Olympiad problems

2021 Tuymaada Olympiad, 3

Tags: inequalities , algebra

Positive real numbers $a_1, \dots, a_k, b_1, \dots, b_k$ are given. Let $A = \sum_{i = 1}^k a_i, B = \sum_{i = 1}^k b_i$. Prove the inequality \[ \left( \sum_{i = 1}^k \frac{a_i b_i}{a_i B + b_i A} - 1 \right)^2 \ge \sum_{i = 1}^k \frac{a_i^2}{a_i B + b_i A} \cdot \sum_{i = 1}^k \frac{b_i^2}{a_i B + b_i A}. \]

2009 Belarus Team Selection Test, 1

Tags: inequalities , algebra

Prove that any positive real numbers a,b,c satisfy the inequlaity $$\frac{1}{(a+b)b}+\frac{1}{(b+c)c}+\frac{1}{(c+a)a}\ge \frac{9}{2(ab+bc+ca)}$$ I.Voronovich

2021 Science ON all problems, 4

Tags: inequality , inequalities , algebra

Consider positive real numbers $x,y,z$. Prove the inequality $$\frac 1x+\frac 1y+\frac 1z+\frac{9}{x+y+z}\ge 3\left (\left (\frac{1}{2x+y}+\frac{1}{x+2y}\right )+\left (\frac{1}{2y+z}+\frac{1}{y+2z}\right )+\left (\frac{1}{2z+x}+\frac{1}{x+2z}\right )\right ).$$ [i] (Vlad Robu \& Sergiu Novac)[/i]

2000 VJIMC, Problem 4

Tags: real analysis , inequalities , measure theory

Let $\mathcal B$ be a family of open balls in $\mathbb R^n$ and $c<\lambda\left(\bigcup\mathcal B\right)$ where $\lambda$ is the $n$-dimensional Lebesgue measure. Show that there exists a finite family of pairwise disjoint balls $\{U_i\}^k_{i=1}\subseteq\mathcal B$ such that $$\sum_{j=1}^k\lambda(U_j)>\frac c{3^n}.$$

1971 IMO Longlists, 38

Tags: ratio , inequalities , analytic geometry , circumcircle , geometry

Let $A,B,C$ be three points with integer coordinates in the plane and $K$ a circle with radius $R$ passing through $A,B,C$. Show that $AB\cdot BC\cdot CA\ge 2R$, and if the centre of $K$ is in the origin of the coordinates, show that $AB\cdot BC\cdot CA\ge 4R$.

2011 Singapore Junior Math Olympiad, 1

Tags: inequalities , algebra

Suppose $a,b,c,d> 0$ and $x = \sqrt{a^2+b^2}, y = \sqrt{c^2+d^2}$. Prove that $xy \ge ac + bd$.

2002 Hungary-Israel Binational, 2

Tags: trigonometry , incenter , inequalities , geometry

Let $A', B' , C'$ be the projections of a point $M$ inside a triangle $ABC$ onto the sides $BC, CA, AB$, respectively. Deﬁne $p(M ) = \frac{MA'\cdot MB'\cdot MC'}{MA \cdot MB \cdot MC}$ . Find the position of point $M$ that maximizes $p(M )$.

2016 Bulgaria National Olympiad, Problem 3

Tags: algebra , inequalities

For $a,b,c,d>0$ prove that $$\frac {a+\sqrt{ab}+\sqrt[3]{abc}+\sqrt[4]{abcd}}{4} \leq \sqrt[4]{a.\frac{a+b}{2}.\frac{a+b+c}{3}.\frac{a+b+c+d}{4}}$$

2013 Bosnia and Herzegovina Junior BMO TST, 2

Tags: inequality , positive real , algebra , inequalities

Let $a$, $b$ and $c$ be positive real numbers such that $a^2+b^2+c^2=3$. Prove the following inequality: $\frac{a}{3c(a^2-ab+b^2)} + \frac{b}{3a(b^2-bc+c^2)} + \frac{c}{3b(c^2-ca+a^2)} \leq \frac{1}{abc}$

1988 All Soviet Union Mathematical Olympiad, 472

Tags: trigonometry , inequalities

$A, B, C$ are the angles of a triangle. Show that $2\frac{\sin A}{A} + 2\frac{\sin B}{B} + 2\frac{\sin C}{C} \le \left(\frac{1}{B} + \frac{1}{C}\right) \sin A + \left(\frac{1}{C} + \frac{1}{A}\right) \sin B + \left(\frac{1}{A} + \frac{1}{B}\right) \sin C$

2003 Indonesia MO, 5

Tags: inequalities

For any real numbers $a,b,c$, prove that \[ 5a^2 + 5b^2 + 5c^2 \ge 4ab + 4ac + 4bc \] and determine when equality occurs.

2016 Korea Junior Math Olympiad, 7

Tags: algebra , inequalities

positive integers $a_1, a_2, . . . , a_9$ satisfying $a_1+a_2+ . . . +a_9 =90$ find maximum of $$\frac{1^{a_1} \cdot 2^{a_2} \cdot . . . \cdot 9^{a_9}}{a_1! \cdot a_2! \cdot . . . \cdot a_9!}$$ [hide=mention] I was really shocked because there are no inequality problems at KJMO and the test difficulty even more lower...[/hide]

2021 South East Mathematical Olympiad, 7

Tags: inequalities , algebra

Let $a,b,c$ be pairwise distinct positive real, Prove that$$\dfrac{ab+bc+ca}{(a+b)(b+c)(c+a)}<\dfrac17(\dfrac{1}{|a-b|}+\dfrac{1}{|b-c|}+\dfrac{1}{|c-a|}).$$

1992 Turkey Team Selection Test, 2

Tags: induction , inequalities , combinatorics

There are $n$ boxes which is numbere from $1$ to $n$. The box with number $1$ is open, and the others are closed. There are $m$ identical balls ($m\geq n$). One of the balls is put into the open box, then we open the box with number $2$. Now, we put another ball to one of two open boxes, then we open the box with number $3$. Go on until the last box will be open. After that the remaining balls will be randomly put into the boxes. In how many ways this arrangement can be done?

