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

2022 Saudi Arabia JBMO TST, 2
Tags: max , algebra , inequalities
Consider non-negative real numbers $a, b, c$ satisfying the condition $a^2 + b^2 + c^2 = 2$ . Find the maximum value of the following expression $$P=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c$$

2007 Nicolae Coculescu, 1
Tags: three variable inequality , geometry , floor , algebra , inequalities
Calculate $ \left\lfloor \frac{(a^2+b^2+c^2)(a+b+c)}{a^3+b^3+c^3} \right\rfloor , $ where $ a,b,c $ are the lengths of the side of a triangle. [i]Costel Anghel[/i]

2012 Junior Balkan Team Selection Tests - Moldova, 2
Tags: inequalities
Let $ a,b,c $ be positive real numbers, prove the inequality: $ (a+b+c)^2+ab+bc+ac\geq 6\sqrt{abc(a+b+c)} $

1972 All Soviet Union Mathematical Olympiad, 169
Tags: minimum , maximum , algebra , inequalities
Let $x,y$ be positive numbers, $s$ -- the least of $$\{ x, (y+ 1/x), 1/y\}$$ What is the greatest possible value of $s$? To what $x$ and $y$ does it correspond?

2010 Today's Calculation Of Integral, 644
Tags: logarithm , integration , calculus computations , inequalities , calculus , trigonometry
For a constant $p$ such that $\int_1^p e^xdx=1$, prove that \[\left(\int_1^p e^x\cos x\ dx\right)^2+\left(\int_1^p e^x\sin x\ dx\right)^2>\frac 12.\] Own

2012 Stars of Mathematics, 3
Tags: algebra , inequalities
For all triplets $a,b,c$ of (pairwise) distinct real numbers, prove the inequality $$ \left | \dfrac {a+b} {a-b} \right | + \left | \dfrac {b+c} {b-c} \right | + \left | \dfrac {c+a} {c-a} \right | \geq 2$$ and determine all cases of equality. Prove that if we also impose $a,b,c \geq 0$, then $$ \left | \dfrac {a+b} {a-b} \right | + \left | \dfrac {b+c} {b-c} \right | + \left | \dfrac {c+a} {c-a} \right | > 3,$$ with the value $3$ being the best constant possible. ([i]Dan Schwarz[/i])

2010 Indonesia MO, 3
Tags: inequalities , combinatorics
A mathematical competition was attended by 120 participants from several contingents. At the closing ceremony, each participant gave 1 souvenir each to every other participants from the same contingent, and 1 souvenir to any person from every other contingents. It is known that there are 3840 souvenirs whom were exchanged. Find the maximum possible contingents such that the above condition still holds? [i]Raymond Christopher Sitorus, Singapore[/i]

1958 AMC 12/AHSME, 34
Tags: inequalities
The numerator of a fraction is $ 6x \plus{} 1$, then denominator is $ 7 \minus{} 4x$, and $ x$ can have any value between $ \minus{}2$ and $ 2$, both included. The values of $ x$ for which the numerator is greater than the denominator are: $ \textbf{(A)}\ \frac{3}{5} < x \le 2\qquad \textbf{(B)}\ \frac{3}{5} \le x \le 2\qquad \textbf{(C)}\ 0 < x \le 2\qquad \\ \textbf{(D)}\ 0 \le x \le 2\qquad \textbf{(E)}\ \minus{}2 \le x \le 2$

2019 Jozsef Wildt International Math Competition, W. 68
Tags: tetrahedron , 3d geometry , exradius , inradius , inequalities
In all tetrahedron $ABCD$ holds [list=1] [*] $\displaystyle{\sum \limits_{cyc}\frac{h_a-r}{h_a+r}\geq \sum \limits_{cyc}\frac{h_a^t-r^t}{(h_a+r)^t}}$ [*] $\displaystyle{\sum \limits_{cyc}\frac{2r_a-r}{2r_a+r}\geq \sum \limits_{cyc}\frac{2r_a^t-r^t}{(2r_a+r)^t}}$ [/list] for all $t\in [0,1]$

2002 Czech-Polish-Slovak Match, 2
Tags: geometry , inequalities
A triangle $ABC$ has sides $BC = a, CA = b, AB = c$ with $a < b < c$ and area $S$. Determine the largest number $u$ and the least number $v$ such that, for every point $P$ inside $\triangle ABC$, the inequality $u \le PD + PE + PF \le v$ holds, where $D,E, F$ are the intersection points of $AP,BP,CP$ with the opposite sides.

1976 Swedish Mathematical Competition, 5
Tags: analysis , algebra , inequalities , function
$f(x)$ is defined for $x \geq 0$ and has a continuous derivative. It satisfies $f(0)=1$, $f'(0)=0$ and $(1+f(x))f''(x)=1+x$. Show that $f$ is increasing and that $f(1) \leq 4/3$.

