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

2017 Federal Competition For Advanced Students, P2, 4
Tags: maximum , inequalities , three variable inequality , rational , algebra
(a) Determine the maximum $M$ of $x+y +z$ where $x, y$ and $z$ are positive real numbers with $16xyz = (x + y)^2(x + z)^2$. (b) Prove the existence of infinitely many triples $(x, y, z)$ of positive rational numbers that satisfy $16xyz = (x + y)^2(x + z)^2$ and $x + y + z = M$. Proposed by Karl Czakler

2014 JBMO Shortlist, 2
Tags: algebra , inequalities , positive real , three variable inequality
Let $a, b, c$ be positive real numbers such that $abc = \dfrac {1} {8}$. Prove the inequality:$$a ^ 2 + b ^ 2 + c ^ 2 + a ^ 2b ^ 2 + b ^ 2c ^ 2 + c ^ 2a ^ 2 \geq \dfrac {15} {16}$$ When the equality holds?

1972 All Soviet Union Mathematical Olympiad, 169
Tags: inequalities , algebra , minimum , maximum
Let $x,y$ be positive numbers, $s$ -- the least of $$\{ x, (y+ 1/x), 1/y\}$$ What is the greatest possible value of $s$? To what $x$ and $y$ does it correspond?

2020 German National Olympiad, 5
Tags: integer , inequalities
Let $a_1,a_2,\dots,a_{22}$ be positive integers with sum $59$. Prove the inequality \[\frac{a_1}{a_1+1}+\frac{a_2}{a_2+1}+\dots+\frac{a_{22}}{a_{22}+1}<16.\]

1957 Poland - Second Round, 4
Tags: algebra , inequalities
Prove that if $ a > 0 $, $ b > 0 $, $ c > 0 $, then $$ \frac{a}{b + c} + \frac{b}{c+ a} + \frac{c}{a+b} \geq \frac{3}{2}.$$

1990 All Soviet Union Mathematical Olympiad, 532
Tags: inequalities , geometric inequality , 3d geometry , tetrahedron , distance , altitude
If every altitude of a tetrahedron is at least $1$, show that the shortest distance between each pair of opposite edges is more than $2$.

The Golden Digits 2024, P1
Tags: algebra , inequalities
Let $n\geqslant 2$ be an integer. Prove that for any positive real numbers $a_1, a_2,\ldots, a_n$, \[\frac{1}{2\sqrt{2}}\sum_{i=1}^{n}2^{i}a_i^2 \geqslant\sum_{1 \leqslant i < j \leqslant n}a_i a_j.\][i]Proposed by Andrei Vila[/i]

2020 Bulgaria National Olympiad, P2
Tags: inequalities , algebra , n-variable inequality , sequence
Let $b_1$, $\dots$ , $b_n$ be nonnegative integers with sum $2$ and $a_0$, $a_1$, $\dots$ , $a_n$ be real numbers such that $a_0=a_n=0$ and $|a_i-a_{i-1}|\leq b_i$ for each $i=1$, $\dots$ , $n$. Prove that $$\sum_{i=1}^n(a_i+a_{i-1})b_i\leq 2$$ [hide]I believe that the original problem was for nonnegative real numbers and it was a typo on the version of the exam paper we had but I'm not sure the inequality would hold[/hide]

2003 Estonia National Olympiad, 4
Tags: inequalities , algebra
Let $a, b$, and $c$ be positive real numbers not greater than $2$. Prove the inequality $\frac{abc}{a + b + c} \le \frac43$

2016 Danube Mathematical Olympiad, 1
Tags: geometric inequality , inequalities
1.Let $ABC$ be a triangle, $D$ the foot of the altitude from $A$ and $M$ the midpoint of the side $BC$. Let $S$ be a point on the closed segment $DM$ and let $P, Q$ the projections of $S$ on the lines $AB$ and $AC$ respectively. Prove that the length of the segment $PQ$ does not exceed one quarter the perimeter of the triangle $ABC$.

