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

Tags were heavily modified to better represent problems.

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Found problems: 6530

2001 VJIMC, Problem 3
Tags: inequalities
Let $n\ge2$ be a natural number. Prove that $$\prod_{k=2}^n\ln k<\frac{\sqrt{n!}}n.$$

2017 Greece JBMO TST, 1
Tags: inequalities
Positive real numbers $a,b,c$ satisfy $a+b+c=1$. Prove that $$(a+1)\sqrt{2a(1-a)} + (b+1)\sqrt{2b(1-b)} + (c+1)\sqrt{2c(1-c)} \geq 8(ab+bc+ca).$$ Also, find the values of $a,b,c$ for which the equality happens.

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

2006 QEDMO 2nd, 3
Tags: inequalities
Prove the inequality $\frac{b^2+c^2-a^2}{a\left(b+c\right)}+\frac{c^2+a^2-b^2}{b\left(c+a\right)}+\frac{a^2+b^2-c^2}{c\left(a+b\right)}\geq\frac32$ for any three positive reals $a$, $b$, $c$. [i]Comment.[/i] This was an attempt of creating a contrast to the (rather hard) inequality at the QEDMO before. However, it turned out to be more difficult than I expected (a wrong solution was presented during the competition). Darij

2006 Thailand Mathematical Olympiad, 6
Tags: inequalities , algebra
Let $a, b, c$ be positive reals. Show that $$1 +\frac{3}{ab + bc + ca}\ge \frac{6}{a + b + c}$$

2005 Today's Calculation Of Integral, 76
Tags: limit , calculus , calculus computations , function , inequalities , integration
The function $f_n (x)\ (n=1,2,\cdots)$ is defined as follows. \[f_1 (x)=x,\ f_{n+1}(x)=2x^{n+1}-x^n+\frac{1}{2}\int_0^1 f_n(t)\ dt\ \ (n=1,2,\cdots)\] Evaluate \[\lim_{n\to\infty} f_n \left(1+\frac{1}{2n}\right)\]

2001 India IMO Training Camp, 2
Tags: inequalities , combinatorics
Two symbols $A$ and $B$ obey the rule $ABBB = B$. Given a word $x_1x_2\ldots x_{3n+1}$ consisting of $n$ letters $A$ and $2n+1$ letters $B$, show that there is a unique cyclic permutation of this word which reduces to $B$.

1995 AIME Problems, 13
Tags: inequalities , floor function
Let $f(n)$ be the integer closest to $\sqrt[4]{n}.$ Find $\sum_{k=1}^{1995}\frac 1{f(k)}.$

2011 AMC 10, 24
Tags: calculus , inequalities , integration , analytic geometry
A lattice point in an $xy$-coordinate system is any point $(x,y)$ where both $x$ and $y$ are integers. The graph of $y=mx+2$ passes through no lattice point with $0<x \le 100$ for all $m$ such that $\frac{1}{2}<m<a$. What is the maximum possible value of $a$? $ \textbf{(A)}\ \frac{51}{101} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{51}{100} \qquad \textbf{(D)}\ \frac{52}{101} \qquad \textbf{(E)}\ \frac{13}{25} $

2014 India Regional Mathematical Olympiad, 2
Tags: inequalities
Find all real $x,y$ such that \[x^2 + 2y^2 + \frac{1}{2} \le x(2y+1) \]

2015 Iran Team Selection Test, 1
Tags: inequalities
$a,b,c,d$ are positive numbers such that $\sum_{cyc} \frac{1}{ab} =1$. Prove that : $abcd+16 \geq 8 \sqrt{(a+c)(\frac{1}{a} + \frac{1}{c})}+8\sqrt{(b+d)(\frac{1}{b}+\frac{1}{d})}$