1999 Junior Balkan Team Selection Tests - Romania, 4
Tags: inequalities , geometry
Let be three discs $ D_1,D_2,D_3. $ For each $ i,j\in\{1,2,3\} , $ denote $ a_{ij} $ as being the area of $ D_i\cap D_j. $ If $ x_1,x_2,x_3\in\mathbb{R} $ such that $ x_1x_2x_3\neq 0, $ then $$ a_{11} x_1^2+a_{22} x_2^2+a_{33} x_3^2+2a_{12} x_1x_2+2a_{23 }x_2x_3+2a_{31} x_3x_1>0. $$ [i]Vasile Pop[/i]

1970 Canada National Olympiad, 5
Tags: inequalities
A quadrilateral has one vertex on each side of a square of side-length 1. Show that the lengths $a$, $b$, $c$ and $d$ of the sides of the quadrilateral satisfy the inequalities \[ 2\le a^2+b^2+c^2+d^2\le 4. \]

1914 Eotvos Mathematical Competition, 2
Tags: inequalities , algebra
Suppose that $$-1 \le ax^2 + bx + c \le 1 \ \ for \ \ -1 \le x \le 1 , $$ where a, b, c are real numbers. Prove that $$-4 \le 2ax + b \le 4 \ \ for \ \ -1 \le x \le 1 , $$

2005 International Zhautykov Olympiad, 1
Tags: inequalities
For the positive real numbers $ a,b,c$ prove the inequality \[ \frac {c}{a \plus{} 2b} \plus{} \frac {a}{b \plus{} 2c} \plus{} \frac {b}{c \plus{} 2a}\ge1. \]

2019 Grand Duchy of Lithuania, 1
Tags: inequalities , algebra
Let $x, y, z$ be positive numbers such that $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1$. Prove that $$\sqrt{x + yz} +\sqrt{y + zx} +\sqrt{z + xy} \ge\sqrt{xyz}+\sqrt{x }+\sqrt{y} +\sqrt{z}$$

1996 Estonia National Olympiad, 2
Tags: min , inequalities , algebra
For which positive $x$ does the expression $x^{1000}+x^{900}+x^{90}+x^6+\frac{1996}{x}$ attain the smallest value?

2005 China Team Selection Test, 1
Tags: inequalities , function , floor function
Find all positive integers $m$ and $n$ such that the inequality: \[ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] \] is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.

2009 AIME Problems, 13
Tags: function , induction , pigeonhole principle , modular arithmetic , inequalities
The terms of the sequence $ (a_i)$ defined by $ a_{n \plus{} 2} \equal{} \frac {a_n \plus{} 2009} {1 \plus{} a_{n \plus{} 1}}$ for $ n \ge 1$ are positive integers. Find the minimum possible value of $ a_1 \plus{} a_2$.

2007 All-Russian Olympiad, 7
Tags: 3d geometry , inequalities , geometry , tetrahedron
Given a tetrahedron $ T$. Valentin wants to find two its edges $ a,b$ with no common vertices so that $ T$ is covered by balls with diameters $ a,b$. Can he always find such a pair? [i]A. Zaslavsky[/i]

MathLinks Contest 6th, 1.1
Tags: inequalities
Let $ a, b, c$ be positive real numbers such that $ bc +ca +b = 1,$ . Prove that $$ \frac {1 +b^2c^2}{(b +c)^2} + \frac {1+ c^2a^2}{(c + a)^2} +\frac {1 +a^2b^2}{(a +b)^2} \geq \frac {5}{2}.$$

2019 Jozsef Wildt International Math Competition, W. 66
Tags: inequalities , inverse trigonometric function , integration , calculus , trigonometry
If $0 < a \leq b$ then$$\frac{2}{\sqrt{3}}\tan^{-1}\left(\frac{2(b^2 - a^2)}{(a^2+2)(b^2+2)}\right)\leq \int \limits_a^b \frac{(x^2+1)(x^2+x+1)}{(x^3 + x^2 + 1) (x^3 + x + 1)}dx\leq \frac{4}{\sqrt{3}}\tan^{-1}\left(\frac{(b-a)\sqrt{3}}{a+b+2(1+ab)}\right)$$

1998 USAMTS Problems, 2
Tags: inequalities
Determine the smallest rational number $\frac{r}{s}$ such that $\frac{1}{k}+\frac{1}{m}+\frac{1}{n}\leq \frac{r}{s}$ whenever $k, m,$ and $n$ are positive integers that satisfy the inequality $\frac{1}{k}+\frac{1}{m}+\frac{1}{n} < 1$.

2010 Baltic Way, 19
Tags: inequalities , number theory
For which $k$ do there exist $k$ pairwise distinct primes $p_1,p_2,\ldots ,p_k$ such that \[p_1^2+p_2^2+\ldots +p_k^2=2010? \]

IV Soros Olympiad 1997 - 98 (Russia), 9.5
Tags: inequalities , algebra
All ordinary proper irreducible fractions whose numerators are two-digit numbers were ordered in ascending order. Between what two consecutive fractions is the number $\frac58$ located?