Find the domain and range of $f(x)=2-|x-5|.$

by Vulch, May 1, 2025, 2:07 AM

Find the domain and range of $f(x)=\frac{1}{1-2\cos x}.$

by Vulch, May 1, 2025, 2:06 AM

Find the domain and range of $f(x)=\frac{1}{1-2\cos x}.$

Find the domain and range of $f(x)=\frac{4-x}{x-4}.$

by Vulch, May 1, 2025, 2:02 AM

Find the domain and range of $f(x)=\frac{4-x}{x-4}.$

Inequalities

by sqing, May 1, 2025, 12:20 AM

Let $a,b,c>0 .$ Prove that
$\left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq 4\left(\frac{a+b}{b+c}+ \frac{b+c}{a+b}\right)$ $\left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{32}{9}\left(\frac{a+b}{b+c}+ \frac{c+a}{a+b}\right)$ $\left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{8}{3}\left( \frac{a+b}{b+c}+ \frac{b+c}{c+a}+ \frac{c+a}{a+b}\right)$ $\left(1 +\frac{a^2}{b^2}\right)\left(1+\frac{b^2}{c^2}\right)\left(1+\frac{c^2}{a^2}\right )\geq \frac{8}{3}\left( \frac{a^2+bc}{b^2+ca}+\frac{b^2+ca}{c^2+ab}+\frac{c^2+ab}{a^2+bc}\right)$

This post has been edited 4 times. Last edited by sqing, Today at 1:02 AM

inequalities

Cool vieta sum

by Kempu33334, Apr 30, 2025, 11:44 PM

Let the roots of $\mathcal{P}(x) = x^{108}+x^{102}+x^{96}+2x^{54}+3x^{36}+4x^{24}+5x^{18}+6$ be $r_1, r_2, \dots, r_{108}$ . Find $\dfrac{r_1^6+r_2^6+\dots+r_{108}^6}{r_1^6r_2^6+r_1^6r_3^6+\dots+r_{107}^6r_{108}^6}$ without Newton Sums.

algebra

polynomial

Vieta

Purple Comet High School Math Meet 2024 P1

by franklin2013, Apr 30, 2025, 11:20 PM

Joe ate one half of a fifth of a pizza. Gale ate one third of a quarter of that pizza. The difference in the amounts that the two ate was $\frac{1}{n}$ of the pizza, where $n$ is a positive integer. Find $n$ .

This post has been edited 1 time. Last edited by franklin2013, Yesterday at 11:20 PM

Purple Comet

Inequality, tougher than it looks

by tom-nowy, Apr 29, 2025, 3:51 AM

Prove that for $a,b \in \mathbb{R}$
$2(a^2+1)(b^2+1) \geq 3(a+b).$ Is there an elegant way to prove this?

Inequality

Amc 10 mock

by Mathsboy100, Oct 9, 2024, 12:07 AM

let $\lfloor x \rfloor$ denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which $x^2 - 20\lfloor x \rfloor + 19 = 0$

This post has been edited 3 times. Last edited by Mathsboy100, Oct 17, 2024, 1:57 AM
Reason: Error

algebra

floor function

Frankenstein FE

by NamelyOrange, Jul 19, 2024, 5:10 AM

My own problem wrote:

Solve the FE $f(x)+f(-x)=2f(x^2)$ over $\mathbb{R}$ . Ignore "pathological" solutions.

How do I solve this? I made this while messing around, and I have no clue as to what to do...

This post has been edited 1 time. Last edited by NamelyOrange, Jul 19, 2024, 5:11 AM

functional equation

Original

Inequalities

by sqing, Jul 12, 2024, 10:41 AM

Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$(1-abc) (1-a)(1-b)(1-c) \ge 208$ $(1+abc) (1-a)(1-b)(1-c) \le -224$ $(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840$