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

2004 Germany Team Selection Test, 1
Tags: calculus , inequalities , sequence , algebra , bounded
Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$. (1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded? (2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$? Justify your answer.

2011 Morocco TST, 3
Tags: geometry , incenter , circumcircle , inequalities
The vertices $X, Y , Z$ of an equilateral triangle $XYZ$ lie respectively on the sides $BC, CA, AB$ of an acute-angled triangle $ABC.$ Prove that the incenter of triangle $ABC$ lies inside triangle $XYZ.$ [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

2008 Mathcenter Contest, 5
Tags: algebra , inequalities
Let $a,b,c$ be positive real numbers where $ab+bc+ca = 3$. Prove that $$\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}\geq\dfrac{3} {2}.$$ [i](dektep)[/i]

2004 Moldova Team Selection Test, 9
Tags: inequalities
Let $a,b$ and $c$ be positive real numbers . Prove that\[\left | \frac{4(b^3-c^3)}{b+c}+ \frac{4(c^3-a^3)}{c+a}+ \frac{4(a^3-b^3)}{a+b} \right |\leq (b-c)^2+(c-a)^2+(a-b)^2.\]

2010 Victor Vâlcovici, 3
Tags: geometry , area , inequalities , center of mass , vector geometry
$ A',B',C' $ are the feet of the heights of an acute-angled triangle $ ABC. $ Calculate $$ \frac{\text{area} (ABC)}{\text{area}\left( A'B'C'\right)} , $$ knowing that $ ABC $ and $ A'B'C' $ have the same center of mass. [i]Carmen[/i] and [i]Viorel Botea[/i]

2011 IFYM, Sozopol, 2
Tags: inequalities
prove that $(\frac{1}{a+c}+\frac{1}{b+d})(\frac{1}{\frac{1}{a}+\frac{1}{c}}+\frac{1}{\frac{1}{b}+\frac{1}{d}}) \leq 1$ for $0 < a < b \leq c < d$ and when $(\frac{1}{a+c}+\frac{1}{b+d})(\frac{1}{\frac{1}{a}+\frac{1}{c}}+\frac{1}{\frac{1}{b}+\frac{1}{d}}) = 1 $

MathLinks Contest 3rd, 2
Tags: inequalities
Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC, CA, AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all three semicircles has radius $t$. Prove that $$\frac{s}{2} < t \le \frac{s}{2} + \left( 1 - \frac{\sqrt3}{2} \right)r.$$

2010 Romanian Master of Mathematics, 4
Tags: inequalities , algebra , polynomial , function , ceiling function
Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions: (i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers); (ii) $|a_1-b_1|+|a_2-b_2|=2010$; (iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$; (iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$. [i]Massimo Gobbino, Italy[/i]

2008 China Team Selection Test, 5
Tags: inequalities , algebra , function , cauchy inequality
For two given positive integers $ m,n > 1$, let $ a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m)$ be nonnegative real numbers, not all zero, find the maximum and the minimum values of $ f$, where \[ f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\sum_{j = 1}^{m}a_{ij})^2 + mn\sum_{i = 1}^{n}\sum_{j=1}^{m}a_{ij}^2}. \]

2000 USA Team Selection Test, 6
Tags: trig identities , trigonometry , circumcircle , geometry , law of sines , inequalities
Let $ ABC$ be a triangle inscribed in a circle of radius $ R$, and let $ P$ be a point in the interior of triangle $ ABC$. Prove that \[ \frac {PA}{BC^{2}} \plus{} \frac {PB}{CA^{2}} \plus{} \frac {PC}{AB^{2}}\ge \frac {1}{R}. \] [i]Alternative formulation:[/i] If $ ABC$ is a triangle with sidelengths $ BC\equal{}a$, $ CA\equal{}b$, $ AB\equal{}c$ and circumradius $ R$, and $ P$ is a point inside the triangle $ ABC$, then prove that $ \frac {PA}{a^{2}} \plus{} \frac {PB}{b^{2}} \plus{} \frac {PC}{c^{2}}\ge \frac {1}{R}$.

2020 Germany Team Selection Test, 1
Tags: algebra , inequalities
Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

2004 France Team Selection Test, 3
Tags: pigeonhole principle , triangle inequality , inequalities , complex number , analytic geometry , geometry
Each point of the plane with two integer coordinates is the center of a disk with radius $ \frac {1} {1000}$. Prove that there exists an equilateral triangle whose vertices belong to distinct disks. Prove that such a triangle has side-length greater than 96.

