An easy FE

by oVlad, Apr 21, 2025, 1:36 PM

Determine all functions $f:\mathbb R\to\mathbb R$ such that $f(xy-1)+f(x)f(y)=2xy-1,$ for any real numbers $x{}$ and $y{}.$

algebra

function

GCD of a sequence

by oVlad, Apr 21, 2025, 1:35 PM

Determine all pairs $(a,b)$ of positive integers with the following property: all of the terms of the sequence $(a^n+b^n+1)_{n\geqslant 1}$ have a greatest common divisor $d>1.$

number theory

GCD

sequece

Maximum with the condition $x^2+y^2+z^2=1$

by hlminh, Apr 21, 2025, 9:20 AM

Let $x,y,z$ be real numbers such that $x^2+y^2+z^2=1,$ find the largest value of $E=|x-2y|+|y-2z|+|z-2x|.$

inequalities

UIL Number Sense problem

by Potato512, Apr 21, 2025, 12:17 AM

I keep seeing a certain type of problem in UIL Number Sense, though I can't figure out how to do it (I aim to do it in my head in about 7-8 seconds).

The problem is x^((p+1)/2) mod p, where p is prime.
For example 11^15 mod 29
I know it technically doesn't work this way, but using fermats little theorem (on √x^(p+1)) always gives either the number itself, x, or the modular inverse, p-x.
By using the theorem i mean √x^28 mod 29 = 1, and then youre left with √x^2 mod 29 or x, but then its + or -.
I was wondering if there is a way to figure out whether its + or -, a slow or fast way if its slow maybe its possible to speed it up.

number theory

Paint and Optimize: A Grid Strategy Problem

by mojyla222, Apr 20, 2025, 4:25 AM

Ali and Shayan are playing a turn-based game on an infinite grid. Initially, all cells are white. Ali starts the game, and in the first turn, he colors one unit square black. In the following turns, each player must color a white square that shares at least one side with a black square. The game continues for exactly 2808 turns, after which each player has made 1404 moves. Let $A$ be the set of black cells at the end of the game. Ali and Shayan respectively aim to minimize and maximise the perimeter of the shape $A$ by playing optimally. (The perimeter of shape $A$ is defined as the total length of the boundary segments between a black and a white cell.)

What are the possible values of the perimeter of $A$ , assuming both players play optimally?

combinatorics

Iran 2nd Round

infinite grid

Mock 22nd Thailand TMO P10

by korncrazy, Apr 13, 2025, 6:57 PM

Prove that there exists infinitely many triples of positive integers $(a,b,c)$ such that $a>b>c,\,\gcd(a,b,c)=1$ and $a^2-b^2,a^2-c^2,b^2-c^2$ are all perfect square.

number theory

n + k are composites for all nice numbers n, when n+1, 8n+1 both squares

by parmenides51, Nov 3, 2022, 1:21 PM

The positive $n > 3$ called ‘nice’ if and only if $n +1$ and $8n + 1$ are both perfect squares. How many positive integers $k \le 15$ such that $4n + k$ are composites for all nice numbers $n$ ?

number theory

Perfect Squares

Nationalist Combo

by blacksheep2003, May 24, 2020, 11:00 PM

Let $\mathcal{P}$ be a regular polygon, and let $\mathcal{V}$ be its set of vertices. Each point in $\mathcal{V}$ is colored red, white, or blue. A subset of $\mathcal{V}$ is patriotic if it contains an equal number of points of each color, and a side of $\mathcal{P}$ is dazzling if its endpoints are of different colors.

Suppose that $\mathcal{V}$ is patriotic and the number of dazzling edges of $\mathcal{P}$ is even. Prove that there exists a line, not passing through any point in $\mathcal{V}$ , dividing $\mathcal{V}$ into two nonempty patriotic subsets.

Ankan Bhattacharya

This post has been edited 5 times. Last edited by v_Enhance, Oct 25, 2020, 6:02 AM
Reason: date

USEMO

Distinct Integers with Divisibility Condition

by tastymath75025, Jul 3, 2017, 12:47 AM

For each integer $C>1$ decide whether there exist pairwise distinct positive integers $a_1,a_2,a_3,...$ such that for every $k\ge 1$ , $a_{k+1}^k$ divides $C^ka_1a_2...a_k$ .

Proposed by Daniel Liu

number theory

IMO Shortlist 2014 N6

by hajimbrak, Jul 11, 2015, 9:13 AM

Let $a_1 < a_2 < \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$ . On the segment $I = [0, a_1 a_2 \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 , \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$ .

Proposed by Serbia

This post has been edited 2 times. Last edited by hajimbrak, Jul 23, 2015, 10:52 AM
Reason: Added proposer