Olympiad problems

1999 Ukraine Team Selection Test, 10

For a natural number $n$, let $w(n)$ denote the number of (positive) prime divisors of $n$. Find the smallest positive integer $k$ such that $2^{w(n)} \le k \sqrt[4]{ n}$ for each $n \in N$.

2015 Azerbaijan National Olympiad, 2

Tags: inequalities , geometry

Let $a,b$ and $c$ be the length of sides of a triangle.Then prove that $S\le\frac{a^2+b^2+c^2}{6}$ where $S$ is the area of triangle.

I Soros Olympiad 1994-95 (Rus + Ukr), 10.8

Tags: trigonometry , inequalities , algebra

Find all $x$ for which the inequality holds $$\sqrt{7+8x-16x^2} \ge 2^{\cos^2 \pi x}+2^{\sin ^2 \pi x}$$

1962 All Russian Mathematical Olympiad, 023

Tags: inequalities , geometry , area , maximum

What maximal area can have a triangle if its sides $a,b,c$ satisfy inequality $0\le a\le 1\le b\le 2\le c\le 3$ ?

2013 Today's Calculation Of Integral, 863

Tags: inequalities , derivative , trigonometry , limit , calculus , integration , calculus computations

For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$ (1) Find $\lim_{t\rightarrow 0} F(t).$ (2) Find the range of $t$ such that $F(t)\geq 1.$

2012 Putnam, 4

Tags: calculus , inequalities , integration , induction , limit , logarithm

Suppose that $a_0=1$ and that $a_{n+1}=a_n+e^{-a_n}$ for $n=0,1,2,\dots.$ Does $a_n-\log n$ have a finite limit as $n\to\infty?$ (Here $\log n=\log_en=\ln n.$)

2019 Tuymaada Olympiad, 4

Tags: inequalities , combinatorics , algebra , operation

A calculator can square a number or add $1$ to it. It cannot add $1$ two times in a row. By several operations it transformed a number $x$ into a number $S > x^n + 1$ ($x, n,S$ are positive integers). Prove that $S > x^n + x - 1$.

2006 Hanoi Open Mathematics Competitions, 9

Tags: minimum , algebra , inequalities

What is the smallest possible value of $x^2 + y^2 - x -y - xy$?

2019 Swedish Mathematical Competition, 5

Tags: functional , algebra , functional inequality , functional equation , inequalities

Let $f$ be a function that is defined for all positive integers and whose values are positive integers. For $f$ it also holds that $f (n + 1)> f (n)$ and $f (f (n)) = 3n$, for each positive integer $n$. Calculate $f (2019)$.

2012 China National Olympiad, 1

Tags: inequalities

Let $f(x)=(x + a)(x + b)$ where $a,b>0$. For any reals $x_1,x_2,\ldots ,x_n\geqslant 0$ satisfying $x_1+x_2+\ldots +x_n =1$, find the maximum of $F=\sum\limits_{1 \leqslant i < j \leqslant n} {\min \left\{ {f({x_i}),f({x_j})} \right\}} $.

1993 China National Olympiad, 2

Tags: inequalities

Given a natural number $k$ and a real number $a (a>0)$, find the maximal value of $a^{k_1}+a^{k_2}+\cdots +a^{k_r}$, where $k_1+k_2+\cdots +k_r=k$ ($k_i\in \mathbb{N} ,1\le r \le k$).

2009 Moldova Team Selection Test, 2

Tags: calculus , derivative , inequalities , rearrangement inequality , function , logarithm

[color=darkred]Let $ m,n\in \mathbb{N}$, $ n\ge 2$ and numbers $ a_i > 0$, $ i \equal{} \overline{1,n}$, such that $ \sum a_i \equal{} 1$. Prove that $ \small{\dfrac{a_1^{2 \minus{} m} \plus{} a_2 \plus{} ... \plus{} a_{n \minus{} 1}}{1 \minus{} a_1} \plus{} \dfrac{a_2^{2 \minus{} m} \plus{} a_3 \plus{} ... \plus{} a_n}{1 \minus{} a_1} \plus{} ... \plus{} \dfrac{a_n^{2 \minus{} m} \plus{} a_1 \plus{} ... \plus{} a_{n \minus{} 2}}{1 \minus{} a_1}\ge n \plus{} \dfrac{n^m \minus{} n}{n \minus{} 1}}$[/color]

2014 USAJMO, 4

Tags: inequalities , function , pigeonhole principle

Let $b\geq 2$ be an integer, and let $s_b(n)$ denote the sum of the digits of $n$ when it is written in base $b$. Show that there are infinitely many positive integers that cannot be represented in the form $n+s_b(n)$, where $n$ is a positive integer.

