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.

AND:
OR:
NO:

Found problems: 6530

2004 Moldova Team Selection Test, 2
Tags: geometry , 3d geometry , tetrahedron , sphere , inequalities
In the tetrahedron $ABCD$ the radius of its inscribed sphere is $r$ and the radiuses of the exinscribed spheres (each tangent with a face of the tetrahedron and with the planes of the other faces) are $r_A, r_B, r_C, r_D.$ Prove the inequality $$\frac{1}{\sqrt{r_A^2-r_Ar_B+r_B^2}}+\frac{1}{\sqrt{r_B^2-r_Br_C+r_C^2}}+\frac{1}{\sqrt{r_C^2-r_Cr_D+r_D^2}}+\frac{1}{\sqrt{r_D^2-r_Dr_A+r_A^2}}\leq\frac{2}{r}.$$

2000 Junior Balkan Team Selection Tests - Moldova, 4
Tags: inequalities , system , algebra
Find the smallest natural number nonzero n so that it exists in real numbers $x_1, x_2,..., x_n$ which simultaneously check the conditions: 1) $x_i \in [1/2 , 2]$ , $i = 1, 2,... , n$ 2) $x_1+x_2+...+x_n \ge \frac{7n}{6}$ 3) $\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}$

1972 Bulgaria National Olympiad, Problem 6
Tags: geometry , 3d geometry , tetrahedron , inequalities , geometrical inequalities
It is given a tetrahedron $ABCD$ for which two points of opposite edges are mutually perpendicular. Prove that: (a) the four altitudes of $ABCD$ intersects at a common point $H$; (b) $AH+BH+CH+DH<p+2R$, where $p$ is the sum of the lengths of all edges of $ABCD$ and $R$ is the radii of the sphere circumscribed around $ABCD$. [i]H. Lesov[/i]

2020 Jozsef Wildt International Math Competition, W56
Tags: inequalities
If $p_k>0,a_k\ge2~(k=1,2,\ldots,n)$ and $$S_n=\sum_{k=1}^na_k,A_n=\prod_{\text{cyc}}a_1^{p_2+p_3+\ldots+p^n},B_n=\prod_{k=1}^na_k^{p_k},$$ then prove that $$\sum_{k=1}^np_k\log_{S_n-a_k}a_k\ge\left(\sum_{k=1}^np_k\right)\log_{A_n}B_n$$ [i]Proposed by Mihály Bencze[/i]

2024 India IMOTC, 2
Tags: inequalities
Let $x_1, x_2 \dots, x_{2024}$ be non-negative real numbers such that $x_1 \le x_2\cdots \le x_{2024}$, and $x_1^3 + x_2^3 + \dots + x_{2024}^3 = 2024$. Prove that \[\sum_{1 \le i < j \le 2024} (-1)^{i+j} x_i^2 x_j \ge -1012.\] [i]Proposed by Shantanu Nene[/i]

2023 Belarus Team Selection Test, 4.3
Tags: inequalities , algebra
Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that \[2^{j-i}x_ix_j>2^{s-3}.\]

2014 Cuba MO, 2
Tags: algebra , inequalities
Let $a$ and $b$ be real numbers with $0 \le a, b \le 1$. (a) Prove that $ \frac{a}{b+1} +\frac{b}{a+1} \le 1.$ (b) Find the case of equality.

1967 AMC 12/AHSME, 7
Tags: inequalities
If $\frac{a}{b}<-\frac{c}{d}$ where $a$, $b$, $c$, and $d$ are real numbers and $bd \not= 0$, then: $ \text{(A)}\ a \; \text{must be negative} \qquad \text{(B)}\ a \; \text{must be positive} \qquad$ $\text{(C)}\ a \; \text{must not be zero} \qquad \text{(D)}\ a \; \text{can be negative or zero, but not positive } \\ \text{(E)}\ a \; \text{can be positive, negative, or zero}$

2012 Putnam, 2
Tags: geometry , inequalities , 3d geometry , sphere , prism
Let $P$ be a given (non-degenerate) polyhedron. Prove that there is a constant $c(P)>0$ with the following property: If a collection of $n$ balls whose volumes sum to $V$ contains the entire surface of $P,$ then $n>c(P)/V^2.$

2023 Olimphíada, 1
Tags: floor function , algebra , inequalities
Let $n \geq 2023$ be an integer. For each real $x$, we say that $\lfloor x \rceil$ is the closest integer to $x$, and if there are two closest integers then it is the greater of the two. Suppose there is a positive real $a$ such that $$\lfloor an \rceil = n + \bigg\lfloor\frac{n}{a} \bigg\rceil.$$ Show that $|a^2 - a - 1| < \frac{n\varphi+1}{n^2}$.

