A factorization
Let be a right triangle at
. Factor
.
Solution
Since we have successively:
Equality
Prove that in any triangle it holds that
Solution
We have successively:
On a Lambert W integral
Prove that
A Fibonacci series
Let denote the
– th Fibonacci number. Prove that
Solution
Let denote the given sum. Then,
A floor series
Let . Prove that
.
.
where
denotes the sign function.
Solution
- We have successively:
However,
and the result follows.
- Let
. Then,
and
. It follows that
. Hence,
since
is integer for all
. The result follows from the previous question as well as the fact that
if-f
is odd;
.
- It follows from the previous question since
.