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: 592

1983 Brazil National Olympiad, 5
Tags: positive integer , number theory , minimum , inequality
Show that $1 \le n^{1/n} \le 2$ for all positive integers $n$. Find the smallest $k$ such that $1 \le n ^{1/n} \le k$ for all positive integers $n$.

1979 IMO Shortlist, 19
Tags: number theory , recurrence relation , sequence , inequality
Consider the sequences $(a_n), (b_n)$ defined by \[a_1=3, \quad b_1=100 , \quad a_{n+1}=3^{a_n} , \quad b_{n+1}=100^{b_n} \] Find the smallest integer $m$ for which $b_m > a_{100}.$

Russian TST 2016, P1
Tags: algebra , polynomial , inequality
For which even natural numbers $d{}$ does there exists a constant $\lambda>0$ such that any reduced polynomial $f(x)$ of degree $d{}$ with integer coefficients that does not have real roots satisfies the inequality $f(x) > \lambda$ for all real numbers?

2009 Philippine MO, 4
Tags: algebra , inequality , minimum value
Let $k$ be a positive real number such that $$\frac{1}{k+a} + \frac{1}{k+b} + \frac{1}{k+c} \leq 1$$ for any positive positive real numbers $a$, $b$ and $c$ with $abc = 1$. Find the minimum value of $k$.

2021 China Team Selection Test, 4
Tags: inequality , algebra
Suppose $x_1,x_2,...,x_{60}\in [-1,1]$ , find the maximum of $$ \sum_{i=1}^{60}x_i^2(x_{i+1}-x_{i-1}),$$ where $x_{i+60}=x_i$.

2020 JBMO TST of France, 4
Tags: algebra , inequality
$a, b, c$ are real positive numbers for which $a+b+c=3$. Prove that $a^{12}+b^{12}+c^{12}+8(ab+bc+ca) \geq 27$

2020 Taiwan APMO Preliminary, P6
Tags: algebra , inequalities , inequality
Let $a,b,c$ be positive reals. Find the minimum value of $$\dfrac{13a+13b+2c}{2a+2b}+\dfrac{24a-b+13c}{2b+2c}+\dfrac{(-a+24b+13c)}{2c+2a}$$. (1)What is the minimum value? (2)If the minimum value occurs when $(a,b,c)=(a_0,b_0,c_0)$,then find $\frac{b_0}{a_0}+\frac{c_0}{b_0}$.

1999 Mongolian Mathematical Olympiad, Problem 2
Tags: inequality , inequalities
Let $a,b,c$ be the real numbers with $a\ge\frac85b>0$ and $a\ge c>0$. Prove the inequality $$\frac45\left(\frac1a+\frac1b\right)+\frac2c\ge\frac{27}2\cdot\frac1{a+b+c}.$$

2007 USAMO, 1
Tags: induction , number theory , inequality
Let $n$ be a positive integer. Define a sequence by setting $a_{1}= n$ and, for each $k > 1$, letting $a_{k}$ be the unique integer in the range $0\leq a_{k}\leq k-1$ for which $a_{1}+a_{2}+...+a_{k}$ is divisible by $k$. For instance, when $n = 9$ the obtained sequence is $9,1,2,0,3,3,3,...$. Prove that for any $n$ the sequence $a_{1},a_{2},...$ eventually becomes constant.

2021 Taiwan TST Round 2, A
Tags: algebra , inequality , inequalities
[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x . \] [i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$, \[ \sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x . \]

2014 Contests, A1
Tags: inequality , algebra
$\boxed{\text{A1}}$Let $a,b,c$ be positive reals numbers such that $a+b+c=1$.Prove that $2(a^2+b^2+c^2)\ge \frac{1}{9}+15abc$

Russian TST 2021, P1
Tags: algebra , inequality , 4-variable inequality , inequalities
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

2024 APMO, 3
Tags: algebra , inequality , induction
Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$

2021 Federal Competition For Advanced Students, P2, 1
Tags: algebra , inequality
Let $a, b$ and $c$ be pairwise different natural numbers. Prove $\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c$. When does equality holds? (Karl Czakler)

