At the age of 12 I discovered a beautiful (I think until now) solution to the following problem. Given two segments AB ans CD in the space. Let M be the midpoint of AB, and N of CD. Prove that |MN| \leq 1/2(|AC|+|BD|).

My solution was: Cosider triangle ABC. Let E be the midpoint of BC. Then |ME|=1/2|AC|. Consider triangle BCD. Then |EN=1/2|BD|. Using triangle inequality for the triangle MEN, we obtain |MN|\leq |ME|+|EN|=1/2(|AC|+|BD|).

Excellent example Vladislav!

For me it was not a "mathematical problem" that I had to solve, but it was the following spontaneous inferred discovery (also around the age of 12-13):

The canonical parabola y=x^2 for integer values of x can be represented as the sum of the n initial odd numbers.

Quite stupid discovery I acknowledge, short thereafter I realised that that is true and that could be demonstrated as

y=sum(2n-1) for n integer

and

sum(2n-1)=sum(2n)-x=2sum(n)-x=2x(x+1)/2-x=x^2

not very high level maths but I was somehow very proud of myself...

thank you for this question

make me to smile and remember that I was young. Now, among of books, papers and my computer, I am thinking on my first big issue regarding of elliptic curves (second year on high school) 

At around 12 I discover my own method to multiply and divide fractions because I did not find intuitive the standard method.  I quickly had to forget it and to learn the standard way.  In my engineering degree in a course where we had to learn to do integral on curve.  I was using the easy and intuitive method that only work on the circle.  We were taught the more general method that work on all curve but I did not learn it. In the exam , we had to do an integral on an ellipse but I did not know the general method.  I then discover in the exam that the elipse was a circle for the y coordinate and another circle with a different radius for the x coordinate and I could solve the problem.  The teacher did not give me the points because I did not use the correct method.  

A nice question. Probably the first 'discovery' was at around 12 when I tried spontaneously read Markushevich's book Infinite series (in a Little Mathematics Library series). The formula for nth term of a Maclaurin series turned out to be a stunning discovery and experience for me.. Another example was a bit later, famous Euler's solution of the Basel problem (in Polyas book).

Another one by me, is the fact the the function

y=ln(x)

can be rewritten as

y=lim(n->0)[(x^n-1)/n]

I noticed this with 14 or so, when we learnt the derivatives of the functions

and for

y=x^n

y'=n*(x^(n-1))

but for

y=ln(x)

y'=1/x

so I did

y'=lim(n->-1)[x^n]

which primitive rewritten is

y=lim(n->0)[(x^n-1)/n]

practically useless...