This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 6530

2012 German National Olympiad, 4
Tags: square root , inequalities , algebra , real number
Let $a,b$ be positive real numbers and $n\geq 2$ a positive integer. Prove that if $x^n \leq ax+b$ holds for a positive real number $x$, then it also satisfies the inequality $x < \sqrt[n-1]{2a} + \sqrt[n]{2b}.$

2016 239 Open Mathematical Olympiad, 4
Tags: algebra , inequalities
Positive real numbers $a,b,c$ are given such that $abc=1$. Prove that$$a+b+c+\frac{3}{ab+bc+ca}\geq4.$$

1991 Tournament Of Towns, (285) 1
Tags: algebra , inequalities
Prove that the product of the $99$ fractions $$\frac{k^3-1}{k^3+1} \,\, , \,\,\,\,\,\, k=2,3,...,100$$ is greater than $2/3$. (D. Fomin, Leningrad)

1976 Chisinau City MO, 131
Tags: algebra , inequalities
The sum of the real numbers $x_1, x_2, ...,x_n$ belonging to the segment $[a, b]$ is equal to zero. Prove that $$x_1^2+ x_2^2+ ...+x_n^2 \le - nab.$$

2008 Bundeswettbewerb Mathematik, 1
Tags: inequalities , algebra
Determine all real $ x$ satisfying the equation \[ \sqrt[5]{x^3 \plus{} 2x} \equal{} \sqrt[3]{x^5\minus{}2x}.\] Odd roots for negative radicands shall be included in the discussion.

2006 ISI B.Stat Entrance Exam, 7
Tags: inequalities , induction , algebra
for any positive integer $n$ greater than $1$, show that \[2^n<\binom{2n}{n}<\frac{2^n}{\prod\limits_{i=0}^{n-1} \left(1-\frac{i}{n}\right)}\]

2001 Singapore Team Selection Test, 2
Tags: inequalities , algebra
Determine all the integers $n > 1$ such that $$\sum_{i=1}^{n}x_i^2 \ge x_n \sum_{i=1}^{n-1}x_i$$ for all real numbers $x_1, x_2, ... , x_n$.

2023 Switzerland - Final Round, 4
Tags: algebra , inequalities
Determine the smallest possible value of the expression $$\frac{ab+1}{a+b}+\frac{bc+1}{b+c}+\frac{ca+1}{c+a}$$ where $a,b,c \in \mathbb{R}$ satisfy $a+b+c = -1$ and $abc \leqslant -3$

2005 China Team Selection Test, 1
Tags: inequalities , algebra
Prove that for any $n$ ($n \geq 2$) pairwise distinct fractions in the interval $(0,1)$, the sum of their denominators is no less than $\frac{1}{3} n^{\frac{3}{2}}$.

2013 Today's Calculation Of Integral, 864
Tags: calculus , integration , inequalities , limit , calculus computations
Let $m,\ n$ be positive integer such that $2\leq m<n$. (1) Prove the inequality as follows. \[\frac{n+1-m}{m(n+1)}<\frac{1}{m^2}+\frac{1}{(m+1)^2}+\cdots +\frac{1}{(n-1)^2}+\frac{1}{n^2}<\frac{n+1-m}{n(m-1)}\] (2) Prove the inequality as follows. \[\frac 32\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq 2\] (3) Prove the inequality which is made precisely in comparison with the inequality in (2) as follows. \[\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}\]

2024 VJIMC, 1
Tags: inequalities , function , exponential , calculus
Suppose that $f:[-1,1] \to \mathbb{R}$ is continuous and satisfies \[\left(\int_{-1}^1 e^xf(x) dx\right)^2 \ge \left(\int_{-1}^1 f(x) dx\right)\left(\int_{-1}^1 e^{2x}f(x) dx\right).\] Prove that there exists a point $c \in (-1,1)$ such that $f(c)=0$.

