Olympiad problems

2024 Korea Junior Math Olympiad (First Round), 19.

For all integers $ {a}_{0},{a}_{1}, \cdot\cdot\cdot {a}_{100}$, find the maximum of ${a}_{5}-2{a}_{40}+3{a}_{60}-4{a}_{95} $ $\bigstar$ 1) ${a}_{0}={a}_{100}=0$ 2) for all $i=0,1,\cdot \cdot \cdot 99, $ $|{a}_{i+1}-{a}_{i}|\le1$ 3) $ {a}_{10}={a}_{90} $

2016 Azerbaijan Junior Mathematical Olympiad, 6

Tags: inequality , algebra , inequalities

For all reals $x,y,z$ prove that $$\sqrt {x^2+\frac {1}{y^2}}+ \sqrt {y^2+\frac {1}{z^2}}+ \sqrt {z^2+\frac {1}{x^2}}\geq 3\sqrt {2}. $$

2022 JBMO TST - Turkey, 6

Tags: real number , inequality

Let $c$ be a real number. If the inequality $$f(c)\cdot f(-c)\ge f(a)$$ holds for all $f(x)=x^2-2ax+b$ where $a$ and $b$ are arbitrary real numbers, find all possible values of $c$.

2001 All-Russian Olympiad, 2

Tags: polynomial , function , inequality , algebra

The two polynomials $(x) =x^4+ax^3+bx^2+cx+d$ and $Q(x) = x^2+px+q$ take negative values on an interval $I$ of length greater than $2$, and nonnegative values outside of $I$. Prove that there exists $x_0 \in \mathbb R$ such that $P(x_0) < Q(x_0)$.

2020 Tuymaada Olympiad, 2

Tags: inequality , n-variable inequality , inequalities

Given positive real numbers $a_1, a_2, \dots, a_n$. Let \[ m = \min \left( a_1 + \frac{1}{a_2}, a_2 + \frac{1}{a_3}, \dots, a_{n - 1} + \frac{1}{a_n} , a_n + \frac{1}{a_1} \right). \] Prove the inequality \[ \sqrt[n]{a_1 a_2 \dots a_n} + \frac{1}{\sqrt[n]{a_1 a_2 \dots a_n}} \ge m. \]

2015 Indonesia MO Shortlist, A3

Tags: inequality , inequalities

Let $a,b,c$ positive reals such that $a^2+b^2+c^2=1$. Prove that $$\frac{a+b}{\sqrt{ab+1}}+\frac{b+c}{\sqrt{bc+1}}+\frac{c+a}{\sqrt{ac+1}}\le 3$$

2022 JBMO Shortlist, A3

Tags: inequality , algebra

Let $a, b,$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove the following inequality $$a \sqrt[3]{\frac{b}{a}} + b \sqrt[3]{\frac{c}{b}} + c \sqrt[3]{\frac{a}{c}} \le ab + bc + ca + \frac{2}{3}.$$ Proposed by [i]Anastasija Trajanova, Macedonia[/i]

2011 Bosnia and Herzegovina Junior BMO TST, 2

Tags: nonnegative , inequality , inequalities

Prove inequality, with $a$ and $b$ nonnegative real numbers: $\frac{a+b}{1+a+b}\leq \frac{a}{1+a} + \frac{b}{1+b} \leq \frac{2(a+b)}{2+a+b}$

2012 Kyiv Mathematical Festival, 2

Tags: 3-variable inequality , inequality , inequalities , algebra

Positive numbers $x, y, z$ satisfy $x + y + z \le 1$. Prove that $\big( \frac{1}{x}-1\big) \big( \frac{1}{y}-1\big)\big( \frac{1}{z}-1\big) \ge 8$.

