The greatest length of a sequence that satisfies a special condition

by EmersonSoriano, Apr 2, 2025, 10:08 PM

Find the largest possible value of the positive integer $N$ given that there exist positive integers $a_1, a_2, \dots, a_N$ satisfying
$a_n = \sqrt{(a_{n-1})^2 + 2018 \, a_{n-2}}\:, \quad \text{for } n = 3,4,\dots,N.$

Sequence

number theory

Summing the GCD of a number and the divisors of another.

by EmersonSoriano, Apr 2, 2025, 10:03 PM

For each pair of positive integers $m$ and $n$ , we define $f_m(n)$ as follows:
$f_m(n) = \gcd(n, d_1) + \gcd(n, d_2) + \cdots + \gcd(n, d_k),$ where $1 = d_1 < d_2 < \cdots < d_k = m$ are all the positive divisors of $m$ . For example,
$f_4(6) = \gcd(6,1) + \gcd(6,2) + \gcd(6,4) = 5$ .

$a)\:$ Find all positive integers $n$ such that $f_{2017}(n) = f_n(2017)$ .

$b)\:$ Find all positive integers $n$ such that $f_6(n) = f_n(6)$ .

GCD

number theory

Locus of a point on the side of a square

by EmersonSoriano, Apr 2, 2025, 9:58 PM

Let $ABCD$ be a fixed square and $K$ a variable point on segment $AD$ . The square $KLMN$ is constructed such that $B$ is on segment $LM$ and $C$ is on segment $MN$ . Let $T$ be the intersection point of lines $LA$ and $ND$ . Find the locus of $T$ as $K$ varies along segment $AD$ .

Locus

geometry

Chess queens on a cylindrical board

by EmersonSoriano, Apr 2, 2025, 9:56 PM

Let $n$ be a positive integer. In an $n \times n$ board, two opposite sides have been joined, forming a cylinder. Determine whether it is possible to place $n$ queens on the board such that no two threaten each other when:

$a)\:$ $n=14$ .

$b)\:$ $n=15$ .

GCD of x^2-y, y^2-z and z^2-x

by EmersonSoriano, Apr 2, 2025, 9:38 PM

Find all positive integers $d$ that can be written in the form
$d = \gcd(|x^2 - y| , |y^2 - z| , |z^2 - x|),$ where $x, y, z$ are pairwise coprime positive integers such that $x^2 \neq y$ , $y^2 \neq z$ , and $z^2 \neq x$ .

This post has been edited 2 times. Last edited by EmersonSoriano, 2 hours ago
Reason: change subject

GCD

number theory

Classic complex number geo

by Ciobi_, Apr 2, 2025, 12:56 PM

Let $M$ be a point in the plane, distinct from the vertices of $\triangle ABC$ . Consider $N,P,Q$ the reflections of $M$ with respect to lines $AB, BC$ and $CA$ , in this order.
a) Prove that $N, P ,Q$ are collinear if and only if $M$ lies on the circumcircle of $\triangle ABC$ .
b) If $M$ does not lie on the circumcircle of $\triangle ABC$ and the centroids of triangles $\triangle ABC$ and $\triangle NPQ$ coincide, prove that $\triangle ABC$ is equilateral.

complex number geometry

Olympiad Geometry problem-second time posting

by kjhgyuio, Apr 2, 2025, 1:03 AM

In trapezium ABCD,AD is parallel to BC and points E and F are midpoints of AB and DC respectively. If
Area of AEFD/Area of EBCF =√3 + 1/3-√3 and the area of triangle ABD is √3 .find the area of trapezium ABCD

kind of well known?

by dotscom26, Apr 1, 2025, 4:11 AM

Let $y_1, y_2, ..., y_{2025}$ be real numbers satisfying
$y_1^2 + y_2^2 + \cdots + y_{2025}^2 = 1.$
Find the maximum value of
$|y_1 - y_2| + |y_2 - y_3| + \cdots + |y_{2025} - y_1|.$

I have seen many problems with the same structure, Id really appreciate if someone could explain which approach is suitable here

This post has been edited 1 time. Last edited by dotscom26, Yesterday at 4:20 AM

algebra

algebra-geometry

2015 solutions for quotient function!

by raxu, Jun 26, 2015, 1:45 AM

Let $\varphi(n)$ denote the number of positive integers less than $n$ that are relatively prime to $n$ . Prove that there exists a positive integer $m$ for which the equation $\varphi(n)=m$ has at least $2015$ solutions in $n$ .

Proposed by Iurie Boreico

This post has been edited 2 times. Last edited by v_Enhance, Aug 23, 2016, 12:47 AM

number theory

relatively prime

function

Analytic Number Theory

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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by EpicSkills32, Apr 26, 2012, 5:11 PM

536 shouts

Seems

by EmersonSoriano, Apr 2, 2025, 10:08 PM

by EmersonSoriano, Apr 2, 2025, 10:03 PM

by EmersonSoriano, Apr 2, 2025, 9:58 PM

by EmersonSoriano, Apr 2, 2025, 9:56 PM

by EmersonSoriano, Apr 2, 2025, 9:38 PM

by Ciobi_, Apr 2, 2025, 12:56 PM

by kjhgyuio, Apr 2, 2025, 1:03 AM

by dotscom26, Apr 1, 2025, 4:11 AM

by raxu, Jun 26, 2015, 1:45 AM

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