2010 Today's Calculation Of Integral, 582
Tags: inequalities , integration , calculus , trigonometry , calculus computations
Prove the following inequality. \[ \frac{\pi}{4}\sqrt{\frac{3}{2}\plus{}\sqrt{2}}<\int_0^{\frac{\pi}{2}} \sqrt{1\minus{}\frac 12\sin ^ 2 x}\ dx<\frac{\sqrt{3}}{4}\pi\]

2018 CMIMC Algebra, 8
Tags: inequalities , algebra , polynomial
Suppose $P$ is a cubic polynomial satisfying $P(0) = 3$ and \[(x^3 - 2x + 1 - P(x))(2x^3 - 5x^2 + 4 - P(x))\leq 0\] for all $x\in\mathbb R$. Determine all possible values of $P(-1)$.

1963 Swedish Mathematical Competition., 6
Tags: inequalities , algebra , function , analysis , functional inequality
The real-valued function $f(x)$ is defined on the reals. It satisfies $|f(x)| \le A$, $|f''(x)| \le B$ for some positive $A, B$ (and all $x$). Show that $|f'(x)| \le C$, for some fixed$ C$, which depends only on $A$ and $B$. What is the smallest possible value of $C$?

1998 North Macedonia National Olympiad, 4
Tags: inequalities , geometry , geometric inequalities , sidelenghts , triangle area
If $P$ is the area of a triangle $ABC$ with sides $a,b,c$, prove that $\frac{ab+bc+ca}{4P} \ge \sqrt3$

2013 SDMO (Middle School), 4
Tags: inequalities
Let $a$, $b$, $c$, and $d$ be positive real numbers such that $a+b=c+d$ and $a^2+b^2>c^2+d^2$. Prove that $a^3+b^3>c^3+d^3$.

1999 South africa National Olympiad, 1
Tags: inequalities , perimeter , ceiling function , triangle inequality , geometry
How many non-congruent triangles with integer sides and perimeter 1999 can be constructed?

2018 Ramnicean Hope, 3
Tags: geometric inequality , inequalities , geometry
Prove that for any noncollinear points $ A,B,C $ and positive real numbers $ x,y, $ the following inequality is true. $$ xAB^2- \frac{xy}{x+y}BC^2 +yCA^2\ge 0 $$ [i]Constantin Rusu[/i]

2005 Germany Team Selection Test, 2
Tags: inequalities
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]

2008 Indonesia TST, 4
Tags: algebra , inequalities
Let $a, b, c$ be positive reals. Prove that $$\left(\frac{a}{a+b}\right)^2+\left(\frac{b}{b+c}\right)^2+\left(\frac{c}{c+a}\right)^2\ge \frac34$$

2006 CHKMO, 3
Tags: algebra , function , inequalities
Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=1$. Prove that\[ 6(a^3+b^3+c^3+d^3)\ge(a^2+b^2+c^2+d^2)+\frac{1}{8} \]

2014 Contests, 3
Tags: function , algebra , quadratic , inequalities , quadratic formula
Let $ x,y,z $ be three non-negative real numbers such that \[x^2+y^2+z^2=2(xy+yz+zx). \] Prove that \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]

2003 AMC 12-AHSME, 25
Tags: parabola , algebra , function , inequalities , conic , domain , calculus
Let $ f(x)\equal{}\sqrt{ax^2\plus{}bx}$. For how many real values of $ a$ is there at least one positive value of $ b$ for which the domain of $ f$ and the range of $ f$ are the same set? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \text{infinitely many}$

2015 JBMO Shortlist, A4
Tags: inequalities
Let $a,b,c$ be positive real numbers such that $a+b+c = 3$. Find the minimum value of the expression \[A=\dfrac{2-a^3}a+\dfrac{2-b^3}b+\dfrac{2-c^3}c.\]

1989 All Soviet Union Mathematical Olympiad, 496
Tags: geometry , perimeter , inequalities , geometric inequality
A triangle with perimeter $1$ has side lengths $a, b, c$. Show that $a^2 + b^2 + c^2 + 4abc <\frac 12$.