Sum of max of some sequences is sum of powers

Function $f:\mathbb{Z}_{\geq 0}\rightarrow\mathbb{Z}$ satisfies
$f(m+n)^2=f(m|f(n)|)+f(n^2)$ for any non-negative integers $m$ and $n$ . Determine the number of possible sets of integers $\{f(0), f(1), \dots, f(2024)\}$ .

4 replies

Kei0923

Jan 9, 2024

CrazyInMath

12 minutes ago

Evaluate: $\lim_{h\to 0^{-}} \frac{-1}{h}.$

Vulch 1

N an hour ago by happyhippos

Respected users,
I am asking for better solution of the following problem with excellent explanation.
Thank you!

Evaluate: $\lim_{h\to 0^{-}} \frac{-1}{h}.$

1 reply

Vulch

2 hours ago

happyhippos

an hour ago

D1022 : This serie converge?

Dattier 1

N 5 hours ago by Alphaamss

Source: les dattes à Dattier

Is this series $\sum \limits_{k\geq 1} \dfrac{\ln\left(1+\dfrac 13\sin(k)\right)} k$ converge?

1 reply

Dattier

Monday at 8:13 PM

Alphaamss

5 hours ago

Irrational Numbers

EthanWYX2009 1

N Yesterday at 6:49 PM by yofro

Let $x_1,\ldots ,x_{2n-1}$ be distinct irrational numbers. Show that there exists $y_1,\ldots ,y_n\in\{x_1,\ldots ,x_{2n-1}\},$ such that $\forall S\subset [n]\neq\varnothing,$
$\sum_{s\in S}x_s\notin\mathbb Q.$

1 reply

EthanWYX2009

Yesterday at 4:59 AM

yofro

Yesterday at 6:49 PM

VJIMC 2015 Category I, Problem 1

ellipticfunction 9

N Yesterday at 6:08 PM by Rohit-2006

Source: VJIMC2015

Problem 1
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be differentiable on $\mathbb{R}$ . Prove that there exists $x \in [0, 1]$ such that
$\frac{4}{\pi} ( f(1) - f(0) ) = (1+x^2) f'(x) \ .$

9 replies

ellipticfunction

Aug 10, 2015

Rohit-2006

Yesterday at 6:08 PM

Matrix Row and column relation.

Schro 5

N Yesterday at 5:01 PM by Etkan

If ith row of a matrix A is dependent,Then ith column of A is also dependent and vice versa .

Am i correct...

5 replies

Schro

Monday at 2:54 PM

Etkan

Yesterday at 5:01 PM

D1023 : MVT 2.0

Dattier 0

Yesterday at 1:04 PM

Source: les dattes à Dattier

Let $f \in C(\mathbb R)$ derivable on $\mathbb R$ with $\forall x \in \mathbb R,\forall h \geq 0, f(x)-3f(x+h)+3f(x+2h)-f(x+3h) \geq 0$
Is it true that $\forall (a,b) \in\mathbb R^2, |f(a)-f(b)|\leq \max\left(\left|f'\left(\dfrac{a+b} 2\right)\right|,\dfrac {|f'(a)+f'(b)|}{2}\right)\times |a-b|$

0 replies

Dattier

Yesterday at 1:04 PM

0 replies

Time required for draining out water

Kunihiko_Chikaya 1

N Yesterday at 12:22 PM by Mathzeus1024

Source: Kyoto University 2nd-stage entry exam 2006 /Science, Problem

Let $H>0,\ R>0$ and $O$ be the origin, $P\ (R,\ 0\ ,H).$ In space given the vessel formed by the revolution of the segment $OP$ about $z$ -axis. Denote the hight from $O$ to the surface of water by $h.$ When the vessel is filled with water, drain the water out such that the displacement per time is $\sqrt{h}$ at the time of $h,$ that is to say, if the total volume of the water drained out till the time $t$ is $V(t),$ then $\frac{dV}{dt}=\sqrt{h}$ holds. Find the time required for draining out water.

1 reply

Kunihiko_Chikaya

Mar 14, 2006

Mathzeus1024

Yesterday at 12:22 PM

D1021 : Does this series converge?

