Found problems: 592
Let $a, b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that
$$\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5 + |(a - b)(b - c)(c - a)|.$$
Let $(a_n)_{n\ge0}$ be a sequence of positive numbers that satisfy the relations $a_{i-1}a_{i+1}\le a_i^2$ for all $i\in\mathbb N$. For any integer $n>1$, prove the inequality
$$\frac{a_0+\ldots+a_n}{n+1}\cdot\frac{a_1+\ldots+a_{n-1}}{n-1}\ge\frac{a_0+\ldots+a_{n-1}}n\cdot\frac{a_1+\ldots+a_n}n.$$
Let $n$ be a positive integer number. Define $S(n)$ to be the least positive integer such that $S(n) \equiv n \pmod{2}$, $S(n) \geq n$, and such that there are [b]not[/b] positive integers numbers $k,x_1,x_2,...,x_k$ such that $n=x_1+x_2+...+x_k$ and $S(n)=x_1^2+x_2^2+...+x_k^2$. Prove that there exists a real constant $c>0$ and a positive integer $n_0$ such that, for all $n \geq n_0$, $S(n) \geq cn^{\frac{3}{2}}$.
Show that
\[\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16\]
for all positive real numbers $a$ and $b$ such that $ab\geq 1$.
For all $\alpha_1, \alpha_2,\alpha_3 \in \mathbb{R}^+$, Prove
\begin{align*} \sum \frac{1}{2\nu \alpha_1 +\alpha_2+\alpha_3} > \frac{2\nu}{2\nu +1} \left( \sum \frac{1}{\nu \alpha_1 + \nu \alpha_2 + \alpha_3} \right) \end{align*}
for every positive real number $\nu$
One morning, each member of Manjul’s family drank an $8$-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank $1/7$-th of the total amount of milk and $2/17$-th of the total amount of coffee. How many people are there in Manjul’s family?
A positive real $M$ is $strong$ if for any positive reals $a$, $b$, $c$ satisfying
$$ \text{max}\left\{ \frac{a}{b+c} , \frac{b}{c+a} , \frac{c}{a+b} \right\} \geqslant M $$
then the following inequality holds:
$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} > 20.$$
(a) Prove that $M=20-\frac{1}{20}$ is not $strong$.
(b) Prove that $M=20-\frac{1}{21}$ is $strong$.
If $x,y,z$ are nonnegative numbers with $x+y+z=3$, prove that
$$\sqrt x+\sqrt y+\sqrt z\ge xy+yz+xz.$$
Let $a,b,c \ge 0$ such that $a+b+c=1$ and $s \ge 5$.
Prove that $s(a^2+b^2+c^2) \le 3(s-3)(a^3+b^3+c^3)+1$
If $x$ is a positive real number, and $n$ is a positive integer, prove that
\[[ nx] > \frac{[ x]}1 + \frac{[ 2x]}2 +\frac{[ 3x]}3 + \cdots + \frac{[ nx]}n,\]
where $[t]$ denotes the greatest integer less than or equal to $t$. For example, $[ \pi] = 3$ and $\left[\sqrt2\right] = 1$.
Let $a,b,c$ be positive real numbers such that $abc\geq 1$. Prove that \[ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq 1. \]
[hide="Remark"]This problem derives from the well known inequality given in [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=185470#p185470]USAMO 1997, Problem 5[/url].
[/hide]
Let $u_1, u_2, \ldots, u_n, v_1, v_2, \ldots, v_n$ be real numbers. Prove that
\[1+ \sum_{i=1}^n (u_i+v_i)^2 \leq \frac 43 \Biggr( 1+ \sum_{i=1}^n u_i^2 \Biggl) \Biggr( 1+ \sum_{i=1}^n v_i^2 \Biggl) .\]
Prove that for every positive integer $n$ holds:
$(n-1)^n+2n^n \leq (n+1)^{n} \leq 2(n-1)^n+2n^{n}$
The numbers $a_{1}, a_{2}, ..., a_{20}$ satisfy the following conditions:
$a_{1} \ge a_{2} \ge ... \ge a_{20} \ge 0$
$a_{1} + a_{2} = 20$
$a_{3} + a_{4} + ... + a_{20} \le 20$ .
What is maximum value of the expression:
$a_{1}^2 + a_{2}^2 + ... + a_{20}^2$ ?
For which values of $a_{1}, a_{2}, ..., a_{20}$ is the maximum value achieved?
Let $n$ be a fixed positive integer and let $x_1,\ldots,x_n$ be positive real numbers. Prove that
$$x_1\left(1-x_1^2\right)+x_2\left(1-(x_1+x_2)^2\right)+\cdots+x_n\left(1-(x_1+...+x_n)^2\right)<\frac{2}{3}.$$
Let $f(x)=x^2+bx+1,$ where $b$ is a real number. Find the number of integer solutions to the inequality $f(f(x)+x)<0.$
For $0 \leq x < \frac{\pi}{2} $ prove the inequality:
$a^2\tan(x)\cdot(\cos(x))^{\frac{1}{3}}+b^2\sin{x}\geq 2xab$
where $a$ and $b$ are real numbers.
A positive real $M$ is $strong$ if for any positive reals $a$, $b$, $c$ satisfying
$$ \text{max}\left\{ \frac{a}{b+c} , \frac{b}{c+a} , \frac{c}{a+b} \right\} \geqslant M $$
then the following inequality holds:
$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} > 20.$$
(a) Prove that $M=20-\frac{1}{20}$ is not $strong$.
(b) Prove that $M=20-\frac{1}{21}$ is $strong$.
Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.
Let $a_0,a_1,\dots$ be a sequence of positive reals such that
$$ a_{n+2} \leq \frac{2023a_n}{a_na_{n+1}+2023}$$ for all integers $n\geq 0$. Prove that either $a_{2023}<1$ or $a_{2024}<1$.
a) Let $a,b$ and $c$ be side lengths of a triangle with perimeter $4$. Show that
\[a^2+b^2+c^2+abc<8.\]
b) Is there a real number $d<8$ such that for all triangles with perimeter $4$ we have
\[a^2+b^2+c^2+abc<d \quad\]
where $a,b$ and $c$ are the side lengths of the triangle?
For positive numbers is true that $$ab+ac+bc=a+b+c$$
Prove $$a+b+c+1 \geq 4abc$$
$a;b\in\mathbb{R^+}$. Prove the following inequality: $$\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{a}}\leq\sqrt[3]{2(a+b)(\frac1{a}+\frac1{b})}$$
Let $a$ and $b$ be two natural numbers that have an equal number $n$ of digits in their decimal expansions. The first $m$ digits (from left to right) of the numbers $a$ and $b$ are equal. Prove that if $m >\frac{n}{2},$ then $a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}$
[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
\]
[i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
\]