# Tag Info

Accepted

### Regarding the solution of finding the remainder of $g(x^{12})$ divided by $g(x)$

I confess i was, am, and will be also hating "artificial" solutions for problems that hide the structure. As a college student, many competitions i took part at had problems that were easily ...
• 24.3k
Accepted

• 42.3k
Accepted

### How many factors of $2400$ are not factors of $3600$?

If you do a prime number decomposition you find $$2400 = 2^5\cdot 3 \cdot 5^2$$ $$3600 = 2^4\cdot 3^2 \cdot 5^2$$ So the only time you can have a factor of $2400$ and not of $3600$ is when $2^5=32$ ...
• 1,198
Accepted

### Prove that there is a circle containing exactly $2018$ points

Your idea is good, and you just need to find the simplest rigorous way forward. One way is as follows: Take any ray $L$ with $2018$ points on its left. (I leave you to figure out how to rigorously ...
• 53.2k

### Prove that $\frac{2022}{n} + 4n$ is a perfect square iff $\frac{2022}{n} - 8n$ is a perfect square

You can greatly reduce the number of cases to check by working modulo $4$. We have $$2022 = 2\cdot 3 \cdot 337 \equiv 2\cdot 3\cdot 1 \pmod 4.$$ Since a square must be $0$ or $1$ mod $4$, we can ...
• 5,493
Accepted

### Help in proving $\sqrt{a-1}+\sqrt{b-1}+\sqrt{c-1}\leq \sqrt{a(bc+1)}$

Hello NiceMathophile welcome to M.S.E. Here's a hint to solve the problem; Can you show that $\sqrt{b-1}+\sqrt{c-1}\leq \sqrt{bc},$ and then show that $\sqrt{a-1}+\sqrt{bc}\leq \sqrt{a(bc+1)}$? Use ...
• 923

### APMO 2020 Geometry Problem | Proving lines to be concurrent

You have a beautiful, neat proof for this problem, and it is absolutely correct, in fact it looks much more neat than the official solutions. There is nothing wrong with your solution. But when exams ...
• 923
Accepted

### Evaluating $\arctan(u)+\arctan(v)+\arctan(w)$, where $u$, $v$, $w$ are the zeros of $P(x) = x^3 - 10x+11$

\begin{multline} \operatorname{Im}[\ln(1+iu)+\ln(1+iv)+\ln(1+iw)] = \operatorname{Im}[\ln((1+iu)(1+iv)(1+iw))] \\= \operatorname{Im}[\ln((-i)^3(i-u)(i-v)(i - w))]= \operatorname{Im}[\ln(iP(i))] \end{...
• 16.7k

### APMO 2020 Geometry Problem | Proving lines to be concurrent

Your proof is fine. By the way, it use with great ingenuity the hypothesis that e 𝐵, 𝐷, 𝐹, 𝐸 are concyclic via the property of radical center.
• 2,501