Olympiad problems

2019 Greece JBMO TST, 3

Tags: inequalities , algebra

Let $a,b,c$ be positive real numbers . Prove that$$ \frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.$$

2020 IMO, 2

Tags: inequalities , algebra , mount inquality

The real numbers $a, b, c, d$ are such that $a\geq b\geq c\geq d>0$ and $a+b+c+d=1$. Prove that \[(a+2b+3c+4d)a^ab^bc^cd^d<1\] [i]Proposed by Stijn Cambie, Belgium[/i]

2019 SAFEST Olympiad, 4

Tags: minimum , sum , algebra , inequalities

Let $a_1, a_2, . . . , a_{2019}$ be any positive real numbers such that $\frac{1}{a_1 + 2019}+\frac{1}{a_2 + 2019}+ ... +\frac{1}{a_{2019} + 2019}=\frac{1}{2019}$. Find the minimum value of $a_1a_2... a_{2019}$ and determine for which values of $a_1, a_2, . . . , a_{2019}$ this minimum occurs

2009 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: system , algebra , minimum , inequalities

Find minimal value of $a \in \mathbb{R}$ such that system $$\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}=a-1$$ $$\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}=a+1$$ has solution in set of real numbers

1998 China Team Selection Test, 3

Tags: trigonometry , inequalities

For a fixed $\theta \in \lbrack 0, \frac{\pi}{2} \rbrack$, find the smallest $a \in \mathbb{R}^{+}$ which satisfies the following conditions: [b]I. [/b] $\frac{\sqrt a}{\cos \theta} + \frac{\sqrt a}{\sin \theta} > 1$. [b]II.[/b] There exists $x \in \lbrack 1 - \frac{\sqrt a}{\sin \theta}, \frac{\sqrt a}{\cos \theta} \rbrack$ such that $\lbrack (1 - x)\sin \theta - \sqrt{a - x^2 \cos^{2} \theta} \rbrack^{2} + \lbrack x\cos \theta - \sqrt{a - (1 - x)^2 \sin^{2} \theta} \rbrack^{2} \leq a$.

1962 All-Soviet Union Olympiad, 13

Tags: sequence , inequalities , algebra

Given are $a_0,a_1, ... , a_n$, satisfying $a_0=a_n = 0$, and $a_{k-1} - 2a_k+a_{k+1}\ge 0$ for $k=0, 1, ... , n-1$. Prove that all the numbers are negative or zero.

2003 Belarusian National Olympiad, 4

Tags: algebra , sum , min , inequalities

Positive numbers $a_1,a_2,...,a_n, b_1, b_2,...,b_n$ satisfy the condition $a_1+a_2+...+a_n=b_1+ b_2+...+b_n=1$. Find the smallest possible value of the sum $$\frac{a_1^2}{a_1+b_1}+\frac{a_2^2}{a_2+b_2}+...+\frac{a_n^2}{a_n+b_n}$$ (V.Kolbun)

1995 Baltic Way, 6

Tags: search , inequalities

Prove that for positive $a,b,c,d$ \[\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a}\ge 4\]

2014 Saudi Arabia Pre-TST, 1.1

Tags: algebra , inequalities

Let $a_1, a_2,...,a_{2n}$ be positive real numbers such that $a_i + a_{n+i} = 1$, for all $i = 1,...,n$. Prove that there exist two different integers $1 \le j, k \le 2n$ for which $$\sqrt{a^2_j-a^2_k} < \frac{1}{\sqrt{n} +\sqrt{n - 1}}$$

2013 SEEMOUS, Problem 3

Tags: calculus , integration , inequalities

Find the maximum value of $$\int^1_0|f'(x)|^2|f(x)|\frac1{\sqrt x}dx$$over all continuously differentiable functions $f:[0,1]\to\mathbb R$ with $f(0)=0$ and $$\int^1_0|f'(x)|^2dx\le1.$$

2008 ITest, 7

Tags: inequalities , quadratic , floor function

Find the number of integers $n$ for which $n^2+10n<2008$.

