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

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Found problems: 6530

1986 All Soviet Union Mathematical Olympiad, 427
Tags: algebra , inequalities
Prove that the following inequality holds for all positive $\{a_i\}$: $$\frac{1}{a_1} + \frac{2}{a_1+a_2} + ... +\frac{ n}{a_1+...+a_n} < 4\left(\frac{1}{a_1} + ... + \frac{1}{a_n}\right)$$

2004 France Team Selection Test, 1
Tags: inequalities , calculus , derivative , function
Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that $a_1 + ... + a_n = b_1 + ... + b_n = 1$. Find the minimal value of $ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.

1985 Balkan MO, 2
Tags: trigonometry , function , inequalities , quadratic , algebra
Let $a,b,c,d \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ be real numbers such that $\sin{a}+\sin{b}+\sin{c}+\sin{d}=1$ and $\cos{2a}+\cos{2b}+\cos{2c}+\cos{2d}\geq \frac{10}{3}$. Prove that $a,b,c,d \in [0, \frac{\pi}{6}]$

2002 Federal Competition For Advanced Students, Part 1, 2
Tags: inequalities
Find the greatest real number $C$ such that, for all real numbers $x$ and $y \neq x$ with $xy = 2$ it holds that \[\frac{((x + y)^2 - 6)((x - y)^2 + 8)}{(x-y)^2}\geq C.\] When does equality occur?

2006 AMC 12/AHSME, 24
Tags: trigonometry , geometry , analytic geometry , 3d geometry , probability , quadratic , inequalities
Let $ S$ be the set of all points $ (x,y)$ in the coordinate plane such that $ 0\le x\le \frac \pi2$ and $ 0\le y\le \frac \pi2$. What is the area of the subset of $ S$ for which \[ \sin^2 x \minus{} \sin x\sin y \plus{} \sin^2 y\le \frac 34? \]$ \textbf{(A) } \frac {\pi^2}9 \qquad \textbf{(B) } \frac {\pi^2}8 \qquad \textbf{(C) } \frac {\pi^2}6\qquad \textbf{(D) } \frac {3\pi^2}{16} \qquad \textbf{(E) } \frac {2\pi^2}9$

2017 ELMO Problems, 6
Tags: inequalities , algebra
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$: (i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$ (ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$ [i]Proposed by Ashwin Sah[/i]

2016 JBMO Shortlist, 5
Tags: algebra , inequalities
Let $x,y,z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Prove that \[x+y+z\geq \sqrt{\frac{xy+1}{2}}+\sqrt{\frac{yz+1}{2}}+\sqrt{\frac{zx+1}{2}} \ .\] [i]Proposed by Azerbaijan[/i] [hide=Second Suggested Version]Let $x,y,z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Prove that \[x+y+z\geq \sqrt{\frac{x^2+1}{2}}+\sqrt{\frac{y^2+1}{2}}+\sqrt{\frac{z^2+1}{2}} \ .\][/hide]

1994 Polish MO Finals, 2
Tags: inequalities , vector , parallelogram , triangle inequality , geometry
A parallelopiped has vertices $A_1, A_2, ... , A_8$ and center $O$. Show that: \[ 4 \sum_{i=1}^8 OA_i ^2 \leq \left(\sum_{i=1}^8 OA_i \right) ^2 \]

2002 China Team Selection Test, 1
Tags: inequalities
Given $ n \geq 3$, $ n$ is a integer. Prove that: \[ (2^n \minus{} 2) \cdot \sqrt{2i\minus{}1} \geq \left( \sum_{j\equal{}0}^{i\minus{}1}C_n^j \plus{} C_{n\minus{}1}^{i\minus{}1} \right) \cdot \sqrt{n}\] where if $ n$ is even, then $ \displaystyle 1 \leq i \leq \frac{n}{2}$; if $ n$ is odd, then $ \displaystyle 1 \leq i \leq \frac{n\minus{}1}{2}$.

2013 Math Prize For Girls Problems, 13
Tags: probability , function , inequalities
Each of $n$ boys and $n$ girls chooses a random number from the set $\{ 1, 2, 3, 4, 5 \}$, uniformly and independently. Let $p_n$ be the probability that every boy chooses a different number than every girl. As $n$ approaches infinity, what value does $\sqrt[n]{p_n}$ approach?

2010 IberoAmerican Olympiad For University Students, 5
Tags: matrix , algebra , polynomial , inequalities , linear algebra
Let $A,B$ be matrices of dimension $2010\times2010$ which commute and have real entries, such that $A^{2010}=B^{2010}=I$, where $I$ is the identity matrix. Prove that if $\operatorname{tr}(AB)=2010$, then $\operatorname{tr}(A)=\operatorname{tr}(B)$.

