Found problems: 6530
2023 Cono Sur Olympiad, 1
A list of \(n\) positive integers \(a_1, a_2,a_3,\ldots,a_n\) is said to be [i]good[/i] if it checks simultaneously:
\(\bullet a_1<a_2<a_3<\cdots<a_n,\)
\(\bullet a_1+a_2^2+a_3^3+\cdots+a_n^n\le 2023.\)
For each \(n\ge 1\), determine how many [i]good[/i] lists of \(n\) numbers exist.
2019 Mathematical Talent Reward Programme, SAQ: P 3
Suppose $a$, $b$, $c$ are three positive real numbers with $a + b + c = 3$. Prove that
$$\frac{a}{b^2 + c}+\frac{b}{c^2 + a}+\frac{c}{a^2 + b}\geq \frac{3}{2}$$
2018 China National Olympiad, 6
China Mathematical Olympiad 2018 Q6
Given the positive integer $n ,k$ $(n>k)$ and $ a_1,a_2,\cdots ,a_n\in (k-1,k)$ ,if positive number $x_1,x_2,\cdots ,x_n$ satisfying:For any set $\mathbb{I} \subseteq \{1,2,\cdots,n\}$ ,$|\mathbb{I} |=k$,have $\sum_{i\in \mathbb{I} }x_i\le \sum_{i\in \mathbb{I} }a_i$ , find the maximum value of $x_1x_2\cdots x_n.$
2022 Macedonian Team Selection Test, Problem 2
Let $n \geq 2$ be a fixed positive integer and let $a_{0},a_{1},...,a_{n-1}$ be real numbers. Assume that all of the roots of the polynomial $P(x) = x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+...+a_{1}x+a_{0}$ are strictly positive real numbers. Determine the smallest possible value of $\frac{a_{n-1}^{2}}{a_{n-2}}$ over all such polynomials.
[i]Proposed by Nikola Velov[/i]
2002 Hungary-Israel Binational, 1
Suppose that positive numbers $x$ and $y$ satisfy $x^{3}+y^{4}\leq x^{2}+y^{3}$. Prove that $x^{3}+y^{3}\leq 2.$
1994 Poland - First Round, 12
The sequence $(x_n)$ is given by
$x_1=\frac{1}{2},$ $x_n=\frac{2n-3}{2n} \cdot x_{n-1}$ for $n=2,3,... .$
Prove that for all natural numbers $n \geq 1$ the following inequality holds
$x_1+x_2+...+x_n < 1$.
2019 Saudi Arabia JBMO TST, 2
Let $a, b, c$ be non-negative reals which satisfy $a+b+c=1$. Prove that $\frac{\sqrt{a}}{b+1}+\frac{\sqrt{b}}{c+1}+\frac{\sqrt{c}}{a+1}>\frac{1}{2}(\sqrt{a}+\sqrt{b}+\sqrt{c})$
2013 Israel National Olympiad, 6
Let $x_1,...,x_n$ be positive real numbers, satisfying $x_1+\dots+x_n=n$. Prove that
$\frac{x_1}{x_2}+\frac{x_2}{x_3}+\dots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}\leq\frac{4}{x_1\cdot x_2\cdot\dots\cdot x_n}+n-4$.
2023 South East Mathematical Olympiad, 6
Let $a_1\geq a_2\geq \cdots \geq a_n >0 .$ Prove that$$
\left( \frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)^2\geq \sum_{k=1}^{n} \frac{k(2k-1)}{a^2_1+a^2_2+\cdots+a^2_k}$$
2022 JHMT HS, 5
A point $(X, Y, Z)$ is chosen uniformly at random from the ball of radius $4$ centered at the origin (i.e., the set $\{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \leq 4^2\}$). Compute the probability that the inequalities $X^2 \leq 1$ and $X^2 + Y^2 + Z^2 \geq 1$ simultaneously hold.