1946 Moscow Mathematical Olympiad, 115
Tags: trigonometry , inequalities , acute
Prove that if $\alpha$ and $\beta$ are acute angles and $\alpha$ < $\beta$ , then $\frac{tan \alpha}{\alpha} < \frac{tan \beta}{\beta} $

2014 ELMO Shortlist, 2
Tags: inequalities
Given positive reals $a,b,c,p,q$ satisfying $abc=1$ and $p \geq q$, prove that \[ p \left(a^2+b^2+c^2\right) + q\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \geq (p+q) (a+b+c). \][i]Proposed by AJ Dennis[/i]

2012 Czech-Polish-Slovak Match, 3
Tags: inequalities
Let $a,b,c,d$ be positive real numbers such that $abcd=4$ and \[a^2+b^2+c^2+d^2=10.\] Find the maximum possible value of $ab+bc+cd+da$.

2005 APMO, 2
Tags: algebra , inequalities
Let $a, b, c$ be positive real numbers such that $abc=8$. Prove that \[ \frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} +\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} +\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3} \]

2013 Spain Mathematical Olympiad, 1
Tags: inequalities
Let $a,b,n$ positive integers with $a>b$ and $ab-1=n^2$. Prove that $a-b \geq \sqrt{4n-3}$ and study the cases where the equality holds.

1987 AIME Problems, 11
Tags: calculus , integration , quadratic , inequalities , algorithm , algebra
Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.

2018 Saudi Arabia IMO TST, 2
Tags: geometric sequence , inequalities , arithmetic sequence , algebra
a) For integer $n \ge 3$, suppose that $0 < a_1 < a_2 < ...< a_n$ is a arithmetic sequence and $0 < b_1 < b_2 < ... < b_n$ is a geometric sequence with $a_1 = b_1, a_n = b_n$. Prove that a_k > b_k for all $k = 2,3,..., n -1$. b) Prove that for every positive integer $n \ge 3$, there exist an integer arithmetic sequence $(a_n)$ and an integer geometric sequence $(b_n)$ such that $0 < b_1 < a_1 < b_2 < a_2 < ... < b_n < a_n$.

2012 China Team Selection Test, 1
Tags: inequalities , induction , number theory
Given an integer $n\ge 2$. Prove that there only exist a finite number of n-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ which simultaneously satisfy the following three conditions: [list] [*] $a_1>a_2>\ldots>a_n$; [*] $\gcd (a_1,a_2,\ldots,a_n)=1$; [*] $a_1=\sum_{i=1}^{n}\gcd (a_i,a_{i+1})$,where $a_{n+1}=a_1$.[/list]

2005 Thailand Mathematical Olympiad, 20
Tags: max , algebra , inequalities
Let $a, b, c, d > 0$ satisfy $36a + 4b + 4c + 3d = 25$. What is the maximum possible value of $ab^{1/2}c^{1/3}d^{1/4}$ ?

2008 China Girls Math Olympiad, 3
Tags: ratio , inequalities , geometry
Determine the least real number $ a$ greater than $ 1$ such that for any point $ P$ in the interior of the square $ ABCD$, the area ratio between two of the triangles $ PAB$, $ PBC$, $ PCD$, $ PDA$ lies in the interval $ \left[\frac {1}{a},a\right]$.

2005 Germany Team Selection Test, 3
Tags: inequalities , geometry
Let $ABC$ be a triangle with area $S$, and let $P$ be a point in the plane. Prove that $AP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}$.

2018 Azerbaijan BMO TST, 1
Tags: inequalities , algebra
Problem Shortlist BMO 2017 Let $ a $,$ b$,$ c$, be positive real numbers such that $abc= 1 $. Prove that $$\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+b^{5}+b^{2}}\leq 1 . $$

1986 Bulgaria National Olympiad, Problem 6
Tags: inequalities , recurrence relation , algebra , sequence
Let $0<k<1$ be a given real number and let $(a_n)_{n\ge1}$ be an infinite sequence of real numbers which satisfies $a_{n+1}\le\left(1+\frac kn\right)a_n-1$. Prove that there is an index $t$ such that $a_t<0$.

2011 Saudi Arabia Pre-TST, 4.3
Tags: algebra , inequalities
Let $x_1,x_2,...,x_n$ be positive real numbers for which $$\frac{1}{1+x_1}+\frac{1}{1+x_2}+...+\frac{1}{1+x_n}=1$$ Prove that $x_1x_2...x_n \ge (n -1)^n$.