Dattier 1

N Yesterday at 9:01 AM by Dattier

Source: les dattes à Dattier

Is this series $\sum \limits_{k\geq 1} \dfrac{\ln(1+\sin(k))} k$ converge?

1 reply

Dattier

Apr 26, 2025

Dattier

Yesterday at 9:01 AM

UC Berkeley Integration Bee 2025 Qualifying Exam

Silver08 18

N Yesterday at 5:29 AM by Silver08

Source: UC Berkeley Integration Bee 2025

Good luck and have fun!!!

1. $\int_{-\sqrt[3]{2}}^{1}(x^3+x^6)(x^2+2x^5)dx$
2. $\int \frac{e^x}{1+e^{2x}}+\frac{x}{1+x^2}dx$
3. $\int_0^\pi x^4\cos(x)dx$
4. $\int_{0}^{\frac{\pi}{2}}\sin^2(x)\cos^2(x)dx$
5. $\int_1^2\frac{\sqrt{x^2+2\sqrt{x^2-1}}}{\sqrt{x^2-1}}dx$
6. $\int_2^4\frac{(2-x)}{(x-1)(x-5)}dx$
7. $\int_0^\infty \frac{dx}{x^2-2x+2}$
8. $\int (e^x+\ln(x))\left(e^x+\frac{1}{x}\right)dx$
9. $\int_{-\infty}^{\infty}\frac{\ln(x^2+1)}{x^2}dx$
10. $\int_0^1 \frac{dx}{\sqrt{1+\sqrt{x}}-\sqrt{1-\sqrt{x}}}$
11. $\int_0^1 2^{\ln(x)}dx$
12. $\int \frac{d}{dx}\left [ \frac{e^x}{\ln(x)} \right] \cdot \frac{\ln(x)}{e^x} dx$
13. $\int_1^\infty \ln(x)-\frac{1}{2}\ln(x^2+1)dx$
14. $\int_0^1 2x\sqrt{x-x^2}dx$
15. $\int_0^1 \ln^3(x)dx$
16. $\int_0^\pi\frac{\sqrt{\csc(x)-\sin(x)}}{\sqrt{\sin(x)}+1}dx$
17. $\int_0^6\frac{x}{\sqrt{x+\sqrt{x+\sqrt{x+...}}}}dx$
18. $\int_0^\infty \frac{\sin(x)\cos(2x)\cos^2(x)}{x}dx$
19. $\int_0^{\infty}\frac{dx}{1+x+x^2+x^3}$
20. $\int_{-\infty}^{\infty}\frac{(x^5-x^3)}{(x^2-1)^4+x^4}dx$

18 replies

Silver08

Apr 27, 2025

Silver08

Yesterday at 5:29 AM

Polynomial Limit

P162008 1

N Yesterday at 4:05 AM by P162008

Let $p = \lim_{y\to\infty} \left(\frac{2}{y^2} \left(\lim_{z\to\infty} \frac{1}{z^4} \left(\lim_{x\to 0} \frac{((y^2 + y + 1)x^k + 1)^{z^2 + z + 1} - ((z^2 + z + 1)x^k + 1)^{y^2 + y + 1}}{x^{2k}}\right)\right)\right)^y$ where $k \in N$ and $q = \lim_{n\to\infty} \left(\frac{\binom{2n}{n}. n!}{n^n}\right)^{1/n}$ where $n \in N$ . Find the value of $p.q.$

1 reply

P162008

Apr 25, 2025

P162008

Yesterday at 4:05 AM

Sum of max of some sequences is sum of powers

Miquel-point 0

Apr 6, 2025

Source: Romanian IMO TST 1981, Day 3 P1

Consider the set $M$ of all sequences of integers $s=(s_1,\ldots,s_k)$ such that $0\leqslant s_i\leqslant n,\; i=1,2,\ldots,k$ and let $M(s)=\max\{s_1,\ldots,s_k\}$ . Show that
$\sum_{s\in A} M(s)=(n+1)^{k+1}-(1^k+2^k+\ldots +(n+1)^k).$
Ioan Tomescu

0 replies