Olympiad problems

2023 Durer Math Competition (First Round), 1

Find all positive integers $n$ such that $$\lfloor \sqrt{n} \rfloor + \left\lfloor \frac{n}{\lfloor \sqrt{n} \rfloor} \right \rfloor> 2\sqrt{n}.$$ If $k$ is a real number, then $\lfloor k \rfloor$ means the floor of $k$, this is the greatest integer less than or equal to $k$.

2025 Czech-Polish-Slovak Junior Match., 5

Tags: algebra , inequalities

For every integer $n\geq 1$ prove that $$\frac{1}{n+1}-\frac{2}{n+2}+\frac{3}{n+3}-\frac{4}{n+4}+...+\frac{2n-1}{3n-1}>\frac{1}{3}.$$

2012 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: inequalities , algebra

For which real numbers $x$ and $\alpha$ inequality holds: $$\log _2 {x}+\log _x {2}+2\cos{\alpha} \leq 0$$

2005 Germany Team Selection Test, 3

Tags: inradius , euler , incenter , circumcircle , geometry , vector , inequalities

Let $ABC$ be a triangle with orthocenter $H$, incenter $I$ and centroid $S$, and let $d$ be the diameter of the circumcircle of triangle $ABC$. Prove the inequality \[9\cdot HS^2+4\left(AH\cdot AI+BH\cdot BI+CH\cdot CI\right)\geq 3d^2,\] and determine when equality holds.

2013 China Team Selection Test, 3

Tags: inequalities , algebra

Let $n>1$ be an integer and let $a_0,a_1,\ldots,a_n$ be non-negative real numbers. Definite $S_k=\sum_{i\equal{}0}^k \binom{k}{i}a_i$ for $k=0,1,\ldots,n$. Prove that\[\frac{1}{n} \sum_{k\equal{}0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k\equal{}0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.\]

1998 French Mathematical Olympiad, Problem 1

Tags: tetrahedron , 3d geometry , geometry , inequalities

A tetrahedron $ABCD$ satisfies the following conditions: the edges $AB,AC$ and $AD$ are pairwise orthogonal, $AB=3$ and $CD=\sqrt2$. Find the minimum possible value of $$BC^6+BD^6-AC^6-AD^6.$$

2021 Kosovo National Mathematical Olympiad, 3

Tags: algebra , inequalities

Let $a,b$ and $c$ be positive real numbers such that $a^5+b^5+c^5=ab^2+bc^2+ca^2$. Prove the inequality: $$\frac{a^2+b^2}{b}+\frac{b^2+c^2}{c}+\frac{c^2+a^2}{a}\geq 2(ab+bc+ca).$$

1993 Flanders Math Olympiad, 3

Tags: inequalities

For $a,b,c>0$ we have: \[ -1 < \left(\dfrac{a-b}{a+b}\right)^{1993} + \left(\dfrac{b-c}{b+c}\right)^{1993} + \left(\dfrac{c-a}{c+a}\right)^{1993} < 1 \]

2004 Regional Olympiad - Republic of Srpska, 2

Tags: inequalities

The positive real numbers $x,y,z$ satisfy $x+y+z=1$. Show that \[\sqrt{3xyz}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{1-x}+\frac{1}{1-y}+\frac{1}{1-z}\right)\geq4+ \frac{4xyz}{(1-x)(1-y)(1-z)}.\]

1991 China Team Selection Test, 3

Tags: graph theory , tetrahedron , 3d geometry , combinatorics , geometry , inequalities

All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called [i]excentric[/i]. The[i] excentricity [/i]of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Prove that there exist two vertices $B$ and $C$ such that $S_B + S_C \leq 4$.

2019 Peru Cono Sur TST, P1

Tags: number theory , fraction , inequalities

Find all a positive integers $a$ and $b$, such that $$\frac{a^b+b^a}{a^a-b^b}$$ is an integer

2024-IMOC, A6

Tags: calculus , inequalities

Given positive real $a,b,c$ satisfying \[\frac{1}{\sqrt{a+1}}+\frac{3}{\sqrt{b+3}}+\frac{3}{\sqrt{c+3}}=\frac72\] Prove that $abc\leq 3$.\\ I was asked to propose a inequality for the condition of $abc<3$ inequality since <3 looks like a heart shape, then I construct a equality and with the help of wolfram, I gave the birth of this bad-looking inequality, I’m glad to see any method besides calculus.

