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.

AND:
OR:
NO:

Found problems: 6530

2005 France Team Selection Test, 2
Tags: geometry , circumcircle , perimeter , inradius , inequalities , trigonometry , pythagorean theorem
Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let $S$ (respectively $S'$) be the area of the first triangle (respectively of the second triangle). Prove that $\frac{S}{S'}\geq 3+2\sqrt{2}$.

2000 Harvard-MIT Mathematics Tournament, 3
Tags: inequalities
Suppose the positive integers $a,b,c$ satisfy $a^n+b^n=c^n$, where $n$ is a positive integer greater than $1$. Prove that $a,b,c>n$. (Note: Fermat's Last Theorem may [i]not[/i] be used)

1983 Austrian-Polish Competition, 1
Tags: inequalities , algebra
Nonnegative real numbers $a, b,x,y$ satisfy $a^5 + b^5 \le $1 and $x^5 + y^5 \le 1$. Prove that $a^2x^3 + b^2y^3 \le 1$.

1997 Vietnam National Olympiad, 2
Tags: inequalities , number theory , relatively prime , algebra
Let n be an integer which is greater than 1, not divisible by 1997. Let $ a_m\equal{}m\plus{}\frac{mn}{1997}$ for all m=1,2,..,1996 $ b_m\equal{}m\plus{}\frac{1997m}{n}$ for all m=1,2,..,n-1 We arrange the terms of two sequence $ (a_i), (b_j)$ in the ascending order to form a new sequence $ c_1\le c_2\le ...\le c_{1995\plus{}n}$ Prove that $ c_{k\plus{}1}\minus{}c_k<2$ for all k=1,2,...,1994+n

1997 China National Olympiad, 1
Tags: function , inequalities
Let $x_1,x_2,\ldots ,x_{1997}$ be real numbers satisfying the following conditions: i) $-\dfrac{1}{\sqrt{3}}\le x_i\le \sqrt{3}$ for $i=1,2,\ldots ,1997$; ii) $x_1+x_2+\cdots +x_{1997}=-318 \sqrt{3}$ . Determine (with proof) the maximum value of $x^{12}_1+x^{12}_2+\ldots +x^{12}_{1997}$ .

2014 ELMO Shortlist, 8
Tags: function , inequalities
Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that \[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]

2012 Junior Balkan Team Selection Tests - Romania, 1
Tags: inequalities , algebra
Show that, for all positive real numbers $a, b, c$ such that $abc = 1$, the inequality $$\frac{1}{1 + a^2 + (b + 1)^2} +\frac{1}{1 + b^2 + (c + 1)^2} +\frac{1}{1 + c^2 + (a + 1)^2} \le \frac{1}{2}$$

2021 Indonesia TST, A
Tags: inequality , inequalities , algebra
A positive real $M$ is $strong$ if for any positive reals $a$, $b$, $c$ satisfying $$ \text{max}\left\{ \frac{a}{b+c} , \frac{b}{c+a} , \frac{c}{a+b} \right\} \geqslant M $$ then the following inequality holds: $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} > 20.$$ (a) Prove that $M=20-\frac{1}{20}$ is not $strong$. (b) Prove that $M=20-\frac{1}{21}$ is $strong$.

2020 USOMO, 6
Tags: inequalities
Let $n \ge 2$ be an integer. Let $x_1 \ge x_2 \ge ... \ge x_n$ and $y_1 \ge y_2 \ge ... \ge y_n$ be $2n$ real numbers such that $$0 = x_1 + x_2 + ... + x_n = y_1 + y_2 + ... + y_n $$ $$\text{and} \hspace{2mm} 1 =x_1^2 + x_2^2 + ... + x_n^2 = y_1^2 + y_2^2 + ... + y_n^2.$$ Prove that $$\sum_{i = 1}^n (x_iy_i - x_iy_{n + 1 - i}) \ge \frac{2}{\sqrt{n-1}}.$$ [i]Proposed by David Speyer and Kiran Kedlaya[/i]

2014 Online Math Open Problems, 24
Tags: floor function , vector , function , inequalities
Let $\mathcal P$ denote the set of planes in three-dimensional space with positive $x$, $y$, and $z$ intercepts summing to one. A point $(x,y,z)$ with $\min \{x,y,z\} > 0$ lies on exactly one plane in $\mathcal P$. What is the maximum possible integer value of $\left(\frac{1}{4} x^2 + 2y^2 + 16z^2\right)^{-1}$? [i]Proposed by Sammy Luo[/i]

2021 Canadian Junior Mathematical Olympiad, 4
Tags: algebra , inequalities , n-variable inequality
Let $n\geq 2$ be some fixed positive integer and suppose that $a_1, a_2,\dots,a_n$ are positive real numbers satisfying $a_1+a_2+\cdots+a_n=2^n-1$. Find the minimum possible value of $$\frac{a_1}{1}+\frac{a_2}{1+a_1}+\frac{a_3}{1+a_1+a_2}+\cdots+\frac{a_n}{1+a_1+a_2+\cdots+a_{n-1}}$$

