Integer representation

by RL_parkgong_0106, Apr 22, 2025, 2:23 PM

Show that for any positive integer $n$ , there exists some positive integer $k$ that makes the following equation have no integer root $(x_1, x_2, x_3, \dots, x_n)$ .

$x_1^{2^1}+x_2^{2^2}+x_3^{2^3}+\dots+x_n^{2^n}=k$

number theory

real+ FE

by pomodor_ap, Apr 21, 2025, 11:24 AM

Let $f : \mathbb{R}^+ \to \mathbb{R}^+$ be a function such that
$f(x)f(x^2 + y f(y)) = f(x)f(y^2) + x^3$ for all $x, y \in \mathbb{R}^+$ . Determine all such functions $f$ .

functional equation

algebra

Arrangement of integers in a row with gcd

by egxa, Apr 18, 2025, 5:09 PM

Let $n$ be a natural number. The numbers $1, 2, \ldots, n$ are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the $n - 1$ GCDs obtained?

This post has been edited 1 time. Last edited by egxa, Apr 18, 2025, 5:18 PM

GCD

number theory

greatest common divisor

Why is the old one deleted?

by EeEeRUT, Apr 16, 2025, 1:33 AM

For a positive integer $N$ , let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$ . Find all $N \geqslant 3$ such that $\gcd( N, c_i + c_{i+1}) \neq 1$ for all $1 \leqslant i \leqslant m-1$

Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$ . Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$ .

Proposed by Paulius Aleknavičius, Lithuania

This post has been edited 2 times. Last edited by EeEeRUT, Apr 18, 2025, 12:56 AM
Reason: Authorship

number theory

EGMO 2025

Chinese Remainder Theorem

FE solution too simple?

by Yiyj1, Apr 9, 2025, 3:26 AM

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the equality $f(f(x)+y) = f(x^2-y)+4f(x)y$ holds for all pairs of real numbers $(x,y)$ .

My solution

I feel like my solution is too simple. Is there something I did wrong or something I missed?

Functional Equations

function

algebra

Factor sums of integers

by Aopamy, Feb 23, 2023, 3:13 AM

Let $n$ be a positive integer. A positive integer $k$ is called a benefactor of $n$ if the positive divisors of $k$ can be partitioned into two sets $A$ and $B$ such that $n$ is equal to the sum of elements in $A$ minus the sum of the elements in $B$ . Note that $A$ or $B$ could be empty, and that the sum of the elements of the empty set is $0$ .

For example, $15$ is a benefactor of $18$ because $1+5+15-3=18$ .

Show that every positive integer $n$ has at least $2023$ benefactors.

factor

number theory

Least integer T_m such that m divides gauss sum

by Al3jandro0000, Nov 17, 2020, 7:24 PM

Let $T_n$ denotes the least natural such that
$n\mid 1+2+3+\cdots +T_n=\sum_{i=1}^{T_n} i$ Find all naturals $m$ such that $m\ge T_m$ .

Proposed by Nicolás De la Hoz

This post has been edited 3 times. Last edited by Al3jandro0000, Nov 18, 2020, 3:54 PM
Reason: Some clarifications

number theory

Iberoamerican

Polynomials in Z[x]

by BartSimpsons, Dec 27, 2017, 12:25 PM

Find all polynomials $P$ with integer coefficients such that $P (0)\ne 0$ and $P^n(m)\cdot P^m(n)$ is a square of an integer for all nonnegative integers $n, m$ .

Remark: For a nonnegative integer $k$ and an integer $n$ , $P^k(n)$ is defined as follows: $P^k(n) = n$ if $k = 0$ and $P^k(n)=P(P(^{k-1}(n))$ if $k >0$ .

Proposed by Adrian Beker.

This post has been edited 1 time. Last edited by BartSimpsons, Dec 27, 2017, 12:26 PM
Reason: added source

algebra

polynomial

Estonian Math Competitions 2005/2006

by STARS, Jul 30, 2008, 1:17 AM

A $9 \times 9$ square is divided into unit squares. Is it possible to fill each unit square with a number $1, 2,..., 9$ in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?

combinatorics unsolved

combinatorics

Sum of whose elements is divisible by p

by nntrkien, Aug 8, 2004, 1:29 AM

Let $p$ be an odd prime number. How many $p$ -element subsets $A$ of $\{1,2,\dots,2p\}$ are there, the sum of whose elements is divisible by $p$ ?

Submit

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math_exploration

by RL_parkgong_0106, Apr 22, 2025, 2:23 PM

by pomodor_ap, Apr 21, 2025, 11:24 AM

by egxa, Apr 18, 2025, 5:09 PM

by EeEeRUT, Apr 16, 2025, 1:33 AM

by Yiyj1, Apr 9, 2025, 3:26 AM

by Aopamy, Feb 23, 2023, 3:13 AM

by Al3jandro0000, Nov 17, 2020, 7:24 PM

by BartSimpsons, Dec 27, 2017, 12:25 PM

by STARS, Jul 30, 2008, 1:17 AM

by nntrkien, Aug 8, 2004, 1:29 AM