2021 Science ON grade VIII, 4
Tags: inequality , algebra , inequalities
Consider positive real numbers $x,y,z$. Prove the inequality $$\frac 1x+\frac 1y+\frac 1z+\frac{9}{x+y+z}\ge 3\left (\left (\frac{1}{2x+y}+\frac{1}{x+2y}\right )+\left (\frac{1}{2y+z}+\frac{1}{y+2z}\right )+\left (\frac{1}{2z+x}+\frac{1}{x+2z}\right )\right ).$$ [i] (Vlad Robu \& Sergiu Novac)[/i]

2018 IFYM, Sozopol, 1
Tags: set , inequality , algebra
$A = \{a_1, a_2, . . . , a_k\}$ is a set of positive integers for which the sum of some (we can have only one number too) different numbers from the set is equal to a different number i.e. there $2^k - 1$ different sums of different numbers from $A$. Prove that the following inequality holds: $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}<2$

2012 Kosovo National Mathematical Olympiad, 2
Tags: geometry , inequality
Let $a,b,c$ be the lengths of the sides of a triangle. Prove that, $\left|\frac {a}{b}+\frac {b}{c}+\frac {c}{a}-\frac {b}{a}-\frac {c}{b}-\frac {a}{c}\right|<1$

KoMaL A Problems 2023/2024, A. 869
Tags: inequality , algebra
Let $A$ and $B$ be given real numbers. Let the sum of real numbers $0\le x_1\le x_2\le\ldots \le x_n$ be $A$, and let the sum of real numbers $0\le y_1\le y_2\le \ldots\le y_n$ be $B$. Find the largest possible value of \[\sum_{i=1}^n (x_i-y_i)^2.\] [i]Proposed by Péter Csikvári, Budapest[/i]

2018 China National Olympiad, 3
Tags: number theory , approximation , inequality , fractional part , diophantine approximation
Let $q$ be a positive integer which is not a perfect cube. Prove that there exists a positive constant $C$ such that for all natural numbers $n$, one has $$\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}}$$ where $\{ x \}$ denotes the fractional part of $x$.

1998 IMO Shortlist, 3
Tags: rearrangement inequality , algebra , inequality
Let $x,y$ and $z$ be positive real numbers such that $xyz=1$. Prove that \[ \frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)} \geq \frac{3}{4}. \]

2024 Nepal Mathematics Olympiad (Pre-TST), Problem 2
Tags: algebra , inequality
Let, $\displaystyle{S =\sum_{i=1}^{k} {n_i}^2}$. Prove that for $n_i \in \mathbb{R}^+$ $$\sum_{i=1}^{k} \frac{n_i}{S-n_i^2} \geq \frac{4}{n_1+n_2+ \cdots+ n_k}$$ [i]Proposed by Kang Taeyoung, South Korea[/i]

2019 Bulgaria National Olympiad, 1
Tags: inequality , inequalities
Let $f(x)=x^2+bx+1,$ where $b$ is a real number. Find the number of integer solutions to the inequality $f(f(x)+x)<0.$

2007 Junior Macedonian Mathematical Olympiad, 4
Tags: algebra , inequality
The numbers $a_{1}, a_{2}, ..., a_{20}$ satisfy the following conditions: $a_{1} \ge a_{2} \ge ... \ge a_{20} \ge 0$ $a_{1} + a_{2} = 20$ $a_{3} + a_{4} + ... + a_{20} \le 20$ . What is maximum value of the expression: $a_{1}^2 + a_{2}^2 + ... + a_{20}^2$ ? For which values of $a_{1}, a_{2}, ..., a_{20}$ is the maximum value achieved?

1987 IMO Longlists, 33
Tags: rearrangement inequality , three variable inequality , triangle inequality , inequality
Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then \[\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N \] [i]Proposed by Greece.[/i]

2001 Moldova National Olympiad, Problem 6
Tags: inequality , inequalities , algebra , sequence
Set $a_n=\frac{2n}{n^4+3n^2+4},n\in\mathbb N$. Prove that $\frac14\le a_1+a_2+\ldots+a_n\le\frac12$ for all $n$.