Olympiad problems

1980 IMO Shortlist, 2

Define the numbers $a_0, a_1, \ldots, a_n$ in the following way: \[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \] Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]

2020 IMC, 5

Tags: calculus , differential equation , inequality , function

Find all twice continuously differentiable functions $f: \mathbb{R} \to (0, \infty)$ satisfying $f''(x)f(x) \ge 2f'(x)^2.$

2008 IMO Shortlist, 4

Tags: function , algebra , functional equation , inequality

For an integer $ m$, denote by $ t(m)$ the unique number in $ \{1, 2, 3\}$ such that $ m \plus{} t(m)$ is a multiple of $ 3$. A function $ f: \mathbb{Z}\to\mathbb{Z}$ satisfies $ f( \minus{} 1) \equal{} 0$, $ f(0) \equal{} 1$, $ f(1) \equal{} \minus{} 1$ and $ f\left(2^{n} \plus{} m\right) \equal{} f\left(2^n \minus{} t(m)\right) \minus{} f(m)$ for all integers $ m$, $ n\ge 0$ with $ 2^n > m$. Prove that $ f(3p)\ge 0$ holds for all integers $ p\ge 0$. [i]Proposed by Gerhard Woeginger, Austria[/i]

2021 IMC, 7

Tags: complex analysis , polynomial , inequality

Let $D \subseteq \mathbb{C}$ be an open set containing the closed unit disk $\{z : |z| \leq 1\}$. Let $f : D \rightarrow \mathbb{C}$ be a holomorphic function, and let $p(z)$ be a monic polynomial. Prove that $$ |f(0)| \leq \max_{|z|=1} |f(z)p(z)| $$

2024 Junior Balkan Team Selection Tests - Romania, P1

Tags: inequality , algebra

For positive real numbers $x,y,z$ with $xy+yz+zx=1$, prove that $$\frac{2}{xyz}+9xyz \geq 7(x+y+z)$$

2018 Korea National Olympiad, 4

Tags: algebra , inequality , inequalities , sequence

Find all real values of $K$ which satisfies the following. Let there be a sequence of real numbers $\{a_n\}$ which satisfies the following for all positive integers $n$. (i). $0 < a_n < n^K$. (ii). $a_1 + a_2 + \cdots + a_n < \sqrt{n}$. Then, there exists a positive integer $N$ such that for all integers $n>N$, $$a^{2018}_1 + a^{2018}_2 + \cdots +a^{2018}_n < \frac{n}{2018}$$

2021 Germany Team Selection Test, 3

Tags: algebra , inequality , 4-variable inequality , inequalities

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

1973 IMO, 3

Tags: sequence , algebra , inequality , construction , geometric sequence

Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that: [i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$ [i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$ [i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$

2021-IMOC, A7

Tags: algebra , inequality , inequalities

For any positive reals $a,b,c,d$ that satisfy $a^2 + b^2 + c^2 + d^2 = 4,$ show that $$\frac{a^3}{a+b} + \frac{b^3}{b+c} + \frac{c^3}{c+d} + \frac{d^3}{d+a} + 4abcd \leq 6.$$

2019 ELMO Shortlist, A1

Tags: algebra , inequality , inequalities

Let $a$, $b$, $c$ be positive reals such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show that $$a^abc+b^bca+c^cab\ge 27bc+27ca+27ab.$$ [i]Proposed by Milan Haiman[/i]

1992 IMO Shortlist, 18

Tags: floor function , algebra , sequence , fibonacci , fibonacci sequence , inequality

Let $ \lfloor x \rfloor$ denote the greatest integer less than or equal to $ x.$ Pick any $ x_1$ in $ [0, 1)$ and define the sequence $ x_1, x_2, x_3, \ldots$ by $ x_{n\plus{}1} \equal{} 0$ if $ x_n \equal{} 0$ and $ x_{n\plus{}1} \equal{} \frac{1}{x_n} \minus{} \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that \[ x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n < \frac{F_1}{F_2} \plus{} \frac{F_2}{F_3} \plus{} \ldots \plus{} \frac{F_n}{F_{n\plus{}1}},\] where $ F_1 \equal{} F_2 \equal{} 1$ and $ F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$ for $ n \geq 1.$

2014 Contests, A2

Tags: algebra , inequality

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]

2007 Junior Macedonian Mathematical Olympiad, 4

Tags: algebra , inequality

The numbers $a_{1}, a_{2}, ..., a_{20}$ satisfy the following conditions: $a_{1} \ge a_{2} \ge ... \ge a_{20} \ge 0$ $a_{1} + a_{2} = 20$ $a_{3} + a_{4} + ... + a_{20} \le 20$ . What is maximum value of the expression: $a_{1}^2 + a_{2}^2 + ... + a_{20}^2$ ? For which values of $a_{1}, a_{2}, ..., a_{20}$ is the maximum value achieved?

