Is this FE solvable?

by Mathdreams, Apr 1, 2025, 6:58 PM

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy$ for all reals $x$ and $y$ .

algebra

functional equation

OFM2021 Senior P1

by medhimdi, Apr 1, 2025, 6:51 PM

Let $a_1, a_2, a_3, \dots$ and $b_1, b_2, b_3, \dots$ be two sequences of integers such that $a_{n+2}=a_{n+1}+a_n$ and $b_{n+2}=b_{n+1}+b_n$ for all $n\geq1$ . Suppose that $a_n$ divides $b_n$ for an infinity of integers $n\geq1$ . Prove that there exist an integer $c$ such that $b_n=ca_n$ for all $n\geq1$

number theory

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

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

Hard NT problem

by tiendat004, Aug 15, 2024, 2:55 PM

Given two odd positive integers $a,b$ are coprime. Consider the sequence $(x_n)$ given by $x_0=2,x_1=a,x_{n+2}=ax_{n+1}+bx_n,$ $\forall n\geq 0$ . Suppose that there exist positive integers $m,n,p$ such that $mnp$ is even and $\dfrac{x_m}{x_nx_p}$ is an integer. Prove that the numerator in its simplest form of $\dfrac{m}{np}$ is an odd integer greater than $1$ .

number theory

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

Ez Number Theory

by IndoMathXdZ, Jul 17, 2019, 12:17 PM

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

This post has been edited 2 times. Last edited by djmathman, Dec 29, 2019, 1:43 AM
Reason: edited to reflect actual wording in the shortlist re post #7

f(n+1) = f(n) + 2^f(n) implies f(n) distinct mod 3^2013

by v_Enhance, Aug 13, 2013, 6:06 AM

Define a function $f: \mathbb N \to \mathbb N$ by $f(1) = 1$ , $f(n+1) = f(n) + 2^{f(n)}$ for every positive integer $n$ . Prove that $f(1), f(2), \dots, f(3^{2013})$ leave distinct remainders when divided by $3^{2013}$ .

function

modular arithmetic

induction

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