2019 Romania Team Selection Test, 1

Tags: algebra , inequalities

Determine the largest value the expression $$ \sum_{1\le i<j\le 4} \left( x_i+x_j \right)\sqrt{x_ix_j} $$ may achieve, as $ x_1,x_2,x_3,x_4 $ run through the non-negative real numbers, and add up to $ 1. $ Find also the specific values of this numbers that make the above sum achieve the asked maximum.

1985 Iran MO (2nd round), 7

Tags: inequalities

Let $a,b$ and $c$ be real numbers with $b,c >0.$ Prove that if $ a<b \ ( a>b),$ then \[\frac{a+c}{b+c} > \frac ab \qquad ( \frac{a+c}{b+c} < \frac ab) \] And then prove that $\frac{a+c}{b+c}$ is between $1$ and $\frac ab.$

2019 Jozsef Wildt International Math Competition, W. 54

Tags: inequalities , summation

Let $x_1, x_2,\geq , x_n$ be a positive numbers, $k \geq 1$. Then the following inequality is true: $$\left(x_1^k+x_2^k+\cdots +x_n^k\right)^{k+1}\geq \left(x_1^{k+1}+x_2^{k+1}\cdots +x_n^{k+1}\right)^k+2\left(\sum \limits_{1\leq i<j\leq n}x_i^kx_j\right)^k$$

2013 Switzerland - Final Round, 2

Tags: prime , number theory , inequalities

Let $n$ be a natural number and $p_1, ..., p_n$ distinct prime numbers. Show that $$p_1^2 + p_2^2 + ... + p_n^2 > n^3$$

1982 All Soviet Union Mathematical Olympiad, 346

Tags: floor function , inequalities , algebra

Prove that the following inequality holds for all real $a$ and natural $n$: $$|a| \cdot |a-1|\cdot |a-2|\cdot ...\cdot |a-n| \ge \frac{n!F(a)}{2n}$$ $F(a)$ is the distance from $a$ to the closest integer.

2017 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , divisor , difference , inequalities

Let $n$ be a positive integer. For each of the numbers $1, 2,.., n$ we compute the difference between the number of its odd positive divisors and its even positive divisors. Prove that the sum of these differences is at least $0$ and at most $n$.

2024 Korea Junior Math Olympiad, 7

Tags: inequalities , combinatorics

Let $A_k$ be the number of pairs $(a_1, a_2, ..., a_{2k})$ for $k\leq 50$, where $a_1, a_2, ..., a_{2k}$ are all different positive integers that satisfy the following. [b]$\cdot$[/b] $a_1, a_2, ..., a_{2k} \leq 100$ [b]$\cdot$[/b] For an odd number less or equal than $2k-1$, we have $a_i > a_{i+1}$ [b]$\cdot$[/b] For an even number less or equal than $2k-2$, we have $a_i < a_{i+1}$ Prove that $A_1 \leq A_2 \leq \cdots \leq A_{49}$.

1999 National Olympiad First Round, 36

Tags: inequalities , trigonometry

Let $ x_{1} ,x_{2} ,\ldots ,x_{9}$ be real numbers on $ \left[ \minus{} 1,1\right]$. If $ \sum _{i \equal{} 1}^{9}x_{i}^{3} \equal{} 0$, then what is the largest possible value of $ \sum _{i \equal{} 1}^{9}x_{i}$? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac {3}{2} \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ \frac {9}{2} \qquad\textbf{(E)}\ \text{None}$

1964 Swedish Mathematical Competition, 4

Tags: combinatorial geometry , min , max , geometric inequalities , inequalities , geometry

Points $H_1, H_2, ... , H_n$ are arranged in the plane so that each distance $H_iH_j \le 1$. The point $P$ is chosen to minimise $\max (PH_i)$. Find the largest possible value of $\max (PH_i)$ for $n = 3$. Find the best upper bound you can for $n = 4$.

2017 Olympic Revenge, 4

Tags: algebra , inequalities

Let $f:\mathbb{R}_{+}^{*}$$\rightarrow$$\mathbb{R}_{+}^{*}$ such that $f'''(x)>0$ for all $x$ $\in$ $\mathbb{R}_{+}^{*}$. Prove that: $f(a^{2}+b^{2}+c^{2})+2f(ab+bc+ac)$ $\geq$ $f(a^{2}+2bc)+f(b^{2}+2ca)+f(c^{2}+2ab)$, for all $a,b,c$ $\in$ $\mathbb{R}_{+}^{*}$.

1976 IMO Longlists, 6

Tags: inequalities , function

For each point $X$ of a given polytope, denote by $f(X)$ the sum of the distances of the point $X$ from all the planes of the faces of the polytope. Prove that if $f$ attains its maximum at an interior point of the polytope, then $f$ is constant.

2015 Postal Coaching, Problem 3

Tags: holder , three variable inequality , inequalities

Let $a,b,c \in \mathbb{R^+}$ such that $abc=1$. Prove that $$\sum_{a,b,c} \sqrt{\frac{a}{a+8}} \ge 1$$