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

1992 Swedish Mathematical Competition, 4
Tags: inequalities , number theory , diophantine
Find all positive integers $a, b, c$ such that $a < b$, $a < 4c$, and $b c^3 \le a c^3 + b$.

2024-IMOC, A8
Tags: algebra , inequalities
$a$, $b$, $c$ are three distinct real numbers, given $\lambda >0$. Proof that \[\frac{1+ \lambda ^2a^2b^2}{(a-b)^2}+\frac{1+ \lambda ^2b^2c^2}{(b-c)^2}+\frac{1+ \lambda ^2c^2a^2}{(c-a)^2} \geq \frac 32 \lambda.\] [hide=Remark]Old problem, can be found [url=https://artofproblemsolving.com/community/c6h588854p3487434]here[/url]. Double post to have a cleaner thread for collection (as the original one contains a messy quote)[/hide]

1966 IMO Shortlist, 26
Tags: inequalities , n-variable inequality , algebra
Prove the inequality [b]a.)[/b] $ \left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left( a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) , $ where $k\geq 1$ is a natural number and $a_{1},$ $a_{2},$ $...,$ $a_{k}$ are arbitrary real numbers. [b]b.)[/b] Using the inequality (1), show that if the real numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ satisfy the inequality \[ a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left( a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) }, \] then all of these numbers $a_{1},$ $a_{2},$ $\ldots,$ $a_{n}$ are non-negative.

1987 Traian Lălescu, 2.1
Tags: inequalities
For any nonegative real $ a $ and natural $ n, $ prove that $$ \sqrt{a+1+\sqrt{a+2+\cdots +\sqrt{a+n}}} <a+3. $$

2018 Costa Rica - Final Round, 2
Tags: algebra , inequalities , max
Let $a, b, c$, and $d$ be real numbers. The six sums of two numbers $x$ and $y$, different from the previous four, are $117$, $510$, $411$, $252$, in no particular order. Determine the maximum possible value of $x + y$.

1993 Italy TST, 1
Tags: inequalities , algebra
Let $x_1,x_2,...,x_n$ ($n \ge 2$) be positive numbers with the sum $1$. Prove that $$\sum_{i=1}^{n} \frac{1}{\sqrt{1-x_i}} \ge n\sqrt{\frac{n}{n-1}} $$

2016 China National Olympiad, 1
Tags: inequalities , sequence
Let $a_1,a_2,\cdots, a_{31} ;b_1,b_2, \cdots, b_{31}$ be positive integers such that $a_1< a_2<\cdots< a_{31}\leq2015$ , $ b_1< b_2<\cdots<b_{31}\leq2015$ and $a_1+a_2+\cdots+a_{31}=b_1+b_2+\cdots+b_{31}.$ Find the maximum value of $S=|a_1-b_1|+|a_2-b_2|+\cdots+|a_{31}-b_{31}|.$

2009 Czech and Slovak Olympiad III A, 3
Tags: inequalities
Find the least value of $x>0$ such that for all positive real numbers $a,b,c,d$ satisfying $abcd=1$, the inequality $a^x+b^x+c^x+d^x\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$ is true.

2020 Tuymaada Olympiad, 2
Tags: inequality , inequalities , n-variable inequality
Given positive real numbers $a_1, a_2, \dots, a_n$. Let \[ m = \min \left( a_1 + \frac{1}{a_2}, a_2 + \frac{1}{a_3}, \dots, a_{n - 1} + \frac{1}{a_n} , a_n + \frac{1}{a_1} \right). \] Prove the inequality \[ \sqrt[n]{a_1 a_2 \dots a_n} + \frac{1}{\sqrt[n]{a_1 a_2 \dots a_n}} \ge m. \]

2014 Silk Road, 4
Tags: induction , inequalities , number theory
Find all $ f:N\rightarrow N$, such that $\forall m,n\in N $ $ 2f(mn) \geq f(m^2+n^2)-f(m)^2-f(n)^2 \geq 2f(m)f(n) $

1997 Slovenia Team Selection Test, 2
Tags: polynomial , functional , functional inequality , algebra , inequalities
Find all polynomials $p$ with real coefficients such that for all real $x$ , $xp(x)p(1-x)+x^3 +100 \ge 0$.

1964 German National Olympiad, 1
Tags: algebra , inequalities
Prove that for all positive, entirely rational numbers $a$ and $b$ always holds $$\frac{a + b}{2} \ge \sqrt[a+b]{a^b \cdot b^a}.$$ When does the equal sign hold?