2012 Mediterranean Mathematics Olympiad, 2
Tags: inequalities , trigonometry
In an acute $\triangle ABC$, prove that \begin{align*}\frac{1}{3}\left(\frac{\tan^2A}{\tan B\tan C}+\frac{\tan^2 B}{\tan C\tan A}+\frac{\tan^2 C}{\tan A\tan B}\right) \\ +3\left(\frac{1}{\tan A+\tan B+\tan C}\right)^{\frac{2}{3}}\ge 2.\end{align*}

2003 Croatia National Olympiad, Problem 3
Tags: inequalities
For positive numbers $a_1,a_2,\ldots,a_n$ ($n\ge2$) denote $s=a_1+\ldots+a_n$. Prove that $$\frac{a_1}{s-a_1}+\ldots+\frac{a_n}{s-a_n}\ge\frac n{n-1}.$$

2000 Korea Junior Math Olympiad, 6
Tags: algebra , inequalities
$x, y, z$ are positive reals which their product is not smaller then their sum. Prove the inequality: $$\sqrt{2x^2+yz}+\sqrt{2y^2+zx}+\sqrt{2z^2+xy} \geq 9$$

2000 Taiwan National Olympiad, 1
Tags: number theory , inequalities
Find all pairs $(x,y)$ of positive integers such that $y^{x^2}=x^{y+2}$.

1993 Miklós Schweitzer, 6
Tags: geometry , inequalities
Let $P_1 , P_2 , ...$ be arbitrary points and A be a connected compact set in the plane with a diameter greater than 4. Show that for some point P in A , $\overline {PP_1} \cdot \overline {PP_2} \cdots \overline {PP_n}>1$. Furthermore, prove that this is no longer necessarily true for compact sets of diameter 4.

2002 China Team Selection Test, 1
Tags: geometry , inequalities
Given triangle $ ABC$ and $ AB\equal{}c$, $ AC\equal{}b$ and $ BC\equal{}a$ satisfying $ a \geq b \geq c$, $ BE$ and $ CF$ are two interior angle bisectors. $ P$ is a point inside triangle $ AEF$. $ R$ and $ Q$ are the projections of $ P$ on sides $ AB$ and $ AC$. Prove that $ PR \plus{} PQ \plus{} RQ < b$.

2014 Silk Road, 4
Tags: number theory , inequalities , induction
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) $

2009 Kosovo National Mathematical Olympiad, 2
Tags: inequalities
If $x_1$ and $x_2$ are the solutions of the equation $x^2-(m+3)x+m+2=0$ Find all real values of $m$ such that the following inequations are valid $\frac {1}{x_1}+\frac {1}{x_2}>\frac{1}{2}$ and $x_1^2+x_2^2<5$

2007 China Team Selection Test, 1
Tags: inequalities , trigonometry , quadratic , function
$ u,v,w > 0$,such that $ u \plus{} v \plus{} w \plus{} \sqrt {uvw} \equal{} 4$ prove that $ \sqrt {\frac {uv}{w}} \plus{} \sqrt {\frac {vw}{u}} \plus{} \sqrt {\frac {wu}{v}}\geq u \plus{} v \plus{} w$

Azerbaijan Al-Khwarizmi IJMO TST 2025, 2
Tags: inequality , inequalities , algebra
For $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2 \geq 3$,show that: $\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$.

2007 German National Olympiad, 6
Tags: algebra , inequalities , derivative , quadratic , calculus
For two real numbers a,b the equation: $x^{4}-ax^{3}+6x^{2}-bx+1=0$ has four solutions (not necessarily distinct). Prove that $a^{2}+b^{2}\ge{32}$

2023 New Zealand MO, 7
Tags: number theory , inequalities , combinatorics
Let $n,m$ be positive integers. Let $A_1,A_2,A_3, ... ,A_m$ be sets such that $A_i \subseteq \{1, 2, 3, . . . , n\}$ and $|A_i| = 3$ for all $i$ (i.e. $A_i$ consists of three different positive integers each at most $n$). Suppose for all $i < j$ we have $|A_i \cap A_j | \le 1$ (i.e. $A_i$ and $A_j$ have at most one element in common). (a) Prove that $m \le \frac{n(n-1)}{ 6}$ . (b) Show that for all $n \ge3$ it is possible to have $m \ge \frac{(n-1)(n-2)}{ 6}$ .

1997 Portugal MO, 6
Tags: combinatorics , algebra , inequalities
$n$ parallel segments of lengths $a_1 \le a_2 \le a_3 \le ... \le a_n$ were painted to mark an airport atrium. However, the architect decided that the $n$ segments should have equal length. If the cost per meter of extending the lines is equal to the cost of reducing them, how long should the lines be in order to minimize costs?