+CN \leq 3(AD+BE+CF)\] When does equality occur?