Not so common problem

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?

9 replies

Yiyj1

Apr 9, 2025

jasperE3

a minute ago

interesting function equation (fe) in IR

skellyrah 2

N 2 minutes ago by jasperE3

Source: mine

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

2 replies

skellyrah

Today at 9:51 AM

jasperE3

2 minutes ago

Complicated FE

XAN4 1

N 4 minutes ago by jasperE3

Source: own

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.

1 reply

XAN4

Today at 11:53 AM

jasperE3

4 minutes ago

IMO Shortlist 2011, G4

WakeUp 125

N 30 minutes ago by Davdav1232

Source: IMO Shortlist 2011, G4

Let $ABC$ be an acute triangle with circumcircle $\Omega$ . Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$ . Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$ . Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$ . Prove that the points $D,G$ and $X$ are collinear.

Proposed by Ismail Isaev and Mikhail Isaev, Russia

125 replies

WakeUp

Jul 13, 2012

Davdav1232

30 minutes ago

Z[x], P(\sqrt[3]5+\sqrt[3]25)=5+\sqrt[3]5

jasperE3 5

N an hour ago by Assassino9931

Source: VJIMC 2013 2.3

Prove that there is no polynomial $P$ with integer coefficients such that $P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5$ .

5 replies

jasperE3

May 31, 2021

Assassino9931

an hour ago

IMO problem 1

iandrei 77

N an hour ago by YaoAOPS

Source: IMO ShortList 2003, combinatorics problem 1

Let $A$ be a $101$ -element subset of the set $S=\{1,2,\ldots,1000000\}$ . Prove that there exist numbers $t_1$ , $t_2, \ldots, t_{100}$ in $S$ such that the sets $A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100$ are pairwise disjoint.

77 replies

iandrei

Jul 14, 2003

YaoAOPS

an hour ago

Divisibility on 101 integers

BR1F1SZ 4

N an hour ago by BR1F1SZ

Source: Argentina Cono Sur TST 2024 P2

There are $101$ positive integers $a_1, a_2, \ldots, a_{101}$ such that for every index $i$ , with $1 \leqslant i \leqslant 101$ , $a_i+1$ is a multiple of $a_{i+1}$ . Determine the greatest possible value of the largest of the $101$ numbers.

4 replies

BR1F1SZ

Aug 9, 2024

BR1F1SZ

an hour ago

2^x+3^x = yx^2

truongphatt2668 2

N an hour ago by CM1910

Prove that the following equation has infinite integer solutions:
$2^x+3^x = yx^2$

2 replies

1 viewing

truongphatt2668

Yesterday at 3:38 PM

CM1910

an hour ago

Prove perpendicular

shobber 29

N 2 hours ago by zuat.e

Source: APMO 2000

Let $ABC$ be a triangle. Let $M$ and $N$ be the points in which the median and the angle bisector, respectively, at $A$ meet the side $BC$ . Let $Q$ and $P$ be the points in which the perpendicular at $N$ to $NA$ meets $MA$ and $BA$ , respectively. And $O$ the point in which the perpendicular at $P$ to $BA$ meets $AN$ produced.

Prove that $QO$ is perpendicular to $BC$ .

29 replies

shobber

Apr 1, 2006

zuat.e

2 hours ago

The smallest of sum of elements

hlminh 1

N 2 hours ago by nguyenhuybao_06

Let $S=\{1,2,...,2014\}$ and $X\subset S$ such that for all $a,b\in X,$ if $a+b\leq 2014$ then $a+b\in X.$ Find the smallest of sum of all elements of $X.$

1 reply

hlminh

2 hours ago

nguyenhuybao_06

2 hours ago

Inequalities

Scientist10 0

2 hours ago

If $x, y, z \in \mathbb{R}$ , then prove that the following inequality holds:
$\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x$

0 replies

+1 w

Scientist10

2 hours ago

0 replies

NT from ukr contest

mshtand1 3

N 2 hours ago by ravengsd

Source: Ukrainian TST for RMM 2021(2) and EGMO 2022 P2

Find the greatest positive integer $n$ such that there exist positive integers $a_1, a_2, ..., a_n$ for which the following holds $a_{k+2} = \dfrac{(a_{k+1}+a_k)(a_{k+1}+1)}{a_k}$ for all $1 \le k \le n-2$ .
Proposed by Mykhailo Shtandenko and Oleksii Masalitin

3 replies

mshtand1

Oct 2, 2021

ravengsd

2 hours ago

Posted before ,but no solution

Nuran2010 1

N 2 hours ago by Nuran2010

Source: 1220 Number Theory Problems

Find all positive integers $n$ where $49n^3+42n^2+11n+1$ is a perfect cube

1 reply

Nuran2010

Apr 11, 2025

Nuran2010

2 hours ago

Not so common problem

m4thbl3nd3r 1

N Apr 15, 2025 by lbh_qys

With each $x\ne -2,-3$ , define $f_1(x)=-\frac{3x+7}{x+2},f_2(x)=f_1(f_1(x))=-\frac{3f_1(x)+7}{f_1(x)+2},f_3(x)=f_1(f_2(x))=-\frac{3f_2(x)+7}{f_2(x)+2},\dots$ Find all integers $x$ such that $f_{269}(x)$ is an integer.