2017-IMOC, A1

Tags: inequalities , algebra

Prove that for all $a,b>0$ with $a+b=2$, we have $$\left(a^n+1\right)\left(b^n+1\right)\ge4$$ for all $n\in\mathbb N_{\ge2}$.

2016 South East Mathematical Olympiad, 5

Tags: complex number , inequalities , algebra , natural logarithm , china southeast mo

Let a constant $\alpha$ as $0<\alpha<1$, prove that: $(1)$ There exist a constant $C(\alpha)$ which is only depend on $\alpha$ such that for every $x\ge 0$, $\ln(1+x)\le C(\alpha)x^\alpha$. $(2)$ For every two complex numbers $z_1,z_2$, $|\ln|\frac{z_1}{z_2}||\le C(\alpha)\left(|\frac{z_1-z_2}{z_2}|^\alpha+|\frac{z_2-z_1}{z_1}|^\alpha\right)$.

2023 China Girls Math Olympiad, 3

Tags: algebra , inequality , inequalities

Let $a,b,c,d \in [0,1] .$ Prove that$$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+d}+\frac{1}{1+d+a}\leq \frac{4}{1+2\sqrt[4]{abcd}}$$

2014 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: partition , number theory , inequalities , set

Let $n \in N$ such that $1 + 2 + ... + n$ is divisible by $3$. Integers $a_1\ge a_2\ge a_3\ge 2$ have sum $n$ and they satisfy $1 + 2 + ... + a_1\le \frac{1}{3}( 1 + 2 + ... + n ) $ and $1 + 2 + ... + (a_1+ a_2) \le \frac{2}{3}( 1 + 2 + ... + n )$. Prove that there is a partition of $\{ 1 , 2 , ... , n\}$ in three subsets $A_1, A_2, A_3$ with cardinals $| A_i| = a_i, i = 1 , 2 , 3$, and with equal sums of their elements .

1996 Turkey MO (2nd round), 2

Tags: inequalities , geometry

Let $ABCD$ be a square of side length 2, and let $M$ and $N$ be points on the sides $AB$ and $CD$ respectively. The lines $CM$ and $BN$ meet at $P$, while the lines $AN$ and $DM$ meet at $Q$. Prove that $\left| PQ \right|\ge 1$.

PEN A Problems, 26

Tags: floor function , inequalities , induction , ratio , divisibility theory

Let $m$ and $n$ be arbitrary non-negative integers. Prove that \[\frac{(2m)!(2n)!}{m! n!(m+n)!}\] is an integer.

2010 South East Mathematical Olympiad, 1

Tags: inequalities , geometry

$ABC$ is a triangle with a right angle at $C$. $M_1$ and $M_2$ are two arbitrary points inside $ABC$, and $M$ is the midpoint of $M_1M_2$. The extensions of $BM_1,BM$ and $BM_2$ intersect $AC$ at $N_1,N$ and $N_2$ respectively. Prove that $\frac{M_1N_1}{BM_1}+\frac{M_2N_2}{BM_2}\geq 2\frac{MN}{BM}$

2017 Spain Mathematical Olympiad, 5

Tags: algebra , inequalities , maximum

Let $a,b,c$ be positive real numbers so that $a+b+c = \frac{1}{\sqrt{3}}$. Find the maximum value of $$27abc+a\sqrt{a^2+2bc}+b\sqrt{b^2+2ca}+c\sqrt{c^2+2ab}.$$

2007 Vietnam Team Selection Test, 3

Tags: trigonometry , search , geometry , inequalities

Given a triangle $ABC$. Find the minimum of \[\frac{\cos^{2}\frac{A}{2}\cos^{2}\frac{B}{2}}{\cos^{2}\frac{C}{2}}+\frac{\cos^{2}\frac{B}{2}\cos^{2}\frac{C}{2}}{\cos^{2}\frac{A}{2}}+\frac{\cos^{2}\frac{C}{2}\cos^{2}\frac{A}{2}}{\cos^{2}\frac{B}{2}}. \]

2025 Kyiv City MO Round 1, Problem 5

Tags: inequalities , algebra

Real numbers $ a, b, c $ satisfy the following conditions: \[ 1000 < |a| < 2000, \quad 1000 < |b| < 2000, \quad 1000 < |c| < 2000, \] and \[ \frac{ab^2}{a+b} + \frac{bc^2}{b+c} + \frac{ca^2}{c+a} = 0. \] What are the possible values of the expression \[ \frac{a}{b} + \frac{b}{c} + \frac{c}{a}? \] [i]Proposed by Vadym Solomka[/i]

MathLinks Contest 6th, 6.1

Tags: algebra , inequalities

Let $p > 1$ and let $a, b, c, d$ be positive numbers such that $$(a + b + c + d) \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)= 16p^2.$$ Find all values of the ratio $ R =\frac{\max \{a, b, c, d\}}{\min \{a, b, c, d\}}$ (depending on the parameter $p$)