2022 Balkan MO Shortlist, A3
Tags: algebra , inequalities
Let $a, b, c, d$ be non-negative real numbers such that \[\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}=3.\] Prove that \[3(ab+bc+ca+ad+bd+cd)+\frac{4}{a+b+c+d}\leqslant 5.\][i]Vasile Cîrtoaje and Leonard Giugiuc[/i]

2006 MOP Homework, 7
Tags: function , inequalities
for real number $a,b,c$ in interval $ (0,1]$ prove that: $\frac{a}{bc+1}+\frac{b}{ac+1}+\frac{c}{ab+1} \leq 2$

2013 Czech-Polish-Slovak Match, 2
Tags: geometry , perimeter , inequalities , combinatorics
Triangular grid divides an equilateral triangle with sides of length $n$ into $n^2$ triangular cells as shown in figure for $n=12$. Some cells are infected. A cell that is not yet infected, ia infected when it shares adjacent sides with at least two already infected cells. Specify for $n=12$, the least number of infected cells at the start in which it is possible that over time they will infected all the cells of the original triangle. [asy] unitsize(0.25cm); path p=polygon(3); for(int m=0; m<=11;++m){ for(int n=0 ; n<= 11-m; ++n){ draw(shift((n+0.5*m)*sqrt(3),1.5*m)*p); } } [/asy]

2021 239 Open Mathematical Olympiad, 4
Tags: algebra , inequalities
Different positive $a, b, c$ are such that $a^{239} = ac- 1$ and $b^{239} = bc- 1$.Prove that $238^2 (ab)^{239} <1$.

2013 Bogdan Stan, 1
Tags: inequalities
Let be three real numbers $ u,v,t $ under the condition $ u+v+t=0. $ Prove that for any positive real number $ a\neq 1 $ the following inequality is true with equality only and only if $ u=v=t=0: $ $$ a^u/a^v+a^v/a^t+a^{v+t}\ge a^u+a^v+1 $$ [i]Ion Tecu[/i]

2012 Serbia Team Selection Test, 2
Tags: floor function , limit , inequalities , number theory
Let $\sigma(x)$ denote the sum of divisors of natural number $x$, including $1$ and $x$. For every $n\in \mathbb{N}$ define $f(n)$ as number of natural numbers $m, m\leq n$, for which $\sigma(m)$ is odd number. Prove that there are infinitely many natural numbers $n$, such that $f(n)|n$.

2004 Romania National Olympiad, 3
Tags: function , integration , calculus , inequalities , polynomial , linear algebra , algebra
Let $f : \left[ 0,1 \right] \to \mathbb R$ be an integrable function such that \[ \int_0^1 f(x) \, dx = \int_0^1 x f(x) \, dx = 1 . \] Prove that \[ \int_0^1 f^2 (x) \, dx \geq 4 . \] [i]Ion Rasa[/i]

1957 AMC 12/AHSME, 34
Tags: inequalities , parameterization , analytic geometry , graphing lines , slope , algebra , function
The points that satisfy the system $ x \plus{} y \equal{} 1,\, x^2 \plus{} y^2 < 25,$ constitute the following set: $ \textbf{(A)}\ \text{only two points} \qquad \\ \textbf{(B)}\ \text{an arc of a circle}\qquad \\ \textbf{(C)}\ \text{a straight line segment not including the end\minus{}points}\qquad \\ \textbf{(D)}\ \text{a straight line segment including the end\minus{}points}\qquad \\ \textbf{(E)}\ \text{a single point}$

Kvant 2025, M2827
Tags: algebra , inequalities
It is known about positive numbers $a, b, c$ that it is possible to form a triangle from segments of length $a^{2024}, b^{2024}, c^{2024}$. Prove that it is possible to reduce one of the numbers $a, b, c$ by $2024$ times and obtain the numbers $a', b', c'$ so that segments with lengths $a', b', c'$ can also be formed into a triangle. [i]L. Shatunov[/i]

2021 Saudi Arabia IMO TST, 11
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]

2010 Contests, 1
Tags: rearrangement inequality , inequalities
Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]

2014 Contests, 2
Tags: inequalities , function , integration , combinatorics
Let $ k\geq 1 $ and let $ I_{1},\dots, I_{k} $ be non-degenerate subintervals of the interval $ [0, 1] $. Prove that \[ \sum \frac{1}{\left | I_{i}\cup I_{j} \right |} \geq k^{2} \] where the summation is over all pairs $ (i, j) $ of indices such that $I_i\cap I_j\neq \emptyset$.

2024 5th Memorial "Aleksandar Blazhevski-Cane", P2
Tags: inequalities , algebra
Let $x,y$ and $z$ be positive real numbers such that $xy+z^2=8$. Determine the smallest possible value of the expression $$\frac{x+y}{z}+\frac{y+z}{x^2}+\frac{z+x}{y^2}.$$

2010 Iran MO (3rd Round), 3
Tags: polynomial , search , inequalities , algebra
prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)

2014 NZMOC Camp Selection Problems, 1
Tags: algebra , inequalities
Prove that for all positive real numbers $a$ and $ b$: $$\frac{(a + b)^3}{4} \ge a^2b + ab^2$$