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

2011 Kazakhstan National Olympiad, 6
Tags: inequalities , integration , algebra , quadratic
Given a positive integer $n$. One of the roots of a quadratic equation $x^{2}-ax +2 n = 0$ is equal to $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}$. Prove that $2\sqrt{2n}\le a\le 3\sqrt{n}$

2007 Pre-Preparation Course Examination, 7
Tags: number theory , modular arithmetic , inequalities
Let $p$ be a prime such that $p \equiv 3 \pmod 4$. Prove that we can't partition the numbers $a,a+1,a+2,\cdots,a+p-2$,($a \in \mathbb Z$) in two sets such that product of members of the sets be equal.

1997 German National Olympiad, 2
Tags: inequalities , number theory , greatest odd divisor , divisor
For a positive integer $k$, let us denote by $u(k)$ the greatest odd divisor of $k$. Prove that, for each $n \in N$, $\frac{1}{2^n} \sum_{k = 1}^{2^n} \frac{u(k)}{k}> \frac{2}{3}$.

2023 Polish Junior Math Olympiad First Round, 5.
Tags: algebra , inequalities
Positive numbers $a$, $b$, $c$ satisfy the inequalities \[a + b \geq ab, \quad b + c \geq bc,\quad\text{and}\quad c+ a \geq ca.\] Prove that $\displaystyle a + b + c \geq \frac34abc$.

2006 AMC 12/AHSME, 24
Tags: analytic geometry , inequalities , algebra , binomial theorem , geometry
The expression \[ (x \plus{} y \plus{} z)^{2006} \plus{} (x \minus{} y \minus{} z)^{2006} \]is simplified by expanding it and combining like terms. How many terms are in the simplified expression? $ \textbf{(A) } 6018 \qquad \textbf{(B) } 671,676 \qquad \textbf{(C) } 1,007,514 \qquad \textbf{(D) } 1,008,016 \qquad \textbf{(E) } 2,015,028$

2013 District Olympiad, 3
Tags: inequalities
Let $n\in {{\mathbb{N}}^{*}}$ and ${{a}_{1}},{{a}_{2}},...,{{a}_{n}}\in \mathbb{R}$ so ${{a}_{1}}+{{a}_{2}}+...+{{a}_{k}}\le k,\left( \forall \right)k\in \left\{ 1,2,...,n \right\}.$Prove that $\frac{{{a}_{1}}}{1}+\frac{{{a}_{2}}}{2}+...+\frac{{{a}_{n}}}{n}\le \frac{1}{1}+\frac{1}{2}+...+\frac{1}{n}$

2001 Federal Math Competition of S&M, Problem 3
Tags: inequalities
Let $p_{1}, p_{2},...,p_{n}$, where $n>2$, be the first $n$ prime numbers. Prove that $\frac{1}{p_{1}^2}+\frac{1}{p_{2}^2}+...+\frac{1}{p_{n}^2}+\frac{1}{p_{1}p_{2}...p_{n}}<\frac{1}{2}$

2013 Romania Team Selection Test, 1
Tags: trigonometry , inequalities
Let $n$ be a positive integer and let $x_1$, $\ldots$, $x_n$ be positive real numbers. Show that: \[ \min\left ( x_1,\frac{1}{x_1}+x_2, \cdots,\frac{1}{x_{n-1}}+x_n,\frac{1}{x_n} \right )\leq 2\cos \frac{\pi}{n+2} \leq\max\left ( x_1,\frac{1}{x_1}+x_2, \cdots,\frac{1}{x_{n-1}}+x_n,\frac{1}{x_n} \right ). \]

2014 VJIMC, Problem 3
Tags: inequalities
Let $n\ge2$ be an integer and let $x>0$ be a real number. Prove that $$\left(1-\sqrt{\tanh x}\right)^n+\sqrt{\tanh(nx)}<1.$$

2013 Math Prize For Girls Problems, 13
Tags: inequalities , function , probability
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?

2002 IMO Shortlist, 2
Tags: geometric inequality , geometry , homothety , triangle , inequalities
Let $ABC$ be a triangle for which there exists an interior point $F$ such that $\angle AFB=\angle BFC=\angle CFA$. Let the lines $BF$ and $CF$ meet the sides $AC$ and $AB$ at $D$ and $E$ respectively. Prove that \[ AB+AC\geq4DE. \]

2004 Alexandru Myller, 3
Tags: trigonometry , inequalities
Consider three real numbers $ x,y,z $ satisfying $ \cos x+\cos y+\cos z =\cos 3x +\cos 3y +\cos 3z=0. $ Show that $ \cos 2x\cdot \cos 2y\cdot\cos 2z\le 0. $ [i]Bogdan Enescu[/i]

2006 Italy TST, 2
Tags: inequalities , trigonometry , geometry , circumcircle
Let $ABC$ be a triangle, let $H$ be the orthocentre and $L,M,N$ the midpoints of the sides $AB, BC, CA$ respectively. Prove that \[HL^{2} + HM^{2} + HN^{2} < AL^{2} + BM^{2} + CN^{2}\] if and only if $ABC$ is acute-angled.

