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MafPrivate's user avatar
MafPrivate
  • Member for 4 years, 10 months
  • Last seen more than a week ago
  • Hong Kong
19 votes

Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction

9 votes
Accepted

If $ab+bc+ca\ge1$, prove that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\ge\frac{\sqrt{3}}{abc}$

8 votes

$n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer.

7 votes
Accepted

Generating function of ordered partitions

6 votes

In how many ways we can distribute 7 distinct balls among 3 students such that everyone gets at least 2 balls?

5 votes
Accepted

Verifying how the sum $\sum_{k=0}^n (-1)^k\binom{n}{k}$ works

5 votes
Accepted

How to find the intersection

5 votes

Geometrically Constructible Angles

4 votes
Accepted

The inequality $\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}\ge \frac{3}{2}$

4 votes
Accepted

Prove it converges: $a_n=\frac{1}{1+1^2}+\frac{1}{1+2^2}+...+\frac{1}{1+n^2}$

4 votes
Accepted

How many positive integer(s) x is / are there so that $\sqrt{x^2 + 4x + 670}$ is an integer?

4 votes

Which is bigger 5^(√3) and 4^(√5)?

4 votes
Accepted

Sum of products of combinatorials

4 votes
Accepted

Proving mean of two odd squares is sum of two squares.

4 votes

$18x\equiv1\pmod{ 25}$. Computing inverses modulo a square.

4 votes
Accepted

How to solve this diophantine equation: $x^2+2y^2=x^2y^2-2000$

4 votes
Accepted

Does the diophantine equation $ax^2+by^2=cz^2+d$ always have solutions?

3 votes
Accepted

is the approach shown below in $\lim_{n \to \infty} [\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\sqrt{n^2+n}} ]=1$ correct?

3 votes

What is $2007^{2008}\pmod{1000}$?

3 votes

Calculating $ \int x^2(x+20)^{2020} dx $

3 votes

Prove that $2 < (1 + \frac{1}{n})^n$ for $n \geq 2$

3 votes

Changing bounds of integration question; spot the mistake.

3 votes
Accepted

Expected length of sequence of positive integers which sums to $2023$

3 votes

Why is $\sqrt{a^2+b^2}\neq a+b$, and is there another rule to simplify the square root?

3 votes
Accepted

Evaluate $\tan 195^{\circ}$ without using the calculator

3 votes

Can I not use QM-AM inequality to solve this?

3 votes

Evaluate $ \int \frac{3x^{2} (\csc{\sqrt{x^{3}-6}})^{2}}{\sqrt{x^{3}-6}} dx $

3 votes

Is there a simple pattern to memorize the sine of $0^\circ$, $15^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $75^\circ$, $90^\circ$?

3 votes

Solve for logarithmic equation. Log on one side.

3 votes
Accepted

Prove the identity $\sin\theta/(1+\cos\theta)=(1-\cos\theta)/\sin\theta$