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

2024 New Zealand MO, 8
Tags: inequalities
Let $a$, $b$ and $c$ be any positive real numbers. Prove that $$\dfrac{a^2+b^2}{2c}+\dfrac{a^2+c^2}{2b}+\dfrac{b^2+c^2}{2a} \geqslant a+b+c.$$

2011 Macedonia National Olympiad, 1
Tags: inequalities
Let $~$ $ a,\,b,\,c,\,d\, >\, 0$ $~$ and $~$ $a+b+c+d\, =\, 1\, .$ $~$ Prove the inequality \[ \frac{1}{4a+3b+c}+\frac{1}{3a+b+4d}+\frac{1}{a+4c+3d}+\frac{1}{4b+3c+d}\; \ge\; 2\, . \]

2018 Stars of Mathematics, 3
Tags: floor function , algebra , function , inequalities
Given a positive integer $n$, determine the largest integer $M$ satisfying $$\lfloor \sqrt{a_1}\rfloor + ... + \lfloor \sqrt{a_n} \rfloor \ge \lfloor\sqrt{ a_1 + ... + a_n +M \cdot min(a_1,..., a_n)}\rfloor $$ for all non-negative integers $a_1,...., a_n$. S. Berlov, A. Khrabrov

2009 Croatia Team Selection Test, 1
Tags: inequalities , algebra
Solve in the set of real numbers: \[ 3\left(x^2 \plus{} y^2 \plus{} z^2\right) \equal{} 1, \] \[ x^2y^2 \plus{} y^2z^2 \plus{} z^2x^2 \equal{} xyz\left(x \plus{} y \plus{} z\right)^3. \]

2005 MOP Homework, 5
Tags: inequalities , algebra
Let $a_1$, $a_2$, ..., $a_{2004}$ be non-negative real numbers such that $a_1+...+ a_{2004} \le 25$. Prove that among them there exist at least two numbers $a_i$ and $a_j$ ($i \neq j$) such that $|\sqrt{a_i}-\sqrt{a_j}| \le \frac{5}{2003}$.

2002 Iran Team Selection Test, 3
Tags: 3d geometry , sphere , inequalities , geometry
A "[i]2-line[/i]" is the area between two parallel lines. Length of "2-line" is distance of two parallel lines. We have covered unit circle with some "2-lines". Prove sum of lengths of "2-lines" is at least 2.

2006 Federal Math Competition of S&M, Problem 2
Tags: inequalities
Let $x,y,z$ be positive numbers with $x+y+z=1$. Show that $$yz+zx+xy\ge4\left(y^2z^2+z^2x^2+x^2y^2\right)+5xyz.$$When does equality hold?

2020 Jozsef Wildt International Math Competition, W45
Tags: inequalities
Let $a_1,a_2,a_3,a_4$ be strictly positive numbers. Then is the following inequality true: $$4\left(a_1a_2^n+a_2a_3^n+a_3a_4^n+a_4a_1^n\right)^n\le\left(a_1^n+a_2^n+a_3^n+a_4^n\right)^{n+1}$$ for each $n\in\mathbb N$? [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

1986 French Mathematical Olympiad, Problem 1
Tags: geometry , 3d geometry , tetrahedron , inequalities
Let $ABCD$ be a tetrahedron. (a) Prove that the midpoints of the edges $AB,AC,BD$, and $CD$ lie in a plane. (b) Find the point in that plane, whose sum of distances from the lines $AD$ and $BC$ is minimal.

2007 Bosnia Herzegovina Team Selection Test, 4
Tags: polynomial , inequalities , vieta , algebra
Let $P(x)$ be a polynomial such that $P(x)=x^3-2x^2+bx+c$. Roots of $P(x)$ belong to interval $(0,1)$. Prove that $8b+9c \leq 8$. When does equality hold?

2019 Jozsef Wildt International Math Competition, W. 59
Tags: tetrahedron , 3d geometry , circumradius , inradius , geometry , inequalities
In the any $[ABCD]$ tetrahedron we denote with $\alpha$, $\beta$, $\gamma$ the measures, in radians, of the angles of the three pairs of opposite edges and with $r$, $R$ the lengths of the rays of the sphere inscribed and respectively circumscribed the tetrahedron. Demonstrate inequality$$\left(\frac{3r}{R}\right)^3\leq \sin \frac{\alpha +\beta +\gamma}{3}$$(A refinement of inequality $R \geq 3r$).

