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9 votes
Accepted

Regular polygon of radius $1$ with diagonals: mysterious ring of radius $1/e$?

I have two graphs. They refer to white dots, which are small discs in the unit circle not crossed by any chord. The first graph shows the size of the largest white dots, for each polygon up to 200 ...
  • 47.9k
1 vote

How to show that $\int_{0}^{1}\sqrt{x}\sqrt{1-x}\,\mathrm dx =\frac{\pi}{8}$

Substitute $x=\frac12(1+t) $ $$\int_{0}^{1}\sqrt{x}\sqrt{1-x}\ dx =\frac14\int_{-1}^1\sqrt{1-t^2}\ dt=\frac{\pi}{8}$$
  • 77k
6 votes

How might I have anticipated that $\frac14(\sqrt{5+2\sqrt5}+\sqrt{10+2\sqrt{5}})$ simplifies to a single surd (namely, $\frac14\sqrt{25+10\sqrt{5}}$)?

$10 + 2 \sqrt 5$ has norm $100 - 5 \cdot 4 = 80.$ $5 + 2 \sqrt 5$ has norm $25 - 5 \cdot 4 = 5.$ The ratio of the norms is $\frac{80}{5} = 16,$ which is an integer and a square, so the ratio ...
  • 133k
8 votes
Accepted

How might I have anticipated that $\frac14(\sqrt{5+2\sqrt5}+\sqrt{10+2\sqrt{5}})$ simplifies to a single surd (namely, $\frac14\sqrt{25+10\sqrt{5}}$)?

Yes. There is a way to formalize this particular type of sum of square roots, similar to the way a determinant is developed for quadratic equations. We can write the generic form of the expression in ...
2 votes

How might I have anticipated that $\frac14(\sqrt{5+2\sqrt5}+\sqrt{10+2\sqrt{5}})$ simplifies to a single surd (namely, $\frac14\sqrt{25+10\sqrt{5}}$)?

One way to (try to) simplify is squaring the sum and see where it leads. Then you get a product of radicals instead of a sum, and it's easier to simplify. $$4A=\sqrt{5+2\sqrt5}+\sqrt{10+2\sqrt5}$$ $$...
0 votes

Find the maximum width and length of a swimming pool under these conditions

This problem can be easily solved by dividing $900$ by $45$. This will allow us to figure out how many one-by-one bricks the gardener have. Doing so, we get $20$. Since he has 20 bricks, that means ...
1 vote
Accepted

Find the area bounded by $y=f(x)$ from $x=1$ to $x=3$ where $f(x)$ satisfies the equation $\int_0^1(x-f(x))f(x)dx=\frac1{12}$

So we have $$\int^{1}_{0}(x-f(x))f(x)dx=\frac{1}{12}.$$ Let's do the following manipulations: $$\int^{1}_{0}(4x-4f(x))f(x)dx=\int^{1}_{0}(4xf(x)-4f(x)^2)dx-\frac{1}{3}=0.$$ Notice that $$\int^{1}_{0}x^...
  • 184
1 vote
Accepted

What is the minimum value of the common area of region M and N?

Either the problem uses minimum instead of maximum, or the solution uses maximum instead of minimum. I've made a quick sketch of the problem. The area you care about is the intersection between $\...
  • 34.5k
0 votes

Element of area in hyperbolic plane with polar coordinates on hyperboloid model

Here's a calculation in change-of-coordinates language. (The use of $r$ throughout may be simpler despite my comment; leaving that as an exercise. Also, it should be straightforward to use ...
3 votes

What is the product of the areas of every regular polygon inscribed in a circle of area $1$?

Not a closed form but some ideas. $$P=\prod\limits_{k=3}^{\infty}\frac{k}{2\pi}\sin{\left(\frac{2\pi}{k}\right)}=\frac{135 \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}}{16 \pi ^4}\,\,\prod\limits_{k=7}^{...
0 votes

Finding the area of a implicit relation

Being a topic of 8 years ago with already a very detailed answer, I certainly don't have the ambition to write innovative things, it's to be understood as a corollary that could be useful to someone. ...
  • 3,455
2 votes
Accepted

How to prove that the triangle areas between squares on a isosceles triangle are equal

The sum of the angles around each vertex is $2\pi$. Since the sum of the angles formed by the squares is $\pi$, so is the sum of the other two angles. The result then follows from the formula for the ...

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