Need hint:''(

by Buh_-1235, Apr 1, 2025, 6:12 PM

Recall that for any positive integer m, φ(m) denotes the number of positive integers less than m which are relatively
prime to m. Let n be an odd positive integer such that both φ(n) and φ(n + 1) are powers of two. Prove n + 1 is power
of two or n = 5.

number theory

inequalities

by Cobedangiu, Apr 1, 2025, 6:10 PM

$a,b>0$ and $a+b=1$ . Find min P:
$P=\sqrt{\frac{1-a}{1+7a}}+\sqrt{\frac{1-b}{1+7b}}$

This post has been edited 1 time. Last edited by Cobedangiu, 36 minutes ago

inequalities

Gut inequality

by giangtruong13, Apr 1, 2025, 3:54 PM

Let $a,b,c>0$ satisfy that $a+b+c=3$ . Find the minimum $\sum_{cyc} \sqrt[4]{\frac{a^3}{b+c}}$

This post has been edited 3 times. Last edited by giangtruong13, 3 hours ago

inequalities

Inspired by old results

by sqing, Apr 1, 2025, 2:09 PM

Let $a, b,c\geq 0$ and $a+2b+3c= 2(\sqrt{6}-1).$ Prove that
$a+ab+abc\leq 3$ Let $a, b,c\geq 0$ and $a+2b+3c= 2\sqrt{6}-1.$ Prove that
$a+ab+abc\leq \frac{25}{8}+\sqrt{ \frac{3}{2}}$ Let $a, b,c\geq 0$ and $a+2b+3c= 2\sqrt{3}-1.$ Prove that
$a+ab+abc\leq \frac{13}{8}+\frac{\sqrt{ 3}}{2}$

inequalities proposed

algebra

Olympiad problem - I can't solve it pls help

by kjhgyuio, Apr 1, 2025, 11:07 AM

It is given that x and y are positive integers such that x>y and
√x + √y=√2000
How many different possible values can x take?

This post has been edited 1 time. Last edited by kjhgyuio, Today at 11:22 AM
Reason: typo in problem

Unsolved NT, 3rd time posting

by GreekIdiot, Mar 26, 2025, 11:40 AM

Solve $5^x-2^y=z^3$ where $x,y,z \in \mathbb Z$
Hint

This post has been edited 2 times. Last edited by GreekIdiot, Mar 26, 2025, 11:41 AM

Minimize Expression Over Permutation

by amuthup, Jul 12, 2022, 12:24 PM

For each integer $n\ge 1,$ compute the smallest possible value of $\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor$ over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$

Proposed by Shahjalal Shohag, Bangladesh

This post has been edited 1 time. Last edited by amuthup, Aug 12, 2022, 3:32 PM

very cute geo

by rafaello, Oct 26, 2021, 7:28 PM

Consider a triangle $ABC$ with incircle $\omega$ . Let $S$ be the point on $\omega$ such that the circumcircle of $BSC$ is tangent to $\omega$ and let the $A$ -excircle be tangent to $BC$ at $A_1$ . Prove that the tangent from $S$ to $\omega$ and the tangent from $A_1$ to $\omega$ (distinct from $BC$ ) meet on the line parallel to $BC$ and passing through $A$ .

geometry

incircle

Let's Invert Some

by Shweta_16, Jan 26, 2020, 12:55 PM

In triangle $\triangle{ABC}$ with incenter $I$ , the incircle $\omega$ touches sides $AC$ and $AB$ at points $E$ and $F$ , respectively. A circle passing through $B$ and $C$ touches $\omega$ at point $K$ . The circumcircle of $\triangle{KEC}$ meets $BC$ at $Q \neq C$ . Prove that $FQ$ is parallel to $BI$ .

Proposed by Anant Mudgal

This post has been edited 2 times. Last edited by Shweta_16, Jan 26, 2020, 1:34 PM
Reason: let's chase some angels