2017 Greece JBMO TST, 1
Tags: inequalities
Positive real numbers $a,b,c$ satisfy $a+b+c=1$. Prove that $$(a+1)\sqrt{2a(1-a)} + (b+1)\sqrt{2b(1-b)} + (c+1)\sqrt{2c(1-c)} \geq 8(ab+bc+ca).$$ Also, find the values of $a,b,c$ for which the equality happens.

2017 Canadian Mathematical Olympiad Qualification, 5
Tags: inequalities
Prove for all real numbers $x, y$, $$(x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \geq 0.$$ Determine when equality holds.

2006 Romania Team Selection Test, 4
Tags: inequalities
Let $x_i$, $1\leq i\leq n$ be real numbers. Prove that \[ \sum_{1\leq i<j\leq n}|x_i+x_j|\geq\frac{n-2}{2}\sum_{i=1}^n|x_i|. \] [i]Discrete version by Dan Schwarz of a Putnam problem[/i]

2004 IMC, 3
Tags: limit , function , inequalities
Let $A_n$ be the set of all the sums $\displaystyle \sum_{k=1}^n \arcsin x_k $, where $n\geq 2$, $x_k \in [0,1]$, and $\displaystyle \sum^n_{k=1} x_k = 1$. a) Prove that $A_n$ is an interval. b) Let $a_n$ be the length of the interval $A_n$. Compute $\displaystyle \lim_{n\to \infty} a_n$.

2002 Vietnam National Olympiad, 1
Tags: algebra , polynomial , inequalities
Let $ a$, $ b$, $ c$ be real numbers for which the polynomial $ x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has three real roots. Prove that \[ 12ab \plus{} 27c \le 6a^3 \plus{} 10\left(a^2 \minus{} 2b\right)^{\frac {3}{2}}\] When does equality occur?

1993 All-Russian Olympiad Regional Round, 9.1
Tags: algebra , inequalities
If $a$ and $b$ are positive numbers, prove the inequality $$a^2 +ab+b^2\ge 3(a+b-1).$$

2021 Israel National Olympiad, P3
Tags: geometric inequalities , inequalities , geometry
Let $ABC$ be a triangle. Let $X$ be the tangency point of the incircle with $BC$. Let $Y$ be the second intersection point of segment $AX$ with the incircle. Prove that \[AX+AY+BC>AB+AC\]

1977 IMO Longlists, 38
Tags: inequalities
Let $m_j > 0$ for $j = 1, 2,\ldots, n$ and $a_1 \leq \cdots \leq a_n < b_1 \leq \cdots \leq b_n < c_1 \leq \cdots \leq c_n$ be real numbers. Prove that \[\Biggl( \sum_{j=1}^{n} m_j(a_j+b_j+c_j) \Biggr)^2 > 3 \Biggl( \sum_{j=1}^{n} m_j \Biggr) \Biggl( \sum_{j=1}^{n} m_j(a_jb_j+b_jc_j+c_ja_j) \Biggr).\]

1968 Swedish Mathematical Competition, 1
Tags: algebra , min , max , inequalities
Find the maximum and minimum values of $x^2 + 2y^2 + 3z^2$ for real $x, y, z$ satisfying $x^2 + y^2 + z^2 = 1$.

2006 AMC 10, 10
Tags: geometry , perimeter , inequalities , triangle inequality
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle? $ \textbf{(A) } 43 \qquad \textbf{(B) } 44 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 46 \qquad \textbf{(E) } 47$

1987 IMO Longlists, 62
Tags: inequalities , geometry
Let $l, l'$ be two lines in $3$-space and let $A,B,C$ be three points taken on $l$ with $B$ as midpoint of the segment $AC$. If $a, b, c$ are the distances of $A,B,C$ from $l'$, respectively, show that $b \leq \sqrt{ \frac{a^2+c^2}{2}}$, equality holding if $l, l'$ are parallel.

2003 Czech And Slovak Olympiad III A, 6
Tags: inequalities
a，b，c>0，abc=1，prove that(a/b)+(b/c)+(c/a)≥a+b+c.

2006 Bulgaria National Olympiad, 2
Tags: function , limit , induction , inequalities , algebra
Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be a function that satisfies for all $x>y>0$ \[f(x+y)-f(x-y)=4\sqrt{f(x)f(y)}\] a) Prove that $f(2x)=4f(x)$ for all $x>0$; b) Find all such functions. [i]Nikolai Nikolov, Oleg Mushkarov [/i]

2010 Thailand Mathematical Olympiad, 7
Tags: inequalities , algebra
Let $a, b, c$ be positive reals. Show that $\frac{a^5}{bc^2} + \frac{b^5}{ca^2} + \frac{c^5}{ab^2} \ge a^2 + b^2 + c^2.$

1992 Czech And Slovak Olympiad IIIA, 2
Tags: inequalities , geometric inequalities , area , 3d geometry , tetrahedron , geometry
Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$