Range of a trigonometric function

by Saucepan_man02, Apr 28, 2025, 4:44 PM

Find the range of the function: $f(x)=\frac{\sin^2 x+\sin x-1}{\sin^2 x-\sin x+2}$ .

Geometry

by AlexCenteno2007, Apr 28, 2025, 3:59 PM

Let ABC be an acute triangle and let D, E and F be the feet of the altitudes from A, B and C respectively. The straight line EF and the circumcircle of ABC intersect at P such that F is between E and P, the straight lines BP and DF intersect at Q. Show that if ED = EP then CQ and DP are parallel.

geometry

Inequalities

by sqing, Apr 28, 2025, 2:16 PM

Let $a,b,c>0 .$ Prove that
$\frac{a}{b}+ \frac{kb^3}{c^3} + \frac{c}{a}\geq 7\sqrt[7]{\frac{k}{729}}$ Where $k >0.$
$\frac{a}{b}+ \frac{729b^3}{c^3} + \frac{c}{a}\geq 7$ $\frac{a}{b}+ \frac{ b^3}{3c^3} + \frac{c}{a}\geq \frac{7}{3}$ $\frac{a}{b}+ \frac{kb^4}{c^4} + \frac{c}{a}\geq \frac{9}{2}\sqrt[9]{\frac{k}{128}}$ Where $k >0.$
$\frac{a}{b}+ \frac{128b^4}{c^4} + \frac{c}{a}\geq \frac{9}{2}$ $\frac{a}{b}+ \frac{ b^4}{4c^4} + \frac{c}{a}\geq \frac{9}{4}$

This post has been edited 1 time. Last edited by sqing, Today at 2:30 PM

inequalities

hmmt quadratic power of a prime

by martianrunner, Apr 28, 2025, 5:11 AM

I was practicing problems and came across one as such:

"Find all integers $x$ such that $2x^2 + x-6$ is a positive integral power of a prime positive integer."

I mean after factoring I don't really know where to go...

A hint would be appreciated, and if you want to solve it, please hide your solutions!

Thanks

Geometry Basic

by AlexCenteno2007, Apr 28, 2025, 12:11 AM

Let $ABC$ be an isosceles triangle such that $AC=BC$ . Let $P$ be a dot on the $AC$ side.
The tangent to the circumcircle of $ABP$ at point $P$ intersects the circumcircle of $BCP$ at $D$ . Prove that CD $\parallel$ AB

This post has been edited 5 times. Last edited by AlexCenteno2007, Today at 12:14 AM
Reason: Error

geometry

trigonogeometry 2024 TMC AIME Mock #15

by parmenides51, Apr 26, 2025, 8:22 PM

Let $\vartriangle ABC$ have angles $\alpha, \beta$ and $\gamma$ such that $\cos (\alpha) = \frac1 3$ and $\cos (\beta) = \frac{1}{17}$ . Moreover, suppose that the product of the side lengths of the triangle is equal to its area. Let $(ABC)$ denote the circumcircle of $ABC$ . Let $AO$ intersect $(BOC)$ at $D$ , $BO$ intersect $(COA)$ at $E$ , and $CO$ intersect $(AOB)$ at $F$ . If the area of $DEF$ can be written as $\frac{p\sqrt{r}}{q}$ for relatively prime integers $p$ and $q$ and squarefree $r$ , find the sum of all prime factors of $q$ , counting multiplicities (so the sum of prime factors of $48$ is $2 + 2 + 2 + 2 + 3 = 11$ ), given that $q$ has $30$ divisors.

geometry

trigonometry

Inequalities

by sqing, Apr 26, 2025, 12:58 PM

Let $x\in(-1,1).$ Prove that
$\dfrac{1}{\sqrt{1-x^2}} + \dfrac{1}{2+ x^2} \geq \dfrac{3}{2}$ $\dfrac{2}{\sqrt{1-x^2}} + \dfrac{1}{1+x^2} \geq 3$

inequalities

algebra

Geometry Angle Chasing

by Sid-darth-vater, Apr 21, 2025, 11:50 PM

Is there a way to do this without drawing obscure auxiliary lines? (the auxiliary lines might not be obscure I might just be calling them obscure)

For example I tried rotating triangle MBC 80 degrees around point C (so the BC line segment would now lie on segment AC) but I couldn't get any results. Any help would be appreciated!

Attachments:

geometry

Inequalities

by sqing, Apr 20, 2025, 1:04 PM

Let $x,y\ge 0$ such that $13(x^3+y^3) \leq 125(1+xy)$ . Prove that
$k(x+y)-xy\leq 5(2k-5)$ Where $k\geq 5.6797.$
$6(x+y)-xy\leq 35$

inequalities

[CMC ARML 2020 I3] Unique Sequence

by franchester, May 29, 2020, 11:52 PM

There is a unique nondecreasing sequence of positive integers $a_1$ , $a_2$ , $\ldots$ , $a_n$ such that $\left(a_1+\frac1{a_1}\right)\left(a_2+\frac1{a_2}\right)\cdots\left(a_n+\frac1{a_n}\right)=2020.$ Compute $a_1+a_2+\cdots+a_n$ .

Proposed by lminsl

ARML