1975 IMO Shortlist, 5

Tags: algebra , digit , series summation , inequality , decimal representation

Let $M$ be the set of all positive integers that do not contain the digit $9$ (base $10$). If $x_1, \ldots , x_n$ are arbitrary but distinct elements in $M$, prove that \[\sum_{j=1}^n \frac{1}{x_j} < 80 .\]

2006 Petru Moroșan-Trident, 2

Tags: inequalities , inequality , induction

Let be an increasing, infinite sequence of natural numbers $ \left( a_n \right)_{n\ge 1} . $ [b]a)[/b] Prove that if $ a_n=n, $ for any natural numbers $ n, $ then $$ -2+2\sqrt{1+n} <\frac{1}{\sqrt{a_1}} +\frac{1}{\sqrt{a_2}} +\cdots +\frac{1}{\sqrt{a_n}} <2\sqrt n , $$ for any natural numbers $ n. $ [b]b)[/b] Disprove the converse of [b]a).[/b] [i]Vasile Radu[/i]

2022 Bulgarian Spring Math Competition, Problem 8.3

Tags: inequality , algebra , inequalities

Given the inequalities: $a)$ $\left(\frac{2a}{b+c}\right)^2+\left(\frac{2b}{a+c}\right)^2+\left(\frac{2c}{a+b}\right)^2\geq \frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ $b)$ $\left(\frac{a+b}{c}\right)^2+\left(\frac{b+c}{a}\right)^2+\left(\frac{c+a}{b}\right)^2\geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+9$ For each of them either prove that it holds for all positive real numbers $a$, $b$, $c$ or present a counterexample $(a,b,c)$ which doesn't satisfy the inequality.

2009 India National Olympiad, 6

Tags: algebra , quadratic , inequality , polynomial

Let $ a,b,c$ be positive real numbers such that $ a^3 \plus{} b^3 \equal{} c^3$.Prove that: $ a^2 \plus{} b^2 \minus{} c^2 > 6(c \minus{} a)(c \minus{} b)$.

2014 Balkan MO Shortlist, A2

Tags: inequality , algebra

Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds. [i]UK - David Monk[/i]

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]

2019 Bulgaria National Olympiad, 1

Tags: inequality , inequalities

Let $f(x)=x^2+bx+1,$ where $b$ is a real number. Find the number of integer solutions to the inequality $f(f(x)+x)<0.$

1967 IMO Shortlist, 2

Tags: inequality , trigonometric inequality , minimum , trigonometry

Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: \[ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. \]

1982 IMO, 3

Tags: minimization , geometric sequence , inequality , geometric series , limit

Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$. [b]a)[/b] Prove that for every such sequence there is an $n\ge1$ such that: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. \] [b]b)[/b] Find such a sequence such that for all $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]

2022 Turkey Junior National Olympiad, 1

Tags: inequality , algebra

$x, y, z$ are positive reals such that $x \leq 1$. Prove that $$xy+y+2z \geq 4 \sqrt{xyz}$$

1978 IMO Shortlist, 6

Tags: algebra , rearrangement inequality , inequality , summation

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

1967 IMO Longlists, 23

Tags: geometric inequality , inequality , 3d geometry , algebra , vector

Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality \[af^2 + bfg +cg^2 \geq 0\] holds if and only if the following conditions are fulfilled: \[a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.\]

1983 IMO, 3

Tags: triangle inequality , algebra , rearrangement inequality , inequality , polynomial

Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that \[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0. \] Determine when equality occurs.

2012 Israel National Olympiad, 3

Tags: inequality , symmetric inequality , algebra , inequalities

Let $a,b,c$ be real numbers such that $a^3(b+c)+b^3(a+c)+c^3(a+b)=0$. Prove that $ab+bc+ca\leq0$.

2021 Federal Competition For Advanced Students, P2, 1

Tags: inequality , algebra

Let $a, b$ and $c$ be pairwise different natural numbers. Prove $\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c$. When does equality holds? (Karl Czakler)

1965 Vietnam National Olympiad, 3

Tags: inequality , algebra

1) Two nonnegative real numbers $x, y$ have constant sum $a$. Find the minimum value of $x^m + y^m$, where m is a given positive integer. 2) Let $m, n$ be positive integers and $k$ a positive real number. Consider nonnegative real numbers $x_1, x_2, . . . , x_n$ having constant sum $k$. Prove that the minimum value of the quantity $x^m_1+ ... + x^m_n$ occurs when $x_1 = x_2 = ... = x_n$.