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Dan
  • Member for 7 years, 4 months
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45 votes
1 answer
891 views

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

44 votes
2 answers
1k views

Intuition is silent: Find the probability that the smallest circle enclosing $n$ random points on a disk lies completely on the disk, as $n\to\infty$.

43 votes
3 answers
4k views

A disc contains $n$ random points. Each point is connected to its nearest neighbor. What does the average cluster size approach as $n\to\infty$?

43 votes
2 answers
1k views

Can a disk of area $1$ be completely covered by six disks of areas $\frac12,\frac13,\frac14,\frac15,\frac16,\frac17$?

38 votes
3 answers
1k views

Crazy fact(?) about circles drawn on base of triangle between cevians: they always fit, no matter what their order?

36 votes
3 answers
1k views

What is the mean value of $|\sin x +\sin (\pi x)|$?

34 votes
6 answers
2k views

Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points.

33 votes
4 answers
7k views

What's wrong with this picture? Impossible circles.

30 votes
2 answers
1k views

What is the largest disk that will be completely covered by randomly placed disks of areas $1,\frac12,\frac13,\dots$ with probability $1$?

29 votes
5 answers
798 views

Conjecture: If matrix $M$ has entries (left to right, then top to botom) $\sin 1,\sin 2,\sin 3,\dots,\sin (n^2)$, where $n\ge 3$, then $\det M = 0$.

29 votes
1 answer
704 views

On average, how many times must a circular pizza be randomly cut, to get a piece with no curved edge?

28 votes
4 answers
659 views

Conjecture: In Pascal's triangle with $n$ rows, the proportion of numbers less than the centre number approaches $e^{-1/\pi}$ as $n\to\infty$.

27 votes
2 answers
715 views

For given $n$, how to find integers $a_0, ..., a_n$ so that $\sum_{k=0}^n a_k x^k$ has $n$ distinct real roots and $\sum_{k=0}^n |a_k|$ is minimized?

26 votes
2 answers
2k views

How to approximate a parameter that gives a tangent line to three circles?

26 votes
0 answers
385 views

Circles of radius $1, 2, 3, ..., n$ all touch a middle circle. How to make the middle circle as small as possible?

26 votes
2 answers
638 views

What is the minimum area of a rectangle containing all circles of radius $1/n$?

26 votes
2 answers
1k views

A curious coincidence: $\prod\limits_{k=1}^\infty\left(1+\int_{k}^{k+1}\left(\frac{\sin (\pi x)}{x}\right)^2\mathrm dx\right)\overset{?}{=}\pi/2$

25 votes
5 answers
972 views

What is $\lim\limits_{n\to\infty}\frac1n \left(\text{maximum value of }\sum\limits_{k=1}^n\sin (kx)\right)$?

25 votes
3 answers
711 views

I tried to kill the central binomial coefficients, but they came back.

25 votes
2 answers
594 views

A square contains many random points. From each point, a disc grows until it hits another disc. What proportion of the square is covered by the discs?

24 votes
2 answers
706 views

Product of areas in a disk

23 votes
2 answers
582 views

Another interesting property of $y=2^{n-1}\prod_{k=0}^n \left(x-\cos{\frac{k\pi}{n}}\right)$: product of arc lengths converges, but to what?

23 votes
2 answers
2k views

Show that $\int_0^{\pi/3}\arccos(2\sin^2 x-\cos x)\mathrm dx=\frac{\pi^2}{5}$

23 votes
1 answer
738 views

Does the interior of Pascal's triangle contain three consecutive integers?

22 votes
5 answers
1k views

Five circles in a rectangle: can the circles move?

21 votes
1 answer
467 views

What is the probability that $\sum_{k=1}^\infty u_k>1$, if $u_1=$ random real number in $(0,1)$, and $u_k=$ random real number in $(0,u_{k-1})$?

21 votes
6 answers
1k views

$\int_0^{\pi/2}\int_0^{\pi/2}\frac{(\tan\alpha)(\tan\beta)}{\tan\alpha+\tan\beta} d\alpha d\beta=(0.9999999913...)(\pi/2)$? Seriously?

20 votes
1 answer
422 views

Take a random walk on Pascal's triangle, without revisits: Does the final number have infinite expectation?

20 votes
1 answer
548 views

Conjecture: If $A,B,C$ are random points on a sphere, then $E\left(\frac{\text{Area}_{\triangle ABC}}{\text{Area}_{\bigcirc ABC}}\right)=\frac14$.

20 votes
5 answers
586 views

If $\sin x+\sin y+\sin z=2$, $\cos x+\cos y+\cos z=11/5$, $\tan x+\tan y+\tan z=17/6$, $x,y,z\in\mathbb{R},$ find $\sin(x+y+z)$ without a calculator

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