Found problems: 2529
Let $a, b, c, x$ and $y$ be positive real numbers such that $ax + by \leq bx + cy \leq cx + ay$.
Prove that $b \leq c$.
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 $abc \le \frac{1}{8}$.
For positive real numbers $x;y;z$ satisfying $0<x,y,z<2$, find the biggest value the following equation could acquire:
$$(2x-yz)(2y-zx)(2z-xy)$$
In the $x$-$y$ plane, for $t>0$, denote by $S(t)$ the area of the part enclosed by the curve $y=e^{t^2x}$, the $x$-axis, $y$-axis and the line $x=\frac{1}{t}.$
Show that $S(t)>\frac 43.$ If necessary, you may use $e^3>20.$
Let $x$ and $y$ be positive integers such that $xy$ divides $x^{2}+y^{2}+1$. Show that \[\frac{x^{2}+y^{2}+1}{xy}=3.\]
Find the largest real number $\lambda$ with the following property: for any positive real numbers $p,q,r,s$ there exists a complex number $z=a+bi$($a,b\in \mathbb{R})$ such that $$ |b|\ge \lambda |a| \quad \text{and} \quad (pz^3+2qz^2+2rz+s) \cdot (qz^3+2pz^2+2sz+r) =0.$$
Let $n \ge 1$ and $x_1, \ldots, x_n \ge 0$. Prove that
$$ (x_1 + \frac{x_2}{2} + \ldots + \frac{x_n}{n}) (x_1 + 2x_2 + \ldots + nx_n) \le \frac{(n+1)^2}{4n} (x_1 + x_2 + \ldots + x_n)^2 .$$
Show that for any vectors $a, b$ in Euclidean space,
\[|a \times b|^3 \leq \frac{3 \sqrt 3}{8} |a|^2 |b|^2 |a-b|^2\]
Remark. Here $\times$ denotes the vector product.
The dimensions of a rectangular box in inches are all positive integers and the volume of the box is $2002\text{ in}^3$. Find the minimum possible sum in inches of the three dimensions.
$\textbf{(A) }36\qquad\textbf{(B) }38\qquad\textbf{(C) }42\qquad\textbf{(D) }44\qquad\textbf{(E) }92$
Let $a,b,c$ be non-negative real numbers such that $a^2+b^2+c^2 \le 3$ then prove that;
$$(a+b+c)(a+b+c-abc)\ge2(a^2b+b^2c+c^2a)$$
Prove that for all real numbers $ a,b$ with $ ab>0$ we have:
$ \sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}10ab\plus{}b^2}{12}$
and find the cases of equality. Hence, or otherwise, prove that for all real numbers $ a,b$
$ \sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}ab\plus{}b^2}{3}$
and find the cases of equality.
Determine the smallest real number $C$ such that the inequality \[ C(x_1^{2005} +x_2^{2005} + \cdots + x_5^{2005}) \geq x_1x_2x_3x_4x_5(x_1^{125} + x_2^{125}+ \cdots + x_5^{125})^{16} \] holds for all positive real numbers $x_1,x_2,x_3,x_4,x_5$.
Let $f(x)=ax^2+bx+c$ be a quadratic function, the graph of which doesn't intersect the x-axis. Prove that
$$a(2a+3b+6c)>0.$$
For the nonnegative numbers $a, b, c$ prove the inequality:
$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge \frac52$$
Given integer $n\geq 2$ and positive real number $a$, find the smallest real number $M=M(n,a)$, such that for any positive real numbers $x_1,x_2,\cdots,x_n$ with $x_1 x_2\cdots x_n=1$, the following inequality holds:
\[\sum_{i=1}^n \frac {1}{a+S-x_i}\leq M\]
where $S=\sum_{i=1}^n x_i$.
Find the smallest real number $M$ with the following property: Given nine nonnegative real numbers with sum $1$, it is possible to arrange them in the cells of a $3 \times 3$ square so that the product of each row or column is at most $M$.
[i]Evan O' Dorney.[/i]
Let $1<n\in\mathbb{N}$ and $1\le a\in\mathbb{R}$ and there are $n$ number of $x_i, i\in\mathbb{N}, 1\le i\le n$ such that $x_1=1$ and $\frac{x_{i}}{x_{i-1}}=a+\alpha _ i$ for $2\le i\le n$, where $\alpha _i\le \frac{1}{i(i+1)}$. Prove that $\sqrt[n-1]{x_n}< a+\frac{1}{n-1}$.
Show that for all positive real numbers $a, b, c$, we have that $$\frac{a+b+c}{3}-\sqrt[3]{abc} \leq \max\{(\sqrt{a}-\sqrt{b})^2, (\sqrt{b}-\sqrt{c})^2, (\sqrt{c}-\sqrt{a})^2\}$$
Let $m, n, p$ be fixed positive real numbers which satisfy $mnp = 8$. Depending on these constants, find the minimum of $$x^2+y^2+z^2+ mxy + nxz + pyz,$$
where $x, y, z$ are arbitrary positive real numbers satisfying $xyz = 8$. When is the equality attained?
Solve the problem for:
[list=a][*]$m = n = p = 2,$
[*] arbitrary (but fixed) positive real numbers $m, n, p.$[/list]
[i]Stijn Cambie[/i]
$\boxed{\text{A3}}$If $a,b$ be positive real numbers, show that:$$ \displaystyle{\sqrt{\dfrac{a^2+ab+b^2}{3}}+\sqrt{ab}\leq a+b}$$
Let $a,b,c$ be positive real numbers for which $a+b+c=3$. Prove the inequality
\[\frac{a^3+2}{b+2}+\frac{b^3+2}{c+2}+\frac{c^3+2}{a+2}\ge3\]
Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$.
For values of $ x$ less than $ 1$ but greater than $ \minus{}4$, the expression
\[ \frac{x^2 \minus{} 2x \plus{} 2}{2x \minus{} 2}
\]
has:
$ \textbf{(A)}\ \text{no maximum or minimum value}\qquad \\
\textbf{(B)}\ \text{a minimum value of }{\plus{}1}\qquad \\
\textbf{(C)}\ \text{a maximum value of }{\plus{}1}\qquad \\
\textbf{(D)}\ \text{a minimum value of }{\minus{}1}\qquad \\
\textbf{(E)}\ \text{a maximum value of }{\minus{}1}$
Let $m$ be a positive integer with $m \leq 2024$. Ana and Banana play a game alternately on a $1\times2024$ board, with squares initially painted white. Ana starts the game. Each move by Ana consists of choosing any $k \leq m$ white squares on the board and painting them all green. Each Banana play consists of choosing any sequence of consecutive green squares and painting them all white. What is the smallest value of $m$ for which Ana can guarantee that, after one of her moves, the entire board will be painted green?
Let $a,b,c,x,y,z$ be positive real numbers, such that $a+b+c=xyz=1$ Prove that:
$$
\frac{x^2}{3a+2}+\frac{y^2}{3b+2}+\frac{z^2}{3c+2} \ge 1
$$
When does equality hold?