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

2017 Saint Petersburg Mathematical Olympiad, 3
Tags: inequalities
Given real numbers $x,y,z,t\in (0,\pi /2]$ such that $$\cos^2 (x)+\cos^2 (y) +\cos^2 (z) +\cos^2 (t)=1.$$ What is the minimum possible value of $$\cot (x) +\cot (y) +\cot (z) +\cot (t)?$$

1973 Poland - Second Round, 1
Tags: algebra , geometry , geometric inequalities , inequalities
Prove that if positive numbers $ x, y, z $ satisfy the inequality $$ \frac{x^2+y^2-z^2}{2xy} + \frac{y^2+z^2-x^2}{2yz} + \frac{z^2+x^2-y ^2}{2xz} > 1,$$ then they are the lengths of the sides of a certain triangle.

2005 Taiwan National Olympiad, 2
Tags: inequalities , geometry
In triangle $ABC$, $D$ is the midpoint of side $AB$. $E$ and $F$ are points arbitrarily chosen on segments $AC$ and $BC$, respectively. Show that $[DEF] < [ADE] + [BDF]$.

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 $\frac{1}{3} n^{\frac{3}{2}}$.

2009 Estonia Team Selection Test, 4
Tags: geometry , geometric inequalities , inequalities , ratio
Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$

1997 Estonia National Olympiad, 1
Tags: number theory , greatest common divisor , inequalities
For positive integers $m$ and $n$ we define $T(m,n) = gcd \left(m, \frac{n}{gcd(m,n)} \right)$ (a) Prove that there are infinitely many pairs $(m,n)$ of positive integers for which $T(m,n) > 1$ and $T(n,m) > 1$. (b) Do there exist positive integers $m,n$ such that $T(m,n) = T(n,m) > 1$?

1952 Moscow Mathematical Olympiad, 214
Tags: inequalities , algebra , absolute value , moscow
Prove that if $|x| < 1$ and $|y| < 1$, then $\left|\frac{x - y}{1 -xy}\right|< 1$.

2017 Serbia JBMO TST, 2
Tags: inequalities
Let $x,y,z$ be positive real numbers.Prove that $(xy^2+yz^2+zx^2)(x^2y+y^2z+z^2x)(xy+yz+zx)\geq 3(x+y+z)^2(xyz)^2.$

2018 Estonia Team Selection Test, 10
Tags: algebra , sequence , recurrence relation , inequalities
A sequence of positive real numbers $a_1, a_2, a_3, ... $ satisfies $a_n = a_{n-1} + a_{n-2}$ for all $n \ge 3$. A sequence $b_1, b_2, b_3, ...$ is defined by equations $b_1 = a_1$ , $b_n = a_n + (b_1 + b_3 + ...+ b_{n-1})$ for even $n > 1$ , $b_n = a_n + (b_2 + b_4 + ... +b_{n-1})$ for odd $n > 1$. Prove that if $n\ge 3$, then $\frac13 < \frac{b_n}{n \cdot a_n} < 1$

2011 Sharygin Geometry Olympiad, 4
Tags: geometric transformation , reflection , inequalities , triangle inequality , geometry
Segments $AA'$, $BB'$, and $CC'$ are the bisectrices of triangle $ABC$. It is known that these lines are also the bisectrices of triangle $A'B'C'$. Is it true that triangle $ABC$ is regular?

2019 BMT Spring, 5
Tags: inequalities , algebra , complex number , geometry
Find the area of the set of all points $ z $ in the complex plane that satisfy $ \left| z - 3i \right| + \left| z - 4 \right| \leq 5\sqrt{2} $.

2003 India Regional Mathematical Olympiad, 4
Tags: function , floor function , inequalities , algebra , combinatorics
Find the number of ordered triples $(x,y,z)$ of non-negative integers satisfying (i) $x \leq y \leq z$ (ii) $x + y + z \leq 100.$

1991 Vietnam Team Selection Test, 2
Tags: inequalities
For a positive integer $ n>2$, let $ \left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a sequence of $ n$ positive reals which is either non-decreasing (this means, we have $ a_{1}\leq a_{2}\leq \ldots \leq a_{n}$) or non-increasing (this means, we have $ a_{1}\geq a_{2}\geq \ldots \geq a_{n}$), and which satisfies $ a_{1}\neq a_{n}$. Let $ x$ and $ y$ be positive reals satisfying $ \frac{x}{y}\geq \frac{a_{1}-a_{2}}{a_{1}-a_{n}}$. Show that: \[ \frac{a_{1}}{a_{2}\cdot x+a_{3}\cdot y}+\frac{a_{2}}{a_{3}\cdot x+a_{4}\cdot y}+\ldots+\frac{a_{n-1}}{a_{n}\cdot x+a_{1}\cdot y}+\frac{a_{n}}{a_{1}\cdot x+a_{2}\cdot y}\geq \frac{n}{x+y}. \]

