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

2023 Romanian Master of Mathematics, 4
Tags: geometry , inequalities , geometry inequality
An acute triangle $ABC$ is given and $H$ and $O$ be its orthocenter and circumcenter respectively. Let $K$ be the midpoint of $AH$ and $\ell$ be a line through $O. $ Let $P$ and $Q$ be the projections of $B$ and $C$ on $\ell. $ Prove that$$KP+KQ\ge BC$$

2021 Iran Team Selection Test, 4
Tags: algebra , inequalities , function , number theory
Find all functions $f : \mathbb{N} \rightarrow \mathbb{R}$ such that for all triples $a,b,c$ of positive integers the following holds : $$f(ac)+f(bc)-f(c)f(ab) \ge 1$$ Proposed by [i]Mojtaba Zare[/i]

2003 All-Russian Olympiad, 2
Tags: inequalities
Let $a, b, c$ be positive numbers with the sum $1$. Prove the inequality \[\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c} \geq \frac{2}{1+a}+\frac{2}{1+b}+\frac{2}{1+c}.\]

1988 All Soviet Union Mathematical Olympiad, 477
Tags: minimum , algebra , inequalities
What is the minimal value of $\frac{b}{c + d} + \frac{c}{a + b}$ for positive real numbers $b$ and $c$ and non-negative real numbers $a$ and $d$ such that $b + c\ge a + d$?

2025 Poland - First Round, 11
Tags: algebra , inequalities
Positive integer $l$ and positive real numbers $a_1, a_2, ..., a_l$ are given. For every positive integer $n$ we define $$c_n=\sum_{k_1+k_2+...+k_l=n}\frac{(2n)!}{(2k_1)!(2k_2)!...(2k_l)!}a_1^{k_1}a_2^{k_2}...a_l^{k_l}.$$ Prove that for every positive integer $n$ the inequality $\sqrt[n]{c_n}\leq \sqrt[n+1]{c_{n+1}}$ holds.

1993 China National Olympiad, 6
Tags: function , inequalities , induction
Let $f: (0,+\infty)\rightarrow (0,+\infty)$ be a function satisfying the following condition: for arbitrary positive real numbers $x$ and $y$, we have $f(xy)\le f(x)f(y)$. Show that for arbitrary positive real number $x$ and natural number $n$, inequality $f(x^n)\le f(x)f(x^2)^{\dfrac{1}{2}}\dots f(x^n)^{\dfrac{1}{n}}$ holds.

2001 Croatia National Olympiad, Problem 3
Tags: inequalities , algebra
Let $a$ and $b$ be positive numbers. Prove the inequality $$\sqrt[3]{\frac ab}+\sqrt[3]{\frac ba}\le\sqrt[3]{2(a+b)\left(\frac1a+\frac1b\right)}.$$

2017 Pan-African Shortlist, A6
Tags: algebra , polynomial , real root , inequalities
Let $n \geq 1$ be an integer, and $a_0, a_1, \dots, a_{n-1}$ be real numbers such that \[ 1 \geq a_{n-1} \geq a_{n-2} \geq \dots \geq a_1 \geq a_0 \geq 0. \] We assume that $\lambda$ is a real root of the polynomial \[ x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0. \] Prove that $|\lambda| \leq 1$.

1997 IMO Shortlist, 26
Tags: inequalities , algebra , minimization , sequence , fibonacci
For every integer $ n \geq 2$ determine the minimum value that the sum $ \sum^n_{i\equal{}0} a_i$ can take for nonnegative numbers $ a_0, a_1, \ldots, a_n$ satisfying the condition $ a_0 \equal{} 1,$ $ a_i \leq a_{i\plus{}1} \plus{} a_{i\plus{}2}$ for $ i \equal{} 0, \ldots, n \minus{} 2.$

2010 BMO TST, 4
Tags: inequalities
Let's consider the inequality $ a^3\plus{}b^3\plus{}c^3<k(a\plus{}b\plus{}c)(ab\plus{}bc\plus{}ca)$ where $ a,b,c$ are the sides of a triangle and $ k$ a real number. [b]a)[/b] Prove the inequality for $ k\equal{}1$. [b]b) [/b]Find the smallest value of $ k$ such that the inequality holds for all triangles.