2012 China Team Selection Test, 2
Tags: vector , inequalities , function
Given two integers $m,n$ which are greater than $1$. $r,s$ are two given positive real numbers such that $r<s$. For all $a_{ij}\ge 0$ which are not all zeroes,find the maximal value of the expression \[f=\frac{(\sum_{j=1}^{n}(\sum_{i=1}^{m}a_{ij}^s)^{\frac{r}{s}})^{\frac{1}{r}}}{(\sum_{i=1}^{m})\sum_{j=1}^{n}a_{ij}^r)^{\frac{s}{r}})^{\frac{1}{s}}}.\]

1989 IberoAmerican, 2
Tags: trigonometry , inequalities
Let $x,y,z$ be real numbers such that $0\le x,y,z\le\frac{\pi}{2}$. Prove the inequality \[\frac{\pi}{2}+2\sin x\cos y+2\sin y\cos z\ge\sin 2x+\sin 2y+\sin 2z.\]

2010 Indonesia TST, 4
Tags: inequalities , algebra
Given a positive integer $n$ and $I = \{1, 2,..., k\}$ with $k$ is a positive integer. Given positive integers $a_1, a_2, ..., a_k$ such that for all $i \in I$: $1 \le a_i \le n$ and $$\sum_{i=1}^k a_i \ge 2(n!).$$ Show that there exists $J \subseteq I$ such that $$n! + 1 \ge \sum_{j \in J}a_j >\sqrt {n! + (n - 1)n}$$

2008 Brazil Team Selection Test, 3
Tags: inequalities , algebra
If $a, b, c$ and $d$ are positive real numbers such that $a + b + c + d = 2$, prove that $$\frac{a^2}{(a^2+1)^2}+\frac{b^2}{(b^2+1)^2}+\frac{c^2}{(c^2+1)^2}+\frac{d^2}{(d^2+1)^2} \le \frac{16}{25}$$

2007 Tournament Of Towns, 1
Tags: inequalities
Let $n$ be a positive integer. In order to find the integer closest to $\sqrt n$, Mary finds $a^2$, the closest perfect square to $n$. She thinks that a is then the number she is looking for. Is she always correct?

2010 Argentina National Olympiad, 3
Tags: number theory , algebra , inequalities
The positive integers $a,b,c$ are less than $99$ and satisfy $a^2+b^2=c^2+99^2$. . Find the minimum and maximum value of $a+b+c$.

2012 Putnam, 4
Tags: arithmetic sequence , modular arithmetic , inequalities
Let $q$ and $r$ be integers with $q>0,$ and let $A$ and $B$ be intervals on the real line. Let $T$ be the set of all $b+mq$ where $b$ and $m$ are integers with $b$ in $B,$ and let $S$ be the set of all integers $a$ in $A$ such that $ra$ is in $T.$ Show that if the product of the lengths of $A$ and $B$ is less than $q,$ then $S$ is the intersection of $A$ with some arithmetic progression.

2001 Abels Math Contest (Norwegian MO), 3b
Tags: geometric inequalities , area , triangle area , inequalities , geometry
The diagonals $AC$ and $BD$ in the convex quadrilateral $ABCD$ intersect in $S$. Let $F_1$ and $F_2$ be the areas of $\vartriangle ABS$ and $\vartriangle CSD$. and let $F$ be the area of the quadrilateral $ABCD$. Show that $\sqrt{ F_1 }+\sqrt{ F_2}\le \sqrt{ F}$

2000 Macedonia National Olympiad, 3
Tags: circumcircle , power of a point , geometry , inequalities
In a triangle with sides $a,b,c,t_a,t_b,t_c$ are the corresponding medians and $D$ the diameter of the circumcircle. Prove that \[\frac{a^2+b^2}{t_c}+\frac{b^2+c^2}{t_a}+\frac{c^2+a^2}{t_b}\le 6D\]

2003 Singapore MO Open, 2
Tags: algebra , max , inequalities
Find the maximum value of $\frac{xyz}{(1 + 5x)(4x + 3y)(5y + 6z)(z + 18)}$ as $x, y$ and $z$ range over the set of all positive real numbers. Justify your answer.

2007 Macedonia National Olympiad, 1
Tags: inequalities
Let $a, b, c$ be positive real numbers. Prove that \[1+\frac{3}{ab+bc+ca}\geq\frac{6}{a+b+c}.\]

2005 National High School Mathematics League, 1
Tags: inequalities
The maximum value of $k$ such that the enequality $\sqrt{x-3}+\sqrt{6-x}\geq k$ has a real solution is $\text{(A)}\sqrt6-\sqrt3\qquad\text{(B)}\sqrt3\qquad\text{(C)}\sqrt3+\sqrt6\qquad\text{(D)}\sqrt6$