2014 Thailand TSTST, 2
Tags: inequalities
Let $a, b, c$ be positive real numbers. Prove that $$\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}\geq\sqrt{\frac{2ab}{3a+b+2c}}+\sqrt{\frac{2bc}{3b+c+2a}}+\sqrt{\frac{2ca}{3c+a+2b}}.$$

2005 USAMO, 6
Tags: induction , modular arithmetic , ceiling function , floor function , pigeonhole principle , inequalities , logarithm
For $m$ a positive integer, let $s(m)$ be the sum of the digits of $m$. For $n\ge 2$, let $f(n)$ be the minimal $k$ for which there exists a set $S$ of $n$ positive integers such that $s\left(\sum_{x\in X} x\right)=k$ for any nonempty subset $X\subset S$. Prove that there are constants $0<C_1<C_2$ with \[C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.\]

2013 Chile National Olympiad, 3
Tags: algebra , inequalities
Given a finite sequence of real numbers $a_1,a_2,...,a_n$ such that $$a_1 + a_2 + ... + a_n > 0.$$ Prove that there is at least one index $ i$ such that $$a_i > 0, a_i + a_{i+1} > 0, ..., a_i + a_{i+1} + ...+ a_n > 0.$$

2010 Today's Calculation Of Integral, 578
Tags: calculus , integration , inequalities , function , algebra , domain , logarithm
Find the range of $ k$ for which the following inequality holds for $ 0\leq x\leq 1$. \[ \int_0^x \frac {dt}{\sqrt {(3 \plus{} t^2)^3}}\geq k\int _0^x \frac {dt}{\sqrt {3 \plus{} t^2}}\] If necessary, you may use $ \ln 3 \equal{} 1.10$.

JOM 2015 Shortlist, G4
Tags: inequalities
Let $ ABC $ be a triangle and let $ AD, BE, CF $ be cevians of the triangle which are concurrent at $ G $. Prove that if $ CF \cdot BE \ge AF \cdot EC + AE \cdot BF + BC \cdot FE $ then $ AG \le GD $.

1992 Miklós Schweitzer, 4
Tags: combinatorics , graph theory , inequalities
show there exist positive constants $c_1$ and $c_2$ such that for any $n\geq 3$, whenever $T_1$ and $T_2$ are two trees on the set of vertices $X = \{1, 2, ..., n\}$, there exists a function $f : X \to \{-1, +1\}$ for which $$\bigg | \sum_ {x \in P} f (x) \bigg | <c_1 \log n$$ for any path P that is a subgraph of $T_1$ or $T_2$ , but with an upper bound $c_2 \log n / \log \log n$ the statement is no longer true.

2004 Austrian-Polish Competition, 2
Tags: inequalities , trigonometry , angle bisector , geometry
In a triangle $ABC$ let $D$ be the intersection of the angle bisector of $\gamma$, angle at $C$, with the side $AB.$ And let $F$ be the area of the triangle $ABC.$ Prove the following inequality: \[2 \cdot \ F \cdot \left( \frac{1}{AD} -\frac{1}{BD} \right) \leq AB.\]

2016 China Team Selection Test, 2
Tags: inequalities , geometric inequality , geometry
Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$.

2001 Moldova National Olympiad, Problem 1
Tags: inequalities
Prove that $y\sqrt{3-2x}+x\sqrt{3-2y}\le x^2+y^2$ for any number $x,y\in\left[1,\frac32\right]$. When does equality occur?

2010 Tournament Of Towns, 3
Tags: invariant , inequalities
Each of $999$ numbers placed in a circular way is either $1$ or $-1$. (Both values appear). Consider the total sum of the products of every $10$ consecutive numbers. $(a)$ Find the minimal possible value of this sum. $(b)$ Find the maximal possible value of this sum.

2016 Dutch IMO TST, 1
Tags: algebra , inequalities
Prove that for all positive reals $a, b,c$ we have: $a +\sqrt{ab}+ \sqrt[3]{abc}\le \frac43 (a + b + c)$

2011 N.N. Mihăileanu Individual, 3
Tags: inequalities , real analysis , numerical analysis
Prove the inequalities $ 0<n\left( \sqrt[n]{2} -1 \right) -\left( \frac{1}{n+1} +\frac{1}{n+2} +\cdots +\frac{1}{n+n}\right) <\frac{1}{2n} , $ where $ n\ge 2. $ [i]Marius Cavachi[/i]

1957 Putnam, A3
Tags: trigonometry , inequalities , cosine
Let $a,b$ be real numbers and $k$ a positive integer. Show that $$ \left| \frac{ \cos kb \cos a - \cos ka \cos b}{\cos b -\cos a} \right|<k^2 -1$$ whenever the left side is defined.