Found problems: 2529
Find all polynomials whose coefficients are all $\pm1$ and whose roots are all real.
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].
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Does there exist an infinite sequence of integers \(a_0\), \(a_1\), \(a_2\), \(\ldots\) such that \(a_0\ne0\) and, for any integer \(n\ge0\), the polynomial \[P_n(x)=\sum_{k=0}^na_kx^k\] has \(n\) distinct real roots?
[i]Proposed by Amol Rama and Espen Slettnes[/i]
Given positive reals $a,b,c,p,q$ satisfying $abc=1$ and $p \geq q$, prove that \[ p \left(a^2+b^2+c^2\right) + q\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \geq (p+q) (a+b+c). \][i]Proposed by AJ Dennis[/i]
Find all quadruplets (x, y, z, w) of positive real numbers that satisfy the following system:
$\begin{cases}
\frac{xyz+1}{x+1}= \frac{yzw+1}{y+1}= \frac{zwx+1}{z+1}= \frac{wxy+1}{w+1}\\
x+y+z+w= 48
\end{cases}$
Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$.
Let $ a, b, c$ be positive real numbers such that $ abc = 1. $ Prove that:
$$ 2 (a^ 2 + b^ 2 + c^ 2) \left (\frac 1 {a^ 2} + \frac 1{b^ 2}+ \frac 1{c^2}\right)\geq 3(a+ b + c + ab + bc + ca).$$
Suppose that $a,b,c,d$ are non-negative real numbers such that $a^2+b^2+c^2+d^2=2$ and $ab+bc+cd+da=1$. Find the maximum value of $a+b+c+d$ and determine all equality cases.
Positive numbers $\alpha ,\beta , x_1, x_2,\ldots, x_n$ ($n \geq 1$) satisfy $x_1+x_2+\cdots+x_n = 1$. Prove that
\[\sum_{i=1}^{n} \frac{x_i^3}{\alpha x_i+\beta x_{i+1}} \geq \frac{1}{n(\alpha+\beta)}.\]
[b]Note.[/b] $x_{n+1}=x_1$.
For positive numbers $x_{1},x_{2},\dots,x_{s}$, we know that $\prod_{i=1}^{s}x_{k}=1$. Prove that for each $m\geq n$ \[\sum_{k=1}^{s}x_{k}^{m}\geq\sum_{k=1}^{s}x_{k}^{n}\]
Let $n$ be a fixed positive integer and let $x_1,\ldots,x_n$ be positive real numbers. Prove that
$$x_1\left(1-x_1^2\right)+x_2\left(1-(x_1+x_2)^2\right)+\cdots+x_n\left(1-(x_1+...+x_n)^2\right)<\frac{2}{3}.$$
The positive real numbers $x,y,z$ satisfy $x+y+z=1$. Show that
\[\sqrt{3xyz}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{1-x}+\frac{1}{1-y}+\frac{1}{1-z}\right)\geq4+
\frac{4xyz}{(1-x)(1-y)(1-z)}.\]
Prove that for all $a,b>0$ with $a+b=2$, we have
$$\left(a^n+1\right)\left(b^n+1\right)\ge4$$
for all $n\in\mathbb N_{\ge2}$.
Find the minimum and maximum values of $ S=\frac{a}{b}+\frac{c}{d} $ where $a$, $b$, $c$, $d$ are positive integers satisfying $a + c = 20202$ and $b + d = 20200$.
Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that
\[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}.
\]
Let $a_1,a_2,...,a_9$ be a sequence of numbers satisfying $0 < p \le a_i \le q$ for each $i = 1,2,..., 9$.
Prove that $\frac{a_1}{a_9}+\frac{a_2}{a_8}+...+\frac{a_9}{a_1} \le 1 + \frac{4(p^2+q^2)}{pq}$
Let $a, b, c$ three positive reals such that
\[\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1}\geq 1. \]
Show that
\[a+b+c\geq ab+bc+ca. \]
Suppose that $a_1, a_2, \dots a_{2021}$ are non-negative numbers such that $\sum_{k=1}^{2021} a_k=1$. Prove that
$$ \sum_{k=1}^{2021}\sqrt[k]{a_1 a_2\dots a_k} \leq 3. $$
Show that for all integers $a \ge 1$,$ \lfloor \sqrt{a}+\sqrt{a+1}+\sqrt{a+2}\rfloor = \lfloor \sqrt{9a+8}\rfloor$
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.
$\boxed{\text{A7}}$ Let $a,b,c$ be positive reals such that $abc=1$.Prove the inequality $\sum\frac{2a^2+\frac{1}{a}}{b+\frac{1}{a}+1}\geq 3$
Let $a,b,c\ge 0$ be three real numbers such that $$ab+bc+ca+2abc=1.$$ Prove that $\sqrt{a}+\sqrt{b}+\sqrt{c}\ge 2$ and determine equality cases.
Solve the equation: $$x \cdot 2^{\dfrac{1}{x}}+\dfrac{1}{x} \cdot 2^x=4$$
Given a triangle $ ABC$ with angle $ C \geq 60^{\circ}$. Prove that:
$ \left(a \plus{} b\right) \cdot \left(\frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c} \right) \geq 4 \plus{} \frac {1}{\sin\left(\frac {C}{2}\right)}.$
For $a,b,c>0: \; abc=1$ prove that
\[a^3+b^3+c^3+6 \ge (a+b+c)^2\]