Found problems: 2529
Given $n$ positive numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ such that $a_{1}\cdot a_{2}\cdot ...\cdot a_{n}=1.$ Prove \[ \left( 1+a_{1}\right) \left( 1+a_{2}\right) ...\left(1+a_{n}\right) \geq 2^{n}.\]
Find all non-negative integers $n$ such that there are $a,b \in \mathbb Z$ satisfying $n^2=a+b$ and $n^3=a^2+b^2$.
[i]Lucian Dragomir[/i]
For all $a, b, c > 0$ and $abc = 1$, prove that
\[\frac{1}{a(a+1)+ab(ab+1)}+\frac{1}{b(b+1)+bc(bc+1)}+\frac{1}{c(c+1)+ca(ca+1)}\ge\frac{3}{4}\]
The number of real triples $(x , y , z )$ that satisfy the equation $x^4 + 4y^4 + z^4 + 4 = 8xyz$ is
(A): $0$, (B): $1$, (C): $2$, (D): $8$, (E): None of the above.
Prove that there exists a constant $C>0$ such that
$$H(a_1)+H(a_2)+\cdots+H(a_m)\leq C\sqrt{\sum_{i=1}^{m}i a_i}$$
holds for arbitrary positive integer $m$ and any $m$ positive integer $a_1,a_2,\cdots,a_m$, where $$H(n)=\sum_{k=1}^{n}\frac{1}{k}.$$
We are given a natural number $d$. Find all open intervals of maximum length $I \subseteq R$ such that for all real numbers $a_0,a_1,...,a_{2d-1}$ inside interval $I$, we have that the polynomial $P(x)=x^{2d}+a_{2d-1}x^{2d-1}+...+a_1x+a_0$ has no real roots.
A polynomial $P(x)$ with real coefficients and degree $2021$ is given. For any real $a$ polynomial $x^{2022}+aP(x)$ has at least one real root.
Find all possible values of $P(0)$
Determine all quadruples $(a, b, c, d)$ of real numbers satisfying the equation
\[256a^3b^3c^3d^3 = (a^6+b^2+c^2+d^2)(a^2+b^6+c^2+d^2)(a^2+b^2+c^6+d^2)(a^2+b^2+c^2+d^6).\]
Let $a,b,c$ be positive real numbers such that $ab+bc+ca\le 3abc$. Prove that
\[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})\]
Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds:
$\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$,
where $T$ is the period of $f(x)$.
[b]i.)[/b] Calculate $x$ if \[ x = \frac{(11 + 6 \cdot \sqrt{2}) \cdot \sqrt{11 - 6 \cdot \sqrt{2}} - (11 - 6 \cdot \sqrt{2}) \cdot \sqrt{11 + 6 \cdot \sqrt{2}}}{(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}) - (\sqrt{\sqrt{5}+1})} \]
[b]ii.)[/b] For each positive number $x,$ let \[ k = \frac{\left( x + \frac{1}{x} \right)^6 - \left( x^6 + \frac{1}{x^6} \right) - 2}{\left( x + \frac{1}{x} \right)^3 - \left( x^3 + \frac{1}{x^3} \right)} \] Calculate the minimum value of $k.$
Let $a,b,c$ be positive real numbers. Prove the inequality
\[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq a+b+c+\frac{4(a-b)^2}{a+b+c}.\]
When does equality occur?
The incircle of a triangle $ A_1A_2A_3$ is centered at $ O$ and meets the segment $ OA_j$ at $ B_j$ , $ j \equal{} 1, 2, 3$. A circle with center $ B_j$ is tangent to the two sides of the triangle having $ A_j$ as an endpoint and intersects the segment $ OB_j$ at $ C_j$. Prove that
\[ \frac{OC_1\plus{}OC_2\plus{}OC_3}{A_1A_2\plus{}A_2A_3\plus{}A_3A_1} \leq \frac{1}{4\sqrt{3}}\]
and find the conditions for equality.
If $a$,$b$,$c$ are positive reals, prove that
$$\frac{a+bc}{a+a^2}+\frac{b+ca}{b+b^2}+\frac{c+ab}{c+c^2} \geq 3$$
A ray emanating from the vertex $A$ of the triangle $ABC$ intersects the side $BC$ at $X$ and the circumcircle of triangle $ABC$ at $Y$. Prove that $\frac{1}{AX}+\frac{1}{XY}\geq \frac{4}{BC}$.
Let $a\geq b\geq c\geq 0$ are real numbers such that $a+b+c=3$. Prove that $ab^{2}+bc^{2}+ca^{2}\leq\frac{27}{8}$ and find cases of equality.
Let real numbers a, b, c satisfy $a^2+2b^2+3c^2= \frac{3}{2}$, prove that $3^{-a}+9^{-b}+27^{-c}\ge1$.
Source: 2017 Canadian Open Math Challenge, Problem C3
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Let $XYZ$ be an acute-angled triangle. Let $s$ be the side-length of the square which has two adjacent vertices on side $YZ$, one vertex on side $XY$ and one vertex on side $XZ$. Let $h$ be the distance from $X$ to the side $YZ$ and let $b$ be the distance from $Y$ to $Z$.
[asy]
pair S, D;
D = 1.27;
S = 2.55;
draw((2, 4)--(0, 0)--(7, 0)--cycle);
draw((1.27,0)--(1.27+2.55,0)--(1.27+2.55,2.55)--(1.27,2.55)--cycle);
label("$X$",(2,4),N);
label("$Y$",(0,0),W);
label("$Z$",(7,0),E);
[/asy]
(a) If the vertices have coordinates $X = (2, 4)$, $Y = (0, 0)$ and $Z = (4, 0)$, find $b$, $h$ and $s$.
(b) Given the height $h = 3$ and $s = 2$, find the base $b$.
(c) If the area of the square is $2017$, determine the minimum area of triangle $XYZ$.
Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$
Let $a,b, c > 0$ and $abc = 1$. Prove that $\frac{a - 1}{c}+\frac{c - 1}{b}+\frac{b - 1}{a} \ge 0$
Let, $\displaystyle{S =\sum_{i=1}^{k} {n_i}^2}$. Prove that for $n_i \in \mathbb{R}^+$
$$\sum_{i=1}^{k} \frac{n_i}{S-n_i^2} \geq \frac{4}{n_1+n_2+ \cdots+ n_k}$$
[i]Proposed by Kang Taeyoung, South Korea[/i]
Suppose $x$ is a non zero real number such that both $x^5$ and $20x+\frac{19}{x}$ are rational numbers. Prove that $x$ is a rational number.
What is the maximum of $|z^3 -z+2|$, where $z$ is a complex number with $|z|=1?$
Prove for any positives $a,b,c$ the inequality $$
\sqrt[3]{\dfrac{a}{b}}+\sqrt[5]{\dfrac{b}{c}}+\sqrt[7]{\dfrac{c}{a}}>\dfrac{5}{2}$$
$\boxed{A7}$Prove that for all $x,y,z>0$ with $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$ and $0\leq a,b,c<1$ the following inequality holds
\[\frac{x^2+y^2}{1-a^z}+\frac{y^2+z^2}{1-b^x}+\frac{z^2+x^2}{1-c^y}\geq \frac{6(x+y+z)}{1-abc}\]