1979 Yugoslav Team Selection Test, Problem 1
Tags: sum , inequalities , number theory
Let $a_1,a_2,...,a_n$ be $n$ different positive integers where $n\ge 1$. Show that $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$

2014 IMO Shortlist, A1
Tags: algebra , inequalities , sequence , unique
Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that \[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\] [i]Proposed by Gerhard Wöginger, Austria.[/i]

2022 Turkey MO (2nd round), 3
Tags: inequalities
Let $a_1, a_2, \cdots, a_{2022}$ be nonnegative real numbers such that $a_1+a_2+\cdots +a_{2022}=1$. Find the maximum number of ordered pairs $(i, j)$, $1\leq i,j\leq 2022$, satisfying $$a_i^2+a_j\ge \frac 1{2021}.$$

2021 CHKMO, 4
Tags: algebra , inequality , inequalities
Let $a,b$ and $c$ be positive real numbers satisfying $abc=1$. Prove that \[\dfrac{1}{a^3+2b^2+2b+4}+\dfrac{1}{b^3+2c^2+2c+4}+\dfrac{1}{c^3+2a^2+2a+4}\leq \dfrac13.\]

2015 Romania Team Selection Tests, 5
Tags: maximization , algebra , inequalities , sum
Given an integer $N \geq 4$, determine the largest value the sum $$\sum_{i=1}^{\left \lfloor{\frac{k}{2}}\right \rfloor+1}\left( \left \lfloor{\frac{n_i}{2}}\right \rfloor+1\right)$$ may achieve, where $k, n_1, \ldots, n_k$ run through the integers subject to $k \geq 3$, $n_1 \geq \ldots\geq n_k\geq 1$ and $n_1 + \ldots + n_k = N$.

2011 Finnish National High School Mathematics Competition, 2
Tags: inequalities , algebra
Find all integers $x$ and $y$ satisfying the inequality \[x^4-12x^2+x^2y^2+30\leq 0.\]

2018 China Northern MO, 2
Tags: three variable inequality , inequalities
Let $a$,$b$,$c$ be nonnegative reals such that $$a^2+b^2+c^2+ab+\frac{2}{3}ac+\frac{4}{3}bc=1$$ Find the maximum and minimum value of $a+b+c$.

1965 Polish MO Finals, 1
Tags: algebra , inequalities , geometry , geometric inequalities
Prove the theorem: the lengths $ a$, $ b $, $ c $ of the sides of a triangle and the arc measures $ \alpha $, $ \beta $, $ \gamma $of its opposite angles satisfy the inequalities $$\frac{\pi}{3}\leq \frac{a \alpha + b \beta +c \gamma}{a+b+c}<\frac{\pi }{ 2}.$$

2018 Sharygin Geometry Olympiad, 3
Tags: inequalities , geometry , angle bisector
Let $ABC$ be a triangle with $\angle A = 60^\circ$, and $AA', BB', CC'$ be its internal angle bisectors. Prove that $\angle B'A'C' \le 60^\circ$.

2009 Princeton University Math Competition, 5
Tags: inequalities
Suppose that for some positive integer $n$, the first two digits of $5^n$ and $2^n$ are identical. Suppose the first two digits are $a$ and $b$ in this order. Find the two-digit number $\overline{ab}$.

2014 France Team Selection Test, 6
Tags: inequalities
Let $n$ be a positive integer and $x_1,x_2,\ldots,x_n$ be positive reals. Show that there are numbers $a_1,a_2,\ldots, a_n \in \{-1,1\}$ such that the following holds: \[a_1x_1^2+a_2x_2^2+\cdots+a_nx_n^2 \ge (a_1x_1+a_2x_2 +\cdots+a_nx_n)^2\]

1983 National High School Mathematics League, 8
Tags: geometry , inequalities , circumcircle
For any $\triangle ABC$, its girth is$l$, its circumradius is$R$, its inscribed radius is $r$.Which one is true? $\text{(A)}l>R+r\qquad\text{(B)}l\leq R+r\qquad\text{(C)}\frac{l}{6}<R+r<6l\qquad\text{(D)}$None above

1999 USAMO, 4
Tags: function , inequalities
Let $a_{1}, a_{2}, \dots, a_{n}$ ($n > 3$) be real numbers such that \[ a_{1} + a_{2} + \cdots + a_{n} \geq n \qquad \mbox{and} \qquad a_{1}^{2} + a_{2}^{2} + \cdots + a_{n}^{2} \geq n^{2}. \] Prove that $\max(a_{1}, a_{2}, \dots, a_{n}) \geq 2$.

1976 Swedish Mathematical Competition, 5
Tags: analysis , inequalities , function , algebra
$f(x)$ 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 $f(1) \leq 4/3$.