User Anders Kaseorg

25 votes

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Is $\sum_{a=0}^m\sum_{b=0}^n\cos(abx)$ always positive?

answered Oct 24, 2021 at 0:54

24 votes

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Show that $\left(\frac{x_1^{x_2}}{x_2}\right)^p+\left(\frac{x_2^{x_3}}{x_3}\right)^p+\cdots+\left(\frac{x_n^{x_1}}{x_1}\right)^p\ge n$ for any $p\ge1$

answered Nov 1, 2021 at 8:03

15 votes

How few $(42^\circ,60^\circ,78^\circ)$ triangles can an equilateral triangle be divided into?

answered Jan 11, 2021 at 10:31

15 votes

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Four Dragon Curves are Edge-covering/Plane-tiling

answered Jan 7, 2019 at 0:25

13 votes

Geometric proof of $\cos A + \cos B + \cos C + \cos (A+B+C) = 4 \cos \frac{A+B}{2} \cos \frac{B+C}{2} \cos \frac{A+C}{2}$

answered Sep 28, 2021 at 19:50

10 votes

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Seems to be the optimal coefficient, $k$ integral inequality $\int_{0}^{1}f^2(x)dx\le \frac{423405}{246064}\left(\int_{0}^{1}f(x)dx\right)^2$

answered Mar 4, 2022 at 9:37

9 votes

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Covering area as much as possible over $[0,1]^2$ keeping the average of each axis above a certain level

answered Sep 19, 2022 at 7:22

8 votes

Possible "clever" ways to solve $x^4+x^3-2x+1=0$, with methodological justification

answered Oct 8, 2022 at 8:41

8 votes

Least square fit of ellipse worsens with increasing border thickness

answered Oct 7, 2021 at 10:28

7 votes

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Can there be a cubical bubble?

answered Aug 24, 2012 at 22:06

7 votes

Prove, If the sum of the first $n$ prime is also a prime then it is also a hypotenuse of a primitive Pythagorean triples

answered Feb 26, 2015 at 4:08

7 votes

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Combinatorial Proof that $p(n)/(1+\epsilon)^n \to 0$

answered Jan 6, 2019 at 11:13

7 votes

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Defining the Cosine Function from First Principles, intuitively

answered Jan 7, 2019 at 22:28

7 votes

Chatting about mathematics (with real-time LaTeX rendering)

answered Mar 29, 2020 at 7:33

7 votes

Limit of $n^2$ and a recurrence relation with ceiling function

answered Aug 4, 2022 at 0:26

6 votes

How do I make this 'intuitive' argument into a rigorous proof?

answered Sep 25, 2022 at 19:39

6 votes

Accepted

Optimal strategy in a coin game has unexpected symmetry?

answered Feb 27, 2022 at 2:45

6 votes

Accepted

Show that ${n\choose k}$ is divisible by a prime greater than $k > 0$ when $n \ge 2k$.

answered Jul 31, 2022 at 20:42

6 votes

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Why does the number of regions in a circle cut by chords joining $n+1$ points equal the number of regions in $\mathbb{R}^4$ cut by $n$ hyperplanes?

answered Jan 11, 2020 at 5:32

5 votes

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How to take the square root of a number in a negative base?

answered Jan 7, 2019 at 11:10

5 votes

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Tied chess matches and the monotonicity of $\sum_{k=0}^n \binom{2n}{k,k,2n-2k} (pq)^k (1-p-q)^{2n-2k}$

answered Oct 2, 2021 at 6:23

5 votes

A combinatorial question about odd and even numbers

answered Oct 3, 2021 at 9:14

5 votes

Accepted

38-sided space-filling polyhedron

answered Sep 5, 2022 at 6:17

5 votes

The least possible number of tokens on a $n×n$ board

answered Aug 3, 2022 at 20:25

5 votes

Accepted

Find all intgers $a,b,c$ such that $a\cdot 4^n+b\cdot 6^n+c\cdot 9^n$ is a perfect square

answered Feb 27, 2022 at 1:17

4 votes

Accepted

Why do all eighteen of these rotations have the same angle of rotation?