2008 Costa Rica - Final Round, 4
Tags: inequalities
Let $ x$, $ y$ and $ z$ be non negative reals, such that there are not two simultaneously equal to $ 0$. Show that $ \frac {x \plus{} y}{y \plus{} z} \plus{} \frac {y \plus{} z}{x \plus{} y} \plus{} \frac {y \plus{} z}{z \plus{} x} \plus{} \frac {z \plus{} x}{y \plus{} z} \plus{} \frac {z \plus{} x}{x \plus{} y} \plus{} \frac {x \plus{} y}{z \plus{} x}\geq\ 5 \plus{} \frac {x^{2} \plus{} y^{2} \plus{} z^{2}}{xy \plus{} yz \plus{} zx}$ and determine the equality cases.

2002 Moldova National Olympiad, 4
Tags: inequalities , geometry , 3d geometry , tetrahedron , circumcircle
The circumradius of a tetrahedron $ ABCD$ is $ R$, and the lenghts of the segments connecting the vertices $ A,B,C,D$ with the centroids of the opposite faces are equal to $ m_a,m_b,m_c$ and $ m_d$, respectively. Prove that: $ m_a\plus{}m_b\plus{}m_c\plus{}m_d\leq \dfrac{16}{3}R$

2006 China Second Round Olympiad, 3
Tags: inequalities
Suppose $A = {x|5x-a \le 0}$, $B = {x|6x-b > 0}$, $a,b \in \mathbb{N}$, and $A \cap B \cap \mathbb{N} = {2,3,4}$. The number of such pairs $(a,b)$ is ${ \textbf{(A)}\ 20\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}} 42\qquad $

2002 China Western Mathematical Olympiad, 2
Tags: modular arithmetic , inequalities , number theory
Given a positive integer $ n$, find all integers $ (a_{1},a_{2},\cdots,a_{n})$ satisfying the following conditions: $ (1): a_{1}\plus{}a_{2}\plus{}\cdots\plus{}a_{n}\ge n^2;$ $ (2): a_{1}^2\plus{}a_{2}^2\plus{}\cdots\plus{}a_{n}^2\le n^3\plus{}1.$

2016 India Regional Mathematical Olympiad, 2
Tags: algebra , inequalities
Let $a,b,c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$ Prove that $abc \le \frac{1}{8}$.

2016 Israel Team Selection Test, 1
Tags: inequalities , calculus , lagrange
Let $a,b,c$ be positive numbers satisfying $ab+bc+ca+2abc=1$. Prove that $4a+b+c \geq 2$.

2007 Italy TST, 1
Tags: inequalities , graph theory , combinatorics
We have a complete graph with $n$ vertices. We have to color the vertices and the edges in a way such that: no two edges pointing to the same vertice are of the same color; a vertice and an edge pointing him are coloured in a different way. What is the minimum number of colors we need?

1985 Brazil National Olympiad, 3
Tags: inequalities , geometry , perimeter , cyclic quadrilateral
A convex quadrilateral is inscribed in a circle of radius $1$. Show that the its perimeter less the sum of its two diagonals lies between $0$ and $2$.

2006 MOP Homework, 1
Tags: inequalities , algebra
Let a,b, and c be positive reals. Prove: $\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2}\ge (a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$

2023 New Zealand MO, 7
Tags: inequalities , combinatorics , number theory
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 $m \ge \frac{(n-1)(n-2)}{ 6}$ .

2013 Polish MO Finals, 6
Tags: analytic geometry , inequalities , combinatorics
For each positive integer $n$ determine the maximum number of points in space creating the set $A$ which has the following properties: $1)$ the coordinates of every point from the set $A$ are integers from the range $[0, n]$ $2)$ for each pair of different points $(x_1,x_2,x_3), (y_1,y_2,y_3)$ belonging to the set $A$ it is satisfied at least one of the following inequalities $x_1< y_1, x_2<y_2, x_3<y_3$ and at least one of the following inequalities $x_1>y_1, x_2>y_2,x_3>y_3$.

2002 Tournament Of Towns, 2
Tags: 3d geometry , geometry , inequalities
A cube is cut by a plane such that the cross section is a pentagon. Show there is a side of the pentagon of length $\ell$ such that the inequality holds: \[ |\ell-1|>\frac{1}{5} \]

1989 Federal Competition For Advanced Students, 2
Tags: inequalities
If $ a$ and $ b$ are nonnegative real numbers with $ a^2\plus{}b^2\equal{}4$, show that: $ \frac{ab}{a\plus{}b\plus{}2} \le \sqrt{2}\minus{}1$ and determine when equality occurs.

2007 Peru IMO TST, 2
Tags: inequalities
Let $a,b,c$ be positive real numbers, such that: $a+b+c \geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$ Prove that: \[a+b+c \geq \frac{3}{a+b+c}+\frac{2}{abc}. \]