MathLinks Contest 7th, 4.1

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

Let $ A,B,C,D,E$ be five distinct points, such that no three of them lie on the same line. Prove that \[ AB\plus{}BC\plus{}CA \plus{} DE < AD \plus{} AE \plus{} BD\plus{}BE \plus{} CD\plus{}CE .\]

2015 Saudi Arabia JBMO TST, 2

Tags: algebra , inequalities

Let $a,b,c$ be positive real numbers. Prove that $$\frac{a}{\sqrt{(2a+b)(2a+c)}} +\frac{b}{\sqrt{(2b+c)(2b+a)}} +\frac{c}{\sqrt{(2c+a)(2c+b)}} \le 1 $$

2003 Turkey Team Selection Test, 2

Tags: limit , geometry , inequalities

Let $K$ be the intersection of the diagonals of a convex quadrilateral $ABCD$. Let $L\in [AD]$, $M \in [AC]$, $N \in [BC]$ such that $KL\parallel AB$, $LM\parallel DC$, $MN\parallel AB$. Show that \[\dfrac{Area(KLMN)}{Area(ABCD)} < \dfrac {8}{27}.\]

2012 Mathcenter Contest + Longlist, 2 sl9

Tags: algebra , inequalities

Let $a,b,c \in \mathbb{R}^+$ where $a^2+b^2+c^2=1$. Find the minimum value of . $$a+b+c+\frac{3}{ab+bc+ca}$$ [i](PP-nine)[/i]

1987 Czech and Slovak Olympiad III A, 4

Tags: inequality , prime number , inequalities , factorial , algebra , number theory

Given an integer $n\ge3$ consider positive integers $x_1,\ldots,x_n$ such that $x_1<x_2<\cdots<x_n<2x_1$. If $p$ is a prime and $r$ is a positive integer such that $p^r$ divides the product $x_1\cdots x_n$, prove that $$\frac{x_1\cdots x_n}{p^r}>n!.$$

2016 South East Mathematical Olympiad, 2

Tags: inequalities

Let $n$ be positive integer,$x_1,x_2,\cdots,x_n$ be positive real numbers such that $x_1x_2\cdots x_n=1 $ . Prove that$$\sum\limits_{i = 1}^{n}x_i\sqrt{x^2_1+x^2_2+\cdots x^2_i}\ge\frac{n+1}{2}\sqrt{n}$$

1998 Irish Math Olympiad, 2

Tags: inequalities

Prove that if $ a,b,c$ are positive real numbers, then: $ \frac{9}{a\plus{}b\plus{}c} \le 2 \left( \frac{1}{a\plus{}b}\plus{}\frac{1}{b\plus{}c}\plus{}\frac{1}{c\plus{}a} \right) \le \frac{1}{a}\plus{}\frac{1}{b}\plus{}\frac{1}{c}.$

2017 South East Mathematical Olympiad, 5

Tags: inequalities

Let $a, b, c$ be real numbers, $a \neq 0$. If the equation $2ax^2 + bx + c = 0$ has real root on the interval $[-1, 1]$. Prove that $$\min \{c, a + c + 1\} \leq \max \{|b - a + 1|, |b + a - 1|\},$$ and determine the necessary and sufficient conditions of $a, b, c$ for the equality case to be achieved.

2018 Benelux, 1

Tags: inequalities , algebra

(a) Determine the minimal value of $\displaystyle\left(x+\dfrac{1}{y}\right)\left(x+\dfrac{1}{y}-2018\right)+\left(y+\dfrac{1}{x}\right)\left(y+\dfrac{1}{x}-2018\right), $ where $x$ and $y$ vary over the positive reals. (b) Determine the minimal value of $\displaystyle\left(x+\dfrac{1}{y}\right)\left(x+\dfrac{1}{y}+2018\right)+\left(y+\dfrac{1}{x}\right)\left(y+\dfrac{1}{x}+2018\right), $ where $x$ and $y$ vary over the positive reals.

2009 Jozsef Wildt International Math Competition, W. 25

Tags: geometry , inequalities , cyclic quadrilateral

Let $ABCD$ be a quadrilateral in which $\widehat{A}=\widehat{C}=90^{\circ}$. Prove that $$\frac{1}{BD}(AB+BC+CD+DA)+BD^2\left (\frac{1}{AB\cdot AD}+\frac{1}{CB\cdot CD}\right )\geq 2\left (2+\sqrt{2}\right )$$

1997 Akdeniz University MO, 2

Tags: easy inequality , inequalities

If $x$ and $y$ are positive reals, prove that $$x^2\sqrt{\frac{x}{y}}+y^2\sqrt{\frac{y}{x}} \geq x^2+y^2$$

2010 Germany Team Selection Test, 2

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]

2013 Vietnam Team Selection Test, 4

Tags: inequalities , inequality

Find the greatest positive integer $k$ such that the following inequality holds for all $a,b,c\in\mathbb{R}^+$ satisfying $abc=1$ \[ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{k}{a+b+c+1}\geqslant 3+\frac{k}{4} \]