# Tag Info

Accepted

### Symmetry in Probability (AMC 12A 2023)

The idea in Solution 1, although not explicitly stated, is that if the frog has not reached or passed $10$, irrespective of how much further Flora needs to go, the probability that the next jump ...
Accepted

Accepted

### Number of words with 8 letters using an alphabet of 3 consonants and 2 vowels with constraints

Recurrence approach. Let $c(n)$ be the number of legal words such that the $n$-th letter is a consonant and let $v(n)$ be the number of legal words such that the $n$-th letter is a vowel. Then the ...
Accepted

### Closed form of $\prod_{n=0}^{\infty}\frac1{1+x^{2^n}}$

This is a telescoping product. $$(1-x^{2^n})(1+x^{2^n}) = 1 - x^{2^{n+1}}.$$ Therefore, $$(1-x) \prod_{n=0}^N (1+x^{2^n}) = 1 - x^{2^{N+1}}.$$ The rest of the details I leave to you as an exercise.
Accepted

Accepted

### How long does it take to get to "Olympiad-Bronze-level" of math problem solving ability from no competition experience?

It's more about the time and effort that you put in. If you can set aside 2-4 hours a day to focus on (pretty much anything), you can make a lot of progress in a year. After that it's having the ...
Accepted

### Number of words with 8 letters using an alphabet of 3 consonants and 2 vowels with constraints

I think your approach is fine. First just strings of c's v's, and separating those with v's at the end of the word and those with no v at the end. 0 v's -- $1$ 1 v not ending in v -- $6$ 1 v ending ...
### Evaluating $\int_{0}^{\pi/2}\frac{1}{1+\tan^{101}x}dx$
$$I=\int_{0}^{\pi/2}\frac{1}{1+\tan^{101}x}dx$$ $$I=\int_0^{\frac{\pi}{2}}\frac{\cos^{101}(x)}{\cos^{101}(x)+\sin^{101}(x)}\,dx$$ Use the King's Property; I=\int_0^{\frac{\pi}{2}}\frac{\sin^{101}(x)}...
### To find all functions which satisfy$f(x^3)+f^3(y)+f^3(z)=3xyz \\ \forall x+y+z=0$ and $x,y,z\in\mathbb R$
I will show how we transform this into a Cauchy's functional equation (since this is what the poster attempts to do). Take $x=y=z=0$, then $f(0)+2f(0)^3=0$. The only real solution is $f(0)=0$. Take \$x=...