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

1996 Romania Team Selection Test, 9
Tags: inequalities
Let $ n\geq 3 $ be an integer and let $ x_1,x_2,\ldots,x_{n-1} $ be nonnegative integers such that \begin{eqnarray*} \ x_1 + x_2 + \cdots + x_{n-1} &=& n \\ x_1 + 2x_2 + \cdots + (n-1)x_{n-1} &=& 2n-2. \end{eqnarray*} Find the minimal value of $ F(x_1,x_2,\ldots,x_n) = \sum_{k=1}^{n-1} k(2n-k)x_k $.

2008 Romania Team Selection Test, 4
Tags: graph theory , induction , combinatorics , inequalities
Let $ n$ be a nonzero positive integer. A set of persons is called a $ n$-balanced set if in any subset of $ 3$ persons there exists at least two which know each other and in each subset of $ n$ persons there are two which don't know each other. Prove that a $ n$-balanced set has at most $ (n \minus{} 1)(n \plus{} 2)/2$ persons.

2010 Junior Balkan MO, 4
Tags: geometry , rectangle , inequalities , rotation , combinatorics
A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares. Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.

2008 Middle European Mathematical Olympiad, 1
Tags: algebra , inequalities
Let $ (a_n)^{\infty}_{n\equal{}1}$ be a sequence of integers with $ a_{n} < a_{n\plus{}1}, \quad \forall n \geq 1.$ For all quadruple $ (i,j,k,l)$ of indices such that $ 1 \leq i < j \leq k < l$ and $ i \plus{} l \equal{} j \plus{} k$ we have the inequality $ a_{i} \plus{} a_{l} > a_{j} \plus{} a_{k}.$ Determine the least possible value of $ a_{2008}.$

2011 Hanoi Open Mathematics Competitions, 4
Tags: algebra , inequalities
Among the five statements on real numbers below, how many of them are correct? "If $a < b < 0$ then $a < b^2$" , "If $0 < a < b$ then $a < b^2$", "If $a^3 < b^3$ then $a < b$", "If $a^2 < b^2$ then $a < b$", "If $|a| < |b|$ then $a < b$", (A) $0$, (B) $1$, (C) $2$, (D) $3$, (E) $4$

2008 Canada National Olympiad, 3
Tags: algebra , inequalities , trigonometry
Let $ a$, $ b$, $ c$ be positive real numbers for which $ a \plus{} b \plus{} c \equal{} 1$. Prove that \[ {{a\minus{}bc}\over{a\plus{}bc}} \plus{} {{b\minus{}ca}\over{b\plus{}ca}} \plus{} {{c\minus{}ab}\over{c\plus{}ab}} \leq {3 \over 2}.\]

2011 Mathcenter Contest + Longlist, 6 sl8
Tags: inequalities , geometric inequality
Let $x,y,z$ represent the side lengths of any triangle, and $s=\dfrac{x+y+z}{2}$ and the area of this triangle be $\sqrt{s}$ square units. Prove that $$s\Big(\frac{1}{x(s-x)^2}+\frac{1}{y(s-y)^2}+\frac{1}{z(s-z)^ 2} \Big)\ge \frac{1}{2} \Big(\frac{1}{s-x}+\frac{1}{s-y}+\frac{1}{s-z}\Big)$$ [i](Zhuge Liang)[/i]

2007 Balkan MO Shortlist, C2
Tags: combinatorics , function , inequalities , algebra , domain
Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find \[\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. \]

2000 Junior Balkan Team Selection Tests - Romania, 3
Tags: inequalities , algebra
Find all real numbers $ a $ such that $ x,y>a\implies x+y+xy>a. $ [i]Gheorghe Iurea[/i]

2005 Korea - Final Round, 2
Tags: induction , quadratic , inequalities
Let $(a_{n})_{n=1}^{\infty}$ be a sequence of positive real numbers and let $\alpha_{n}$ be the arithmetic mean of $a_{1},..., a_{n}$ . Prove that for all positive integers $N$ , \[\sum_{n=1}^{N}\alpha_{n}^{2}\leq 4\sum_{n=1}^{N}a_{n}^{2}. \]

1970 Bulgaria National Olympiad, Problem 5
Tags: inequalities , geometry , geometric inequalities , polygon
Prove that for $n\ge5$ the side of regular inscribable $n$-gon is bigger than the side of regular $n+1$-gon circumscribed around the same circle and if $n\le4$ the opposite statement is true.

2011 Brazil Team Selection Test, 1
Tags: number theory , quadratic , inequalities
Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.