2021 European Mathematical Cup, 1

Tags: inequality , algebra

We say that a quadruple of nonnegative real numbers $(a,b,c,d)$ is [i]balanced [/i]if $$a+b+c+d=a^2+b^2+c^2+d^2.$$ Find all positive real numbers $x$ such that $$(x-a)(x-b)(x-c)(x-d)\geq 0$$ for every balanced quadruple $(a,b,c,d)$. \\ \\ (Ivan Novak)

1982 IMO Shortlist, 15

Tags: inequality , calculus , geometric progression

Show that \[ \frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}\] holds for every $1 \neq s > 0$ real and $0 < a \leq 1$ rational.

2017 Vietnamese Southern Summer School contest, Problem 1

Tags: inequality , algebra , inequalities

Let $x,y,z$ be the non-negative real numbers satisfying $xy+yz+zx\leq 1$. Prove that: $$1-xy-yz-zx\leq (6-2\sqrt{6})(1-\min\{x,y,z\}).$$

1979 IMO Shortlist, 13

Tags: trigonometry , inequality , geometric inequality , trigonometric identities , inequalities

Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$

2017 Junior Balkan Team Selection Tests - Romania, 2

Tags: algebra , inequality

a) Find : $A=\{(a,b,c) \in \mathbb{R}^{3} | a+b+c=3 , (6a+b^2+c^2)(6b+c^2+a^2)(6c+a^2+b^2) \neq 0\}$ b) Prove that for any $(a,b,c) \in A$ next inequality hold : \begin{align*} \frac{a}{6a+b^2+c^2}+\frac{b}{6b+c^2+a^2}+\frac{c}{6c+a^2+b^2} \le \frac{3}{8} \end{align*}

1994 All-Russian Olympiad, 5

Tags: least common multiple , inequality , number theory

Prove that, for any natural numbers $k,m,n$: $[k,m] \cdot [m,n] \cdot [n,k] \ge [k,m,n]^2$

2005 Korea Junior Math Olympiad, 7

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

If positive reals $ x_1,x_2,\cdots,x_n $ satisfy $\sum_{i=1}^{n}x_i=1.$ Prove that$$\sum_{i=1}^{n}\frac{1}{1+\sum_{j=1}^{i}x_j}<\sqrt{\frac{2}{3}\sum_{i=1}^{n}\frac{1}{x_i}} $$

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]

2000 Moldova National Olympiad, Problem 2

Tags: inequality , inequalities

Show that if real numbers $x<1<y$ satisfy the inequality $$2\log x+\log(1-x)\ge3\log y+\log(y-1),$$then $x^3+y^3<2$.

1971 IMO Shortlist, 17

Tags: inequality , 4-variable inequality , four variables

Prove the inequality \[ \frac{a_1+ a_3}{a_1 + a_2} + \frac{a_2 + a_4}{a_2 + a_3} + \frac{a_3 + a_1}{a_3 + a_4} + \frac{a_4 + a_2}{a_4 + a_1} \geq 4, \] where $a_i > 0, i = 1, 2, 3, 4.$

2017 Bosnia and Herzegovina EGMO TST, 4

Tags: triplets , equality , inequality , algebra

Let $a$, $b$, $c$, $d$ and $e$ be distinct positive real numbers such that $a^2+b^2+c^2+d^2+e^2=ab+ac+ad+ae+bc+bd+be+cd+ce+de$ $a)$ Prove that among these $5$ numbers there exists triplet such that they cannot be sides of a triangle $b)$ Prove that, for $a)$, there exists at least $6$ different triplets

2010 Sharygin Geometry Olympiad, 3

Tags: geometry , convex polygon , inequality , geometric inequality , diagonal

All sides of a convex polygon were decreased in such a way that they formed a new convex polygon. Is it possible that all diagonals were increased?