Complicated FE

by XAN4, Apr 23, 2025, 11:53 AM

Find all solutions for the functional equation $f(xyz)+\sum_{cyc}f(\frac{yz}x)=f(x)\cdot f(y)\cdot f(z)$ , in which $f$ : $\mathbb R^+\rightarrow\mathbb R^+$
Note: the solution is actually quite obvious - $f(x)=x^n+\frac1{x^n}$ , but the proof is important.
Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.

algebra

Find maximum area of right triangle

by jl_, Apr 23, 2025, 10:33 AM

Given a right-angled triangle with hypothenuse $2024$ , find the maximal area of the triangle.

algebra

interesting function equation (fe) in IR

by skellyrah, Apr 23, 2025, 9:51 AM

find all function F: IR->IR such that $xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy)$

The old one is gone.

by EeEeRUT, Apr 16, 2025, 1:37 AM

An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$ .

Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots,$ there are infinitely many positive integers $n$ with $a_n = b_n$ .

This post has been edited 2 times. Last edited by EeEeRUT, Apr 16, 2025, 1:39 AM

algebra

EGMO 2025

A game optimization on a graph

by Assassino9931, Apr 8, 2025, 1:59 PM

Let $X_0, X_1, \dots, X_{n-1}$ be $n \geq 2$ given points in the plane, and let $r > 0$ be a real number. Alice and Bob play the following game. Firstly, Alice constructs a connected graph with vertices at the points $X_0, X_1, \dots, X_{n-1}$ , i.e., she connects some of the points with edges so that from any point you can reach any other point by moving along the edges.Then, Alice assigns to each vertex $X_i$ a non-negative real number $r_i$ , for $i = 0, 1, \dots, n-1$ , such that $\sum_{i=0}^{n-1} r_i = 1$ . Bob then selects a sequence of distinct vertices $X_{i_0} = X_0, X_{i_1}, \dots, X_{i_k}$ such that $X_{i_j}$ and $X_{i_{j+1}}$ are connected by an edge for every $j = 0, 1, \dots, k-1$ . (Note that the length $k \geq 0$ is not fixed and the first selected vertex always has to be $X_0$ .) Bob wins if
$\frac{1}{k+1} \sum_{j=0}^{k} r_{i_j} \geq r;$ otherwise, Alice wins. Depending on $n$ , determine the largest possible value of $r$ for which Bob has a winning strategy.

This post has been edited 1 time. Last edited by Assassino9931, 4 hours ago

Bounding number of solutions for floor function equation

by Ciobi_, Apr 2, 2025, 12:39 PM

Let $n \geq 2$ be a positive integer. Consider the following equation: $\{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor$ a) For $n=2$ , solve the given equation in $\mathbb{R}$ .
b) Prove that, for any $n \geq 2$ , the equation has at most $2$ real solutions.

floor function

algebra

Bounding

Inequalities

by idomybest, Oct 15, 2021, 9:16 PM

The problem is in the attachment below.

Attachments:

Inequality

inequalities

inequalities unsolved

algebra unsolved

algebra

Factor of P(x)

by Brut3Forc3, Apr 4, 2010, 2:45 AM

If $P(x),Q(x),R(x)$ , and $S(x)$ are all polynomials such that $P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),$ prove that $x-1$ is a factor of $P(x)$ .

Composite sum

by rohitsingh0812, Jun 3, 2006, 5:39 AM

Let $a$ , $b$ , $c$ , $d$ , $e$ , $f$ be positive integers and let $S = a+b+c+d+e+f$ .
Suppose that the number $S$ divides $abc+def$ and $ab+bc+ca-de-ef-df$ . Prove that $S$ is composite.

Number theory or function ?

by matematikator, Mar 18, 2005, 2:10 PM

Does there exist a function $s\colon \mathbb{Q} \rightarrow \{-1,1\}$ such that if $x$ and $y$ are distinct rational numbers satisfying ${xy=1}$ or ${x+y\in \{0,1\}}$ , then ${s(x)s(y)=-1}$ ? Justify your answer.

Proposed by Dan Brown, Canada