2000 Mongolian Mathematical Olympiad, Problem 5
Tags: combinatorics , inequalities
Given a natural number $n$, find the number of quadruples $(x,y,u,v)$ of integers with $1\le x,y,y,v\le n$ satisfy the following inequalities: \begin{align*} &1\le v+x-y\le n,\\ &1\le x+y-u\le n,\\ &1\le u+v-y\le n,\\ &1\le v+x-u\le n. \end{align*}

1987 IMO Longlists, 75
Tags: inequalities
Let $a_k$ be positive numbers such that $a_1 \geq 1$ and $a_{k+1} -a_k \geq 1 \ (k = 1, 2, . . . )$. Prove that for every $n \in \mathbb N,$ \[\sum_{k=1}^{1987}\frac{1}{a_{k+1} \sqrt[1987]{a_k}} <1987\]

1972 IMO Longlists, 35
Tags: function , inequalities
$(a)$ Prove that for $a, b, c, d \in\mathbb{R}, m \in [1,+\infty)$ with $am + b =-cm + d = m$, \[(i)\sqrt{a^2 + b^2}+\sqrt{c^2 + d^2}+\sqrt{(a-c)^2 + (b-d)^2}\ge \frac{4m^2}{1+m^2},\text{ and}\] \[(ii) 2 \le \frac{4m^2}{1+m^2} < 4.\] $(b)$ Express $a, b, c, d$ as functions of $m$ so that there is equality in $(i).$

2024 Serbia JBMO TST, 2
Tags: inequalities
Let $a, b, c$ be positive reals such that $ab+bc+ca=\frac{3}{4}$. Show that $$(a+b+c)^6 \geq (\frac{9} {8})^3(1+(a+b)^2)(1+(b+c)^2)(1+(c+a)^2).$$ When does equality hold?

2018 Serbia JBMO TST, 2
Tags: inequalities , algebra
Show that for $a,b,c > 0$ the following inequality holds: $\frac{\sqrt{ab}}{a+b+2c}+\frac{\sqrt{bc}}{b+c+2a}+\frac{\sqrt{ca}}{c+a+2b} \le \frac {3}{4}$.

MathLinks Contest 2nd, 2.1
Tags: inequalities
Given are six reals $a, b, c, x, y, z$ such that $(a + b + c)(x + y + z) = 3$ and $(a^2 + b^2 + c^2)(x^2 + y^2 + z^2) = 4$. Prove that $ax + by + cz \ge 0$.

2023 Bulgaria National Olympiad, 5
Tags: algebra , inequalities , n-variable inequality
For every positive integer $n$ determine the least possible value of the expression \[|x_{1}|+|x_{1}-x_{2}|+|x_{1}+x_{2}-x_{3}|+\dots +|x_{1}+x_{2}+\dots +x_{n-1}-x_{n}|\] given that $x_{1}, x_{2}, \dots , x_{n}$ are real numbers satisfying $|x_{1}|+|x_{2}|+\dots+|x_{n}| = 1$.

2013 China Team Selection Test, 3
Tags: inequalities , algebra
Let $n>1$ be an integer and let $a_0,a_1,\ldots,a_n$ be non-negative real numbers. Definite $S_k=\sum_{i\equal{}0}^k \binom{k}{i}a_i$ for $k=0,1,\ldots,n$. Prove that\[\frac{1}{n} \sum_{k\equal{}0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k\equal{}0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.\]

2012 AMC 8, 20
Tags: inequalities
What is the correct ordering of the three numbers $\frac5{19}$, $\frac7{21}$, and $\frac9{23}$, in increasing order? $\textbf{(A)}\hspace{.05in}\dfrac9{23} < \dfrac7{21} < \dfrac5{19}$ $\textbf{(B)}\hspace{.05in}\dfrac5{19} < \dfrac7{21} < \dfrac9{23}$ $\textbf{(C)}\hspace{.05in}\dfrac9{23} < \dfrac5{19} < \dfrac7{21}$ $\textbf{(D)}\hspace{.05in}\dfrac5{19} < \dfrac9{23} < \dfrac7{21}$ $\textbf{(E)}\hspace{.05in}\dfrac7{21} < \dfrac5{19} < \dfrac9{23}$

1996 China National Olympiad, 2
Tags: inequalities , trigonometry , derivative , integration , calculus
Let $n$ be a natural number. Suppose that $x_0=0$ and that $x_i>0$ for all $i\in\{1,2,\ldots ,n\}$. If $\sum_{i=1}^nx_i=1$ , prove that \[1\leq\sum_{i=1}^{n} \frac{x_i}{\sqrt{1+x_0+x_1+\ldots +x_{i-1}}\sqrt{x_i+\ldots+x_n}} < \frac{\pi}{2} \]

2014 Singapore MO Open, 3
Tags: inequalities , n-variable inequality
Let $0<a_1<a_2<\cdots <a_n$ be real numbers. Prove that \[\left (\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots +\frac{1}{1+a_n}\right )^2 \leq \frac{1}{a_1}+\frac{1}{a_2-a_1}+\cdots +\frac{1}{a_n-a_{n-1}}.\]

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

2014 Saudi Arabia Pre-TST, 4.2
Tags: inequalities , algebra
Given $x \ge 0$, prove that $$\frac{(x^2 + 1)^6}{2^7}+\frac12 \ge x^5 - x^3 + x$$