2012 Iran MO (2nd Round), 1

Tags: inequalities , combinatorics

[b]a)[/b] Do there exist $2$-element subsets $A_1,A_2,A_3,...$ of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number $n$, sum of the elements of $A_n$ equals $1391+n$? [b]b)[/b] Do there exist $2$-element subsets $A_1,A_2,A_3,...$ of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number $n$, sum of the elements of $A_n$ equals $1391+n^2$? [i]Proposed by Morteza Saghafian[/i]

2019 Junior Balkan Team Selection Tests - Moldova, 4

Tags: algebra , inequalities

Let $n(n\geq2)$ be a natural number and $a_1,a_2,...,a_n$ natural positive real numbers. Determine the least possible value of the expression $$E_n=\frac{(1+a_1)\cdot(a_1+a_2)\cdot(a_2+a_3)\cdot...\cdot(a_{n-1}+a_n)\cdot(a_n+3^{n+1})} {a_1\cdot a_2\cdot a_3\cdot...\cdot a_n}$$

2014 Indonesia MO Shortlist, A3

Tags: algebra , inequalities , three variable inequality

Prove for each positive real number $x, y, z$, $$\frac{x^2y}{x+2y}+\frac{y^2z}{y+2z}+\frac{z^2x}{z+2x}<\frac{(x+y+z)^2}{8}$$

2013 Tournament of Towns, 5

Tags: function , algebra , inequalities

Do there exist two integer-valued functions $f$ and $g$ such that for every integer $x$ we have (a) $f(f(x)) = x, g(g(x)) = x, f(g(x)) > x, g(f(x)) > x$ ? (b) $f(f(x)) < x, g(g(x)) < x, f(g(x)) > x, g(f(x)) > x$ ?

2023 China Team Selection Test, P14

Tags: inequalities , set

For any nonempty, finite set $B$ and real $x$, define $$d_B(x) = \min_{b\in B} |x-b|$$ (1) Given positive integer $m$. Find the smallest real number $\lambda$ (possibly depending on $m$) such that for any positive integer $n$ and any reals $x_1,\cdots,x_n \in [0,1]$, there exists an $m$-element set $B$ of real numbers satisfying $$d_B(x_1)+\cdots+d_B(x_n) \le \lambda n$$ (2) Given positive integer $m$ and positive real $\epsilon$. Prove that there exists a positive integer $n$ and nonnegative reals $x_1,\cdots,x_n$, satisfying for any $m$-element set $B$ of real numbers, we have $$d_B(x_1)+\cdots+d_B(x_n) > (1-\epsilon)(x_1+\cdots+x_n)$$

2013 Brazil Team Selection Test, 2

Tags: modular arithmetic , inequalities , quadratic , function , algebra , floor function

Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

2013 BMT Spring, 7

Tags: inequalities , algebra

If $x,y$ are positive real numbers satisfying $x^3-xy+1=y^3$, find the minimum possible value of $y$.

2011 Morocco National Olympiad, 2

Tags: geometry , perimeter , trigonometry , inequalities

Let $\alpha , \beta ,\gamma$ be the angles of a triangle $ABC$ of perimeter $ 2p $ and $R$ is the radius of its circumscribed circle. $(a)$ Prove that \[\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right).\] $(b)$ When do we have equality?

2014 Math Prize For Girls Problems, 15

Tags: inequalities

There are two math exams called A and B. 2014 students took the A exam and/or the B exam. Each student took one or both exams, so the total number of exam papers was between 2014 and 4028, inclusive. The score for each exam is an integer from 0 through 40. The average score of all the exam papers was 20. The grade for a student is the best score from one or both exams that she took. The average grade of all 2014 students was 14. Let $G$ be the [i]greatest[/i] possible number of students who took both exams. Let $L$ be the [i]least[/i] possible number of students who took both exams. Compute $G - L$.

1950 Moscow Mathematical Olympiad, 176

Tags: sidelengths , geometric inequality , inequalities , angle

Let $a, b, c$ be the lengths of the sides of a triangle and $A, B, C$, the opposite angles. Prove that $$Aa + Bb + Cc \ge \frac{Ab + Ac + Ba + Bc + Ca + Cb}{2}$$

PEN E Problems, 24

Tags: floor function , limit , number theory , prime number , logarithm , inequalities

Let $p_{n}$ again denote the $n$th prime number. Show that the infinite series \[\sum^{\infty}_{n=1}\frac{1}{p_{n}}\] diverges.

2017 Iran MO (3rd round), 2

Tags: inequalities , mixing

Let $a,b,c$ and $d$ be positive real numbers such that $a^2+b^2+c^2+d^2 \ge 4$. Prove that $$(a+b)^3+(c+d)^3+2(a^2+b^2+c^2+d^2) \ge 4(ab+bc+cd+da+ac+bd)$$

2018 Malaysia National Olympiad, A2

Tags: algebra , inequalities , max

The product of $10$ integers is $1024$. What is the greatest possible sum of these $10$ integers?

2018 Ukraine Team Selection Test, 5

Tags: combinatorial geometry , geometry , inequalities , geometric inequality

Find the smallest positive number $\lambda$ such that for an arbitrary $12$ points on the plane $P_1,P_2,...P_{12}$ (points may coincide), with distance between arbitrary two of them does not exceeds $1$, holds the inequality $\sum_{1\le i\le j\le 12} P_iP_j^2 \le \lambda$