and 
are in the coordinate plane such that 
.
denote the locus of all points 
in the coordinate plane satisfying 
,
and 
are on 
and 
.
can be expressed in the form 
,
and 
are relatively prime positive integers. Find 
.
Prove that
Let 
Prove that
is differentiable and 
for all 
.
,
.
2. 
3. 
Bracket 2
2. 
3. 
Bracket 3![$$\int_0^{1}\frac{dx}{1-\sqrt[3]{1-\sqrt{x}}}$$](http://latex.artofproblemsolving.com/d/5/e/d5e8db00985944b2eca341e37a94e34661cb5c86.png)
2. 
3. 
Bracket 4
2. 
3. 
Bracket 5
2. 
3. 
Bracket 6
2. 
3. 
Bracket 7
2. 
3. 
Bracket 8
2. 
3. 
![$$\int_0^{2\pi}\sqrt[4]{\cos^4(x)-\cos(2x)}dx$$](http://latex.artofproblemsolving.com/7/e/e/7eebb4a488fec702e0130a253ef87f0f6dfedb2d.png)
2. ![$$\int_0^1 \frac{x^5}{\sqrt[3]{1-x^3}}dx$$](http://latex.artofproblemsolving.com/6/0/5/605bb9d81d0ccff54a276c9f7bf1c1726c136398.png)
3. 
Bracket 2
2. 
3. 
Bracket 3
2. ![$$\int_0^1 \frac{dx}{(\sqrt{x}+\sqrt[3]{x})^2}$$](http://latex.artofproblemsolving.com/6/9/8/69869c0ed754c0dfacf412ac6411742ffa164a6d.png)
3. 
Bracket 4
2. 
3. 
2. 
3. 
Bracket 2
2. 
3. 
2. 
3. 
2. 
3. 
4. 
5. 
a prime number, with 
and 
,
vector space.
is a group?
and the distribution function is 
then
and find the expression of 
![\[
\int_0^1 \int_{x^2}^x (x^2 + y^2)^{-1/2} \, dy \, dx
\]](http://latex.artofproblemsolving.com/e/a/9/ea9f3a093b1d4c0fa39e5b6d42afa2f71dace1e0.png)
has no roots in the disc of unity.
Y by Adventure10, Rounak_iitr
Let 
be a right triangle and isosceles, with 
. Let 
be the midpoint of 
and 
the midpoint of 
. Let 
be such that 
is an equilateral triangle with inside the quadrilateral 
. Calculate the measure of 
be a right triangle and isosceles, with 
. Let 
be the midpoint of 
and 
the midpoint of 
. Let 
be such that 
is an equilateral triangle with inside the quadrilateral 
. Calculate the measure of 
This post has been edited 1 time. Last edited by parmenides51, Sep 27, 2021, 5:39 PM
Reason: latex
Reason: latex
.
.
and 
.
,