2023 Bulgaria National Olympiad, 3
Tags: algebra , polynomial , abstract algebra , inequalities
Let $f(x)$ be a polynomial with positive integer coefficients. For every $n\in\mathbb{N}$, let $a_{1}^{(n)}, a_{2}^{(n)}, \dots , a_{n}^{(n)}$ be fixed positive integers that give pairwise different residues modulo $n$ and let \[g(n) = \sum\limits_{i=1}^{n} f(a_{i}^{(n)}) = f(a_{1}^{(n)}) + f(a_{2}^{(n)}) + \dots + f(a_{n}^{(n)})\] Prove that there exists a constant $M$ such that for all integers $m>M$ we have $\gcd(m, g(m))>2023^{2023}$.

2018 Estonia Team Selection Test, 3
Tags: sum , algebra , inequalities , max , min
Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.

2020 Jozsef Wildt International Math Competition, W59
Tags: inequalities
If $a_k>0~(k=1,2,\ldots,n)$ then prove that $$\sum_{\text{cyc}}\left(\frac{(a_1+a_2+\ldots+a_{n-1})^2}{a_n}+\frac{a_n^2}{a_1}\right)\ge\frac{n^2}2\sum_{k=1}^na_k$$ [i]Proposed by Mihály Bencze[/i]

2014 Middle European Mathematical Olympiad, 8
Tags: modular arithmetic , quadratic , inequalities , geometry , 3d geometry , number theory
Determine all quadruples $(x,y,z,t)$ of positive integers such that \[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\]

1990 Romania Team Selection Test, 2
Tags: function , geometry , circumcircle , inequalities
Prove that in any triangle $ABC$ the following inequality holds: \[ \frac{a^{2}}{b+c-a}+\frac{b^{2}}{a+c-b}+\frac{c^{2}}{a+b-c}\geq 3\sqrt{3}R. \] [i]Laurentiu Panaitopol[/i]

2023 Romania EGMO TST, P3
Tags: geometry , inequalities
Let $D{}$ be a point inside the triangle $ABC$. Let $E{}$ and $F{}$ be the projections of $D{}$ onto $AB$ and $AC$, respectively. The lines $BD$ and $CD$ intersect the circumcircle of $ABC$ the second time at $M{}$ and $N{}$, respectively. Prove that \[\frac{EF}{MN}\geqslant \frac{r}{R},\]where $r{}$ and $R{}$ are the inradius and circumradius of $ABC$, respectively.

2022 Regional Competition For Advanced Students, 1
Tags: algebra , inequalities
Let $a$ and $b$ be positive real numbers with $a^2 + b^2 =\frac12$. Prove that $$\frac{1}{1 - a}+\frac{1}{1-b}\ge 4.$$ When does equality hold? [i](Walther Janous)[/i]

2019 Belarusian National Olympiad, 9.3
Tags: polynomial , inequalities , algebra , function
Positive real numbers $a$ and $b$ satisfy the following conditions: the function $f(x)=x^3+ax^2+2bx-1$ has three different real roots, while the function $g(x)=2x^2+2bx+a$ doesn't have real roots. Prove that $a-b>1$. [i](V. Karamzin)[/i]

2012 Indonesia MO, 2
Tags: inequalities , n-variable inequality , ineqstd
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that \[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

1951 AMC 12/AHSME, 21
Tags: inequalities
Given: $ x > 0, y > 0, x > y$ and $ z\not \equal{} 0$. The inequality which is not always correct is: $ \textbf{(A)}\ x \plus{} z > y \plus{} z \qquad\textbf{(B)}\ x \minus{} z > y \minus{} z \qquad\textbf{(C)}\ xz > yz$ $ \textbf{(D)}\ \frac {x}{z^2} > \frac {y}{z^2} \qquad\textbf{(E)}\ xz^2 > yz^2$

2019 Regional Olympiad of Mexico Center Zone, 2
Tags: inequalities , functional inequality , functional , algebra
Find all functions $ f: \mathbb {R} \rightarrow \mathbb {R} $ such that $ f (x + y) \le f (xy) $ for every pair of real $ x $, $ y$.

1966 IMO Shortlist, 2
Tags: inequalities , algebra
Given $n$ positive real numbers $a_1, a_2, \ldots , a_n$ such that $a_1a_2 \cdots a_n = 1$, prove that \[(1 + a_1)(1 + a_2) \cdots (1 + a_n) \geq 2^n.\]