2019 China Western Mathematical Olympiad, 6
Tags: inequalities
Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be positive numbers such that $a_1\leq a_2 \leq \cdots \leq a_n .$ Prove that $$\sum_{1\leq i< j \leq n} (a_i+a_j)^2\left(\frac{1}{i^2}+\frac{1}{j^2}\right)\geq 4(n-1)\sum_{i=1}^{n}\frac{a^2_i}{i^2}.$$

Today's calculation of integrals, 870
Tags: calculus , integration , ellipse , hyperbola , geometry , inequalities , conic
Consider the ellipse $E: 3x^2+y^2=3$ and the hyperbola $H: xy=\frac 34.$ (1) Find all points of intersection of $E$ and $H$. (2) Find the area of the region expressed by the system of inequality \[\left\{ \begin{array}{ll} 3x^2+y^2\leq 3 &\quad \\ xy\geq \frac 34 , &\quad \end{array} \right.\]

1970 IMO Longlists, 36
Tags: inequalities
Let $x, y, z$ be non-negative real numbers satisfying \[x^2 + y^2 + z^2 = 5 \quad \text{ and } \quad yz + zx + xy = 2.\] Which values can the greatest of the numbers $x^2 -yz, y^2 - xz$ and $z^2 - xy$ have?

1992 Vietnam National Olympiad, 2
Tags: geometry , geometric transformation , rotation , rectangle , inequalities
Let $H$ be a rectangle with angle between two diagonal $\leq 45^{0}$. Rotation $H$ around the its center with angle $0^{0}\leq x\leq 360^{0}$ we have rectangle $H_{x}$. Find $x$ such that $[H\cap H_{x}]$ minimum, where $[S]$ is area of $S$.

2020 Saint Petersburg Mathematical Olympiad, 4.
Tags: inequalities , number theory
Let $m$ be a given positive integer. Prove that there exists a positive integer $k$ such that it holds $$1\leq \frac{1^m+2^m+3^m+\ldots +(k-1)^m}{k^m}<2.$$

1983 All Soviet Union Mathematical Olympiad, 365
Tags: geometry , rectangle , geometric inequality , inequalities , area
One side of the rectangle is $1$ cm. It is known that the rectangle can be divided by two orthogonal lines onto four rectangles, and each of the smaller rectangles has the area not less than $1$ square centimetre, and one of them is not less than $2$ square centimetres. What is the least possible length of another side of big rectangle?

1981 Miklós Schweitzer, 6
Tags: function , inequalities , trigonometry , integration , real analysis
Let $ f$ be a strictly increasing, continuous function mapping $ I=[0,1]$ onto itself. Prove that the following inequality holds for all pairs $ x,y \in I$: \[ 1-\cos (xy) \leq \int_0^xf(t) \sin (tf(t))dt + \int_0^y f^{-1}(t) \sin (tf^{-1}(t)) dt .\] [i]Zs. Pales[/i]

PEN E Problems, 15
Tags: inequalities , number theory , logarithm
Show that there exist two consecutive squares such that there are at least $1000$ primes between them.

2009 Estonia Team Selection Test, 1
Tags: inequalities , algebra
For arbitrary pairwise distinct positive real numbers $a, b, c$, prove the inequality $$\frac{(a^2- b^2)^3 + (b^2-c^2)^3+(c^2-a^2)^3}{(a- b)^3 + (b-c)^3+(c-a)^3}> 8abc$$

2003 AMC 12-AHSME, 25
Tags: algebra , function , domain , parabola , calculus , inequalities , conic
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}$

2009 USA Team Selection Test, 9
Tags: inequalities , search , function , calculus , derivative , algebra , polynomial
Prove that for positive real numbers $x$, $y$, $z$, \[ x^3(y^2+z^2)^2 + y^3(z^2+x^2)^2+z^3(x^2+y^2)^2 \geq xyz\left[xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2\right].\] [i]Zarathustra (Zeb) Brady.[/i]

2010 Contests, 2
Tags: function , inequalities , cauchy inequality , absolute value , algebra
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying \[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\] [i]Marko Radovanović, Serbia[/i]

2013 Bosnia And Herzegovina - Regional Olympiad, 1
Tags: algebra , inequalities
If $x$ and $y$ are real numbers such that $x^{2013}+y^{2013}>x^{2012}+y^{2012}$, prove that $x^{2014}+y^{2014}>x^{2013}+y^{2013}$

2000 All-Russian Olympiad Regional Round, 10.5
Tags: algebra , inequalities , function , functional , trigonometry
Is there a function $f(x)$ defined for all $x \in R$ and for all $x, y \in R $ satisfying the inequality $$|f(x + y) + \sin x + \sin y| < 2?$$

2016 Czech And Slovak Olympiad III A, 4
Tags: inequalities , maximum value , maximum , algebra
For positive numbers $a, b, c$ holds $(a + c) (b^2 + a c) = 4a$. Determine the maximum value of $b + c$ and find all triplets of numbers $(a, b, c)$ for which expression takes this value