2010 Contests, 2
Let $a,b,c$ be positive real numbers for which $a+b+c=3$. Prove the inequality
\[\frac{a^3+2}{b+2}+\frac{b^3+2}{c+2}+\frac{c^3+2}{a+2}\ge3\]
2014 Turkey MO (2nd round), 6
$5$ airway companies operate in a country consisting of $36$ cities. Between any pair of cities exactly one company operates two way flights. If some air company operates between cities $A, B$ and $B, C$ we say that the triple $A, B, C$ is [i]properly-connected[/i]. Determine the largest possible value of $k$ such that no matter how these flights are arranged there are at least $k$ properly-connected triples.
2011 Balkan MO, 4
Let $ABCDEF$ be a convex hexagon of area $1$, whose opposite sides are parallel. The lines $AB$, $CD$ and $EF$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $BC$, $DE$ and $FA$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least $3/2$.
2007 IMC, 4
Let $ n > 1$ be an odd positive integer and $ A = (a_{ij})_{i, j = 1..n}$ be the $ n \times n$ matrix with
\[ a_{ij}= \begin{cases}2 & \text{if }i = j \\ 1 & \text{if }i-j \equiv \pm 2 \pmod n \\ 0 & \text{otherwise}\end{cases}.\]
Find $ \det A$.
1997 Tournament Of Towns, (544) 5
Prove that $$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+a} <1$$ where $a, b$ and $c$ are positive numbers such that $abc = 1$.
(G Galperin)
1995 All-Russian Olympiad, 1
Can the numbers $1,2,3,\ldots,100$ be covered with $12$ geometric progressions?
[i]A. Golovanov[/i]
2023 Polish Junior Math Olympiad First Round, 5.
Positive numbers $a$, $b$, $c$ satisfy the inequalities
\[a + b \geq ab, \quad b + c \geq bc,\quad\text{and}\quad c+ a \geq ca.\]
Prove that $\displaystyle a + b + c \geq \frac34abc$.
2012 Greece Team Selection Test, 3
Let $a,b,c$ be positive real numbers satisfying $a+b+c=3$.Prove that $\sum_{sym} \frac{a^{2}}{(b+c)^{3}}\geq \frac{3}{8}$
2009 Mathcenter Contest, 4
Let $x,y,z\in \mathbb{R}^+_0$ such that $xy+yz+zx=1$. Prove that $$\frac{1}{\sqrt{x+y}}+\frac{1}{\sqrt{y+z}}+\frac{1}{\sqrt{z+x}}\ge 2+\frac{1}{\sqrt{2}}.$$
[i](Anonymous314)[/i]
1974 Swedish Mathematical Competition, 2
Show that
\[
1 - \frac{1}{k} \leq n\left(\sqrt[n]{k}-1\right) \leq k - 1
\]
for all positive integers $n$ and positive reals $k$.
2014 Switzerland - Final Round, 6
Let $a,b,c\in \mathbb{R}_{\ge 0}$ satisfy $a+b+c=1$. Prove the inequality :
\[ \frac{3-b}{a+1}+\frac{a+1}{b+1}+\frac{b+1}{c+1}\ge 4 \]
2012 Morocco TST, 3
$a_1,…,a_n$ are real numbers such that $a_1+…+a_n=0$ and $|a_1|+…+|a_n|=1$. Prove that :
$$|a_1+2a_2+…+na_n| \leq \frac{n-1}{2}$$
2011 Morocco National Olympiad, 1
Given positive reals $a,b,c;$ show that we have
\[\left(a+\frac 1b\right)\left(b+\frac 1c\right)\left(c+\frac 1a\right)\geq 8.\]
2014 USAMTS Problems, 4:
Nine distinct positive integers are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of $n$ and the product of any two adjacent numbers in the circle is not a multiple of $n$, where $n$ is a fixed positive integer. Find the smallest possible value for $n$.
2017 Turkey Junior National Olympiad, 4
If real numbers $a>b>1$ satisfy the inequality$$(ab+1)^2+(a+b)^2\leq 2(a+b)(a^2-ab+b^2+1)$$what is the minimum possible value of $\dfrac{\sqrt{a-b}}{b-1}$