nice integral
by Martin.s, May 27, 2025, 11:24 AM
ISI UGB 2025
by Entrepreneur, May 27, 2025, 9:19 AM
1.)
Suppose
is differentiable and 
Show that for some 
3.)
Suppose![$f:[0,1]\to\mathbb R$](//latex.artofproblemsolving.com/e/a/0/ea02b990406104357c11912a8bf57bea7eeb519d.png)
is differentiable with 
If ![$|f'(x)|\le f(x)\;\forall\;x\in[0,1],$](//latex.artofproblemsolving.com/b/9/d/b9d6088657cd85b37190b98abac1d94a18bb5916.png)
then show that 
4.)
Let
be the unit circle in the complex plane. Let 
be the map given by 
We define 
and 
for 
The smallest positive integer 
such that 
is called period of 
Determine the total number of points 
of period 
6.)
Let
denote the set of natural numbers, and let 
be nine distinct tuples in 
Show that there are 
distinct elements in the set 
whose product is a perfect cube.
8.)
Let
and let 
be positive integers such that 
Prove that 
and determine when equality holds.
Suppose
is differentiable and 
Show that for some 
3.)
Suppose
![$f:[0,1]\to\mathbb R$](http://latex.artofproblemsolving.com/e/a/0/ea02b990406104357c11912a8bf57bea7eeb519d.png)
is differentiable with 
If ![$|f'(x)|\le f(x)\;\forall\;x\in[0,1],$](http://latex.artofproblemsolving.com/b/9/d/b9d6088657cd85b37190b98abac1d94a18bb5916.png)
then show that 
4.)
Let
be the unit circle in the complex plane. Let 
be the map given by 
We define 
and 
for 
The smallest positive integer 
such that 
is called period of 
Determine the total number of points 
of period 
6.)
Let
denote the set of natural numbers, and let 
be nine distinct tuples in 
Show that there are 
distinct elements in the set 
whose product is a perfect cube.8.)
Let
and let 
be positive integers such that 
Prove that 
and determine when equality holds.This post has been edited 2 times. Last edited by Entrepreneur, Today at 9:34 AM
Evaluate: $\int_{0}^{\pi} \text{min}\{2\sin x,1-\cos x,1\} dx$
by Vulch, May 27, 2025, 9:11 AM
Evaluate: 
Sequence and Series
by P162008, May 27, 2025, 6:22 AM
Let 
and 
If
for some integers 
and 
Find the value of 
and 
If
for some integers 
and 
Find the value of 
2024 Putnam A1
by KevinYang2.71, Dec 10, 2024, 4:58 PM
Determine all positive integers for which there exists positive integers 
, 
, and 
satisfying
![\[
2a^n+3b^n=4c^n.
\]](//latex.artofproblemsolving.com/0/2/9/0295405b01494edc05ce9867b2ec48a4e069bf26.png)
for which there exists positive integers 
, 
, and 
satisfying![\[
2a^n+3b^n=4c^n.
\]](http://latex.artofproblemsolving.com/0/2/9/0295405b01494edc05ce9867b2ec48a4e069bf26.png)
IMC 2009 Day 1 P2
by joybangla, Jul 15, 2014, 3:20 PM
Let 
be real square matrices of the same order, and suppose is invertible. Prove that
![\[ (A-B)C=BA^{-1}\implies C(A-B)=A^{-1}B \]](//latex.artofproblemsolving.com/0/6/0/06007271017c2c48faa708366f1adc67db91f245.png)
be real square matrices of the same order, and suppose is invertible. Prove that![\[ (A-B)C=BA^{-1}\implies C(A-B)=A^{-1}B \]](http://latex.artofproblemsolving.com/0/6/0/06007271017c2c48faa708366f1adc67db91f245.png)
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