Tangent.

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 $f$ .

4 replies

pomodor_ap

Yesterday at 11:24 AM

jasperE3

37 minutes ago

FE solution too simple?

Yiyj1 8

N 44 minutes ago by lksb

Source: 101 Algebra Problems from the AMSP

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

Clearly, $f(x)=0$ is an obvious solution. Now, let $y=0$ . Then, we have $f(f(x)) = f(x^2),$ or $f(x)=x^2$ . Therefore, the solutions are $f(x)=0, x^2$ .

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

8 replies

Yiyj1

Apr 9, 2025

lksb

44 minutes ago

Polynomials in Z[x]

BartSimpsons 16

N an hour ago by bin_sherlo

Source: European Mathematical Cup 2017 Problem 4

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.

16 replies

BartSimpsons

Dec 27, 2017

bin_sherlo

an hour ago

Why is the old one deleted?

EeEeRUT 13

N an hour ago by EVKV

Source: EGMO 2025 P1

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

13 replies

EeEeRUT

Apr 16, 2025

EVKV

an hour ago

Factor sums of integers

Aopamy 2

N 2 hours ago by cadaeibf

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.

2 replies

Aopamy

Feb 23, 2023

cadaeibf

2 hours ago

Least integer T_m such that m divides gauss sum

Al3jandro0000 33

N 2 hours ago by NerdyNashville

Source: 2020 Iberoamerican P2

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

33 replies

Al3jandro0000

Nov 17, 2020

NerdyNashville

2 hours ago

Estonian Math Competitions 2005/2006

STARS 2

N 2 hours ago by jasperE3

Source: Juniors Problem 4

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?

2 replies

STARS

Jul 30, 2008

jasperE3

2 hours ago

Sum of whose elements is divisible by p

nntrkien 43

N 2 hours ago by lpieleanu

Source: IMO 1995, Problem 6, Day 2, IMO Shortlist 1995, N6

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 $p$ ?

43 replies

nntrkien

Aug 8, 2004

lpieleanu

2 hours ago

Arrangement of integers in a row with gcd

egxa 2

N 2 hours ago by Qing-Cloud

Source: All Russian 2025 10.5 and 11.5

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?

2 replies

egxa

Apr 18, 2025

Qing-Cloud

2 hours ago

Integer representation

RL_parkgong_0106 1

N 2 hours ago by Jackson0423

Source: Own

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

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

1 reply

RL_parkgong_0106

4 hours ago

Jackson0423

2 hours ago

Tangent.

steven_zhang123 2

N Apr 5, 2025 by AshAuktober

Source: China TST 2001 Quiz 6 P1

In $\triangle ABC$ with $AB > BC$ , a tangent to the circumcircle of $\triangle ABC$ at point $B$ intersects the extension of $AC$ at point $D$ . $E$ is the midpoint of $BD$ , and $AE$ intersects the circumcircle of $\triangle ABC$ at $F$ . Prove that $\angle CBF = \angle BDF$ .