answered Feb 17, 2022 at 21:09

4 votes

Accepted

Prove that $X,G$ and $B$ are collinear

answered Sep 6, 2022 at 6:35

4 votes

Accepted

Two players roll the dice until the first repetition. Who is more likely to win?

answered Sep 13, 2022 at 21:39

4 votes

Accepted

Location-privacy-preserving protocol for finding relative direction?

answered Oct 30, 2021 at 20:35

4 votes

Accepted

Can the formula $\frac\pi2=(\frac21)^{1/2}(\frac{2^2}{1\cdot3})^{1/4}(\frac{2^3\cdot4}{1\cdot3^3})^{1/8}\cdots$ prove the irrationality of $\pi$?

answered Aug 14, 2021 at 6:59

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99 Answers

Is $\sum_{a=0}^m\sum_{b=0}^n\cos(abx)$ always positive?

Show that $\left(\frac{x_1^{x_2}}{x_2}\right)^p+\left(\frac{x_2^{x_3}}{x_3}\right)^p+\cdots+\left(\frac{x_n^{x_1}}{x_1}\right)^p\ge n$ for any $p\ge1$

How few $(42^\circ,60^\circ,78^\circ)$ triangles can an equilateral triangle be divided into?

Four Dragon Curves are Edge-covering/Plane-tiling

Geometric proof of $\cos A + \cos B + \cos C + \cos (A+B+C) = 4 \cos \frac{A+B}{2} \cos \frac{B+C}{2} \cos \frac{A+C}{2}$

Seems to be the optimal coefficient, $k$ integral inequality $\int_{0}^{1}f^2(x)dx\le \frac{423405}{246064}\left(\int_{0}^{1}f(x)dx\right)^2$

Covering area as much as possible over $[0,1]^2$ keeping the average of each axis above a certain level

Possible "clever" ways to solve $x^4+x^3-2x+1=0$, with methodological justification

Least square fit of ellipse worsens with increasing border thickness

Can there be a cubical bubble?

Prove, If the sum of the first $n$ prime is also a prime then it is also a hypotenuse of a primitive Pythagorean triples

Combinatorial Proof that $p(n)/(1+\epsilon)^n \to 0$

Defining the Cosine Function from First Principles, intuitively

Chatting about mathematics (with real-time LaTeX rendering)

Limit of $n^2$ and a recurrence relation with ceiling function

How do I make this 'intuitive' argument into a rigorous proof?

Optimal strategy in a coin game has unexpected symmetry?

Show that ${n\choose k}$ is divisible by a prime greater than $k > 0$ when $n \ge 2k$.

Why does the number of regions in a circle cut by chords joining $n+1$ points equal the number of regions in $\mathbb{R}^4$ cut by $n$ hyperplanes?

How to take the square root of a number in a negative base?

Tied chess matches and the monotonicity of $\sum_{k=0}^n \binom{2n}{k,k,2n-2k} (pq)^k (1-p-q)^{2n-2k}$

A combinatorial question about odd and even numbers

38-sided space-filling polyhedron

The least possible number of tokens on a $n×n$ board

Find all intgers $a,b,c$ such that $a\cdot 4^n+b\cdot 6^n+c\cdot 9^n$ is a perfect square

Why do all eighteen of these rotations have the same angle of rotation?

Prove that $X,G$ and $B$ are collinear

Two players roll the dice until the first repetition. Who is more likely to win?

Location-privacy-preserving protocol for finding relative direction?

Can the formula $\frac\pi2=(\frac21)^{1/2}(\frac{2^2}{1\cdot3})^{1/4}(\frac{2^3\cdot4}{1\cdot3^3})^{1/8}\cdots$ prove the irrationality of $\pi$?