2012 Indonesia TST, 3
Tags: inequalities
Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be positive reals such that \[a_1 + b_1 = a_2 + b_2 = \ldots + a_n + b_n\] and \[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge n.\] Prove that \[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge \dfrac{a_1+a_2+\ldots+a_n}{b_1+b_2+\ldots+b_n}.\]

2000 Romania National Olympiad, 2b
Tags: geometry , inequalities , geometric inequality
If $a, b, c$ represent the lengths of the sides of a triangle, prove that: $$\frac{a}{b-a+c}+ \frac{b}{b-a+c}+ \frac{c}{b-a+c} \ge 3$$

2024 Macedonian Mathematical Olympiad, Problem 5
Tags: function , inequalities
Let $f:\mathbb{N} \rightarrow \mathbb{N} \setminus \left \{ 1 \right \}$ be a function which satisfies both the inequality $f(a+f(a)) \leq 2a+3$ and the equation $$f(f(a)+b) = f(a+f(b)),$$ for any two $a,b \in \mathbb{N}$. Let $g:\mathbb{N} \rightarrow \mathbb{N}$ be defined with: $g(a)$ is the largest prime divisor of $f(a)$. Prove that there exist integers $a>b>2024$ such that $b|a$ and $g(a) = g(b)$.

1974 Putnam, B5
Tags: factorial , inequalities
Show that $$1+\frac{n}{1!} + \frac{n^{2}}{2!} +\ldots+ \frac{n^{n}}{n!} > \frac{e^{n}}{2}$$ for every integer $n\geq 0.$

2016 Stars of Mathematics, 3
Tags: inequalities
Let $ n $ be a natural number, and $ 2n $ nonnegative real numbers $ a_1,a_2,\ldots ,a_{2n} $ such that $ a_1a_2\cdots a_{2n}=1. $ Show that $$ 2^{n+1} +\left( a_1^2+a_2^2 \right)\left( a_3^2+a_4^2 \right)\cdots\left( a_{2n-1}^2+a_{2n}^2 \right) \ge 3\left( a_1+a_2 \right)\left( a_3+a_4 \right)\cdots\left( a_{2n-1}+a_{2n} \right) , $$ and specify in which circumstances equality happens.

1984 IMO Longlists, 5
Tags: inequalities
For a real number $x$, let $[x]$ denote the greatest integer not exceeding $x$. If $m \ge 3$, prove that \[\left[\frac{m(m+1)}{2(2m-1)}\right]=\left[\frac{m+1}{4}\right]\]

2018 JBMO TST-Turkey, 8
Tags: algebra , triangle inequality , positive real numbers , inequalities
Let $x, y, z$ be positive real numbers such that $\sqrt {x}, \sqrt {y}, \sqrt {z}$ are sides of a triangle and $\frac {x}{y}+\frac {y}{z}+\frac {z}{x}=5$. Prove that $\frac {x(y^2-2z^2)}{z}+\frac {y(z^2-2x^2)}{x}+\frac {z(x^2-2y^2)}{y}\geqslant0$

2012 Balkan MO, 4
Tags: function , inequalities , functional equation
Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold: (i) $f(n!)=f(n)!$ for every positive integer $n$, (ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.

VMEO II 2005, 8
Tags: inequalities
If a,b,c>0, prove that: \[ \frac{1}{a\sqrt{(a+b)}}+\frac{1}{b\sqrt{(b+c)}}+\frac{1}{c\sqrt{(c+a)}} \geq \frac{3}{\sqrt{2abc}} \] thank u for ur help :oops:

2021-IMOC, A7
Tags: algebra , inequality , inequalities
For any positive reals $a,b,c,d$ that satisfy $a^2 + b^2 + c^2 + d^2 = 4,$ show that $$\frac{a^3}{a+b} + \frac{b^3}{b+c} + \frac{c^3}{c+d} + \frac{d^3}{d+a} + 4abcd \leq 6.$$

2007 Hong Kong TST, 1
Tags: inequalities , hyperbola , algebra , conic
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 1 Let $p,q,r$ and $s$ be real numbers such that $p^{2}+q^{2}+r^{2}-s^{2}+4=0$. Find the maximum value of $3p+2q+r-4|s|$.

2013 Hanoi Open Mathematics Competitions, 12
Tags: algebra , quadratic , inequalities , root
If $f(x) = ax^2 + bx + c$ satisfies the condition $|f(x)| < 1; \forall x \in [-1, 1]$, prove that the equation $f(x) = 2x^2 - 1$ has two real roots.

2006 Iran MO (3rd Round), 1
Tags: floor function , inequalities , ceiling function , combinatorics
Let $A$ be a family of subsets of $\{1,2,\ldots,n\}$ such that no member of $A$ is contained in another. Sperner’s Theorem states that $|A|\leq{n\choose{\lfloor\frac{n}{2}\rfloor}}$. Find all the families for which the equality holds.

2013 Turkey MO (2nd round), 2
Tags: inequalities
Find the maximum value of $M$ for which for all positive real numbers $a, b, c$ we have \[ a^3+b^3+c^3-3abc \geq M(ab^2+bc^2+ca^2-3abc) \]