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

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, 4 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

Modular Arithmetic and Integers

by steven_zhang123, Mar 28, 2025, 12:28 PM

Integers $n, a, b \in \mathbb{Z}^+$ satisfies $n + a + b = 30$ . If $\alpha < b, \alpha \in \mathbb{Z^+}$ , find the maximum possible value of $\sum_{k=1}^{\alpha} \left \lfloor \frac{kn^2 \bmod a }{b-k} \right \rfloor$ .

This post has been edited 1 time. Last edited by steven_zhang123, Mar 29, 2025, 3:01 AM

number theory

modular arithmetic

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

f(x+y)f(z)=f(xz)+f(yz)

by dangerousliri, Jun 25, 2020, 6:15 PM

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all irrational numbers $x, y$ and $z$ ,
$f(x+y)f(z)=f(xz)+f(yz)$
Some stories about this problem. This problem it is proposed by me (Dorlir Ahmeti) and Valmir Krasniqi. We did proposed this problem for IMO twice, on 2018 and on 2019 from Kosovo. None of these years it wasn't accepted and I was very surprised that it wasn't selected at least for shortlist since I think it has a very good potential. Anyway I hope you will like the problem and you are welcomed to give your thoughts about the problem if it did worth to put on shortlist or not.

functional equation

algebra

IMO Proposal

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

disjoint subsets

by nayel, Apr 18, 2007, 3:23 PM

Let $n\ge 3$ be an integer and let $A_{1}, A_{2},\dots, A_{n}$ be $n$ distinct subsets of $S=\{1, 2,\dots, n\}$ . Show that there exists $x\in S$ such that the n subsets $A_{i}-\{x\}, i=1,2,\dots n$ are also disjoint.

what i have is this