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MafPrivate
  • Member for 4 years, 11 months
  • Last seen more than a month ago
  • Hong Kong
21 votes
4 answers
3k views

Prove a trigonometric identity: $\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C=1$ when $A+B+C=\pi$

16 votes
2 answers
425 views

The best $n$-digit password?

12 votes
1 answer
164 views

Closed form of $\sum^{\infty}_{n=1} \dfrac{1}{n^a{(n+1)}^a}$ where $a$ is a positive integer

8 votes
5 answers
268 views

Algebraic proof of $\sum_{k}\binom{n}{2k}\binom{2k}{k}2^{n-2k}=\binom{2n}n$ (Combinatoric proof is given)

8 votes
2 answers
171 views

Are $a=3, b=7$ the only solutions to $\sqrt{3}+\sqrt[3]{7}=\sqrt{a}+\sqrt[3]{b}$ for $a,b \in \mathbb{Q}$?

7 votes
3 answers
797 views

Solve Diophantine equation: $2^x=5^y+3$ for non-negative integers $x,y$.

7 votes
2 answers
463 views

Find the greatest common divisor of $2^m+1$ and $2^n+1$ that $m,n$ are positive integers.

6 votes
3 answers
283 views

Find the sum of the number of all continuous runs of all possible sequences with $2019$ ones and $2019$ zeros

5 votes
3 answers
132 views

Given $x, y$ that $xy-\frac{x}{y^2}-\frac{y}{x^2}=3$, work out $xy-x-y$.

5 votes
4 answers
155 views

Expression of $x^n+\frac1{x^n}$ by $x+\frac1{x}$ where $n$ is a positive odd number.

5 votes
0 answers
134 views

Find all positive integer solutions of $(a,b,n)$ s.t. $(2^a-1)(2^b-1)=n^2$

4 votes
2 answers
504 views

Use coordinate method to solve a pretty hard geometry problem

4 votes
4 answers
139 views

Maximizing $\frac{a^2+6b+1}{a^2+a}$, where $a=p+q+r=pqr$ and $ab=pq+qr+rp$ for positive reals $p$, $q$, $r$

3 votes
1 answer
302 views

Find the closed form of $\sum\limits_{k=0}^n \frac{1}{n\choose k}$: Incomplete Beta function in a combinatoric question

3 votes
1 answer
80 views

A combinatorial question with an easy rolling dice game

3 votes
2 answers
763 views

How many non-empty subsets of $1,2,3,\cdots,p-2,p-1$ have a sum which is divisible by $p$ that $p$ is an odd prime?

3 votes
1 answer
879 views

Probability and real roots of a polynomial

2 votes
1 answer
245 views

Possible for $e^{\log\left(1+x\right)}=1+x$ in Taylor expansion? [duplicate]

2 votes
2 answers
82 views

Given $a,b,c\ni \mathbb {R_{>0}}$ s.t. $abc\left(a+b+c\right)=1$, find the maximum of $\left(a+b\right)\left(a+c\right)$

2 votes
1 answer
105 views

Is $\sum_{k=0}^n \left(k+1\right)\left(C^n_k\right)^2 = \frac{n+2}{2} C^{2n}_n$ for any positive integer $n$? [duplicate]

2 votes
0 answers
130 views

Functional Calculus Inequality?

1 vote
0 answers
61 views

Questions about positive integers $a,b,c$ s.t. $ab+1,bc+1,ca+1$ are perfect squares [duplicate]

1 vote
1 answer
194 views

Polydivisible Pandigital Number in different bases