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Dan's user avatar
Dan
  • Member for 7 years, 6 months
  • Last seen this week

31 Offered bounties for 2,200 reputation

14 votes
2 answers
464 views
+100

Conjectured connection between $e$ and $\pi$ in a semidisk.

20 votes
3 answers
560 views
+50

A mysterious limit: probability that a triangle captures the centre of a circle.

16 votes
1 answer
631 views
+50

A probability involving areas in a random pentagram inscribed in a circle: Is it really just $\frac12$?

15 votes
2 answers
520 views
+50

The vertices of a hexagon are random points on a unit circle; $a,b,c$ are the lengths of three random sides. Conjecture: $P(ab<c)=\frac35$.

19 votes
3 answers
1k views
+50

Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\sin y}{\sin(x+y)}$ and the $x$-axis.

10 votes
4 answers
301 views
+100

Draw tangents at 3 random points on a circle to form a triangle. Show that the probability that a random side is shorter than the diameter is $1/2$.

20 votes
1 answer
1k views
+50

Prove $\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12$.

16 votes
2 answers
820 views
+50

The vertices of a triangle are three random points on a unit circle. The side lengths are $a,b,c$. Show that $P(ab>c)=\frac12$.

9 votes
1 answer
302 views
+50

A walk on a $2D$ Poisson process in which every step goes to the nearest unvisited point: expected distance from origin after $365$ steps?

20 votes
2 answers
768 views
+100

What is the expected area of a triangle in which each side is a random real number between $0$ and $1$?

20 votes
2 answers
768 views
+50

What is the expected area of a triangle in which each side is a random real number between $0$ and $1$?

20 votes
1 answer
558 views
+50

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
1 answer
438 views
+50

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

8 votes
3 answers
343 views
+50

How many numbers in the interior of Pascal's triangle are Mersenne numbers?

13 votes
0 answers
403 views
+50

Can this be done? Split Pascal's triangle (without the $1$s) with a straight line into two regions of equal sums.

23 votes
1 answer
744 views
+50

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

34 votes
6 answers
2k views
+100

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

9 votes
0 answers
336 views
+100

Conjecture: If $a_1\in\mathbb{Z}$ and $a_n=\sec (a_{n-1})$ then the proportion of positive terms approaches $\frac{1}{\sqrt{2\pi}}$ as $n\to\infty$.

13 votes
2 answers
308 views
+50

Maximum volume enclosed by a piece of cloth in the shape of a unit circle

19 votes
2 answers
471 views
+50

You have $n$ rectangles of area $1$ (and variable height). Pack as many as possible in a semicircle of area $n$. How many leftovers will there be?

20 votes
2 answers
623 views
+100

A regular $n$-gon contains a regular $(n+1)$-gon, with no sides coinciding. What is the maximum number of points of contact between them?

11 votes
1 answer
360 views
+50

True or false: For every $N\in\mathbb{N}$, there exists $k\in\mathbb{N}$ such that $\sin{(k^n)}<0$ for $n=1,2,3,...,N$.

27 votes
2 answers
718 views
+50

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?

3 votes
1 answer
154 views
+50

Asymptotic expressions for the coordinates of the turning point in $x\in(0,1)$ on $y=|x(x-1)(x-2)\dots(x-n)|$ as $n\to\infty$?

17 votes
0 answers
439 views
+100

Regular polygon with diagonals: bounds on area of largest cell?

13 votes
7 answers
917 views
+100

Prove or disprove: If $f(x)$ is continuous in $(0,1]$ and $f(x)\to\infty$ as $x\to 0^+$, then $\lim_{n\to\infty}\sum_{k=1}^n f(k/n)$ does not exist.

23 votes
2 answers
621 views
+100

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?

18 votes
1 answer
362 views
+50

Inscribe a triangle in a circle to produce four areas. Can the four areas be integer values simultaneously?

13 votes
3 answers
563 views
+200

Without a calculator, determine whether chords $AB, AC \text{ and }BD$ can divide a circle into five equal-area regions.

4 votes
1 answer
221 views
+100

A circle and two perpendicular lines enclose four regions. Can the regions have distinct rational areas?