Checking a summand property for integers sufficiently large.

by DinDean, Apr 22, 2025, 5:21 PM

For any fixed integer $m\geqslant 2$ , prove that there exists a positive integer $f(m)$ , such that for any integer $n\geqslant f(m)$ , $n$ can be expressed by a sum of positive integers $a_i$ 's as
$n=a_1+a_2+\dots+a_m,$ where $a_1\mid a_2$ , $a_2\mid a_3$ , $\dots$ , $a_{m-1}\mid a_m$ .

number theory

combinatorics

Combo problem

by soryn, Apr 22, 2025, 6:33 AM

The school A has m1 boys and m2 girls, and ,the school B has n1 boys and n2 girls. Each school is represented by one team formed by p students,boys and girls. If f(k) is the number of cases for which,the twice schools has,togheter k girls, fund f(k) and the valute of k, for which f(k) is maximum.

combinatorics

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

Calculate the distance of chess king!!

by egxa, Apr 18, 2025, 9:58 AM

A chess king was placed on a square of an $8 \times 8$ board and made $64$ moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called $\emph{pleasant}$ if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)

combinatorics

grid

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

hard problem

by Cobedangiu, Apr 2, 2025, 6:11 PM

Let $x,y,z>0$ and $xy+yz+zx=3$ : Prove that :
$\sum \ \frac{x}{y+z}\ge\sum \frac{1}{\sqrt{x+3}}$

inequalities

As some nations like to say "Heavy theorems mostly do not help"

by Assassino9931, Dec 20, 2022, 12:02 AM

We say that a positive integer $n$ is lovely if there exist a positive integer $k$ and (not necessarily distinct) positive integers $d_1$ , $d_2$ , $\ldots$ , $d_k$ such that $n = d_1d_2\cdots d_k$ and $d_i^2 \mid n + d_i$ for $i=1,2,\ldots,k$ .

a) Are there infinitely many lovely numbers?

b) Is there a lovely number, greater than $1$ , which is a perfect square of an integer?

This post has been edited 1 time. Last edited by Assassino9931, Dec 20, 2022, 12:02 AM

number theory

Divisors

greatest common divisor

Divisibility

Circumcircle excircle chaos

by CyclicISLscelesTrapezoid, Jul 12, 2022, 12:32 PM

Let $ABC$ be a triangle with circumcircle $\omega$ and let $\Omega_A$ be the $A$ -excircle. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_A$ . Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_A$ at $X$ and $Y$ respectively. The tangent line at $P$ to the circumcircle of the triangle $APX$ intersects the tangent line at $Q$ to the circumcircle of the triangle $AQY$ at a point $R$ . Prove that $\overline{AR} \perp \overline{BC}$ .

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

congruence

by moldovan, Jun 26, 2009, 7:19 AM

Let $p$ be an odd prime. Prove that:
$\displaystyle\sum_{k=1}^{p-1}k^{2p-1} \equiv \frac{p(p+1)}{2} \pmod{p^2}$

modular arithmetic

number theory proposed

number theory

Submit

um this does seem slightly similar to ai
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59 shouts

Pi in the Sky

by DinDean, Apr 22, 2025, 5:21 PM

by soryn, Apr 22, 2025, 6:33 AM

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

by egxa, Apr 18, 2025, 9:58 AM

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

by Cobedangiu, Apr 2, 2025, 6:11 PM

by Assassino9931, Dec 20, 2022, 12:02 AM

by CyclicISLscelesTrapezoid, Jul 12, 2022, 12:32 PM

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

by moldovan, Jun 26, 2009, 7:19 AM