What was the mathematical problem that you managed to solve in your early education (say, school education) that made the most impression on you? If possible, please share the statement of the problem.
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|).
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|).
Dear @Vladislav, my diploma work title in high school was "CIRCLE" I was impressed by PI and its properties! It is inspiring me now!
http://mathworld.wolfram.com/Pi.html
When I was in 9th grade I managed to solve the following problem: describe all possible triples of natural numbers a, b, c such that there exists a triangle whose sides are a, b and c.
To solve this problem I proved the following statment:
Length {a; b; c} of sides of all integer triangles that contains an angle \alpha,
located opposite to the side c and such that \cos\alpha = m/n , where m – integer number and n – natural number, |m| \leq n, can be derived using following formulas
a=d \times \frac {nf^2-nl^2} {D(f,l)} ,
b=d \times \frac {2lnf-2ml^2} {D(f,l)} ,
c=d \times \frac {l^2n-2mlf+nf^2} {D(f,l)},
where d – arbitrary natural number, while l and f – arbitrary natural numbers such that l < f, and D(f,l) - the greatest common divisor of numbers nf^2-nl^2, 2lnf-2ml^2, and l^2n-2mlf+nf^2.
This is a good question. It is good to look back and trace the genesis of various ideas and perceptions in mathematics.
When I was young, I was inspired and fascinated by Euclid's Elements (his postulates, theorems and proofs) that made geometry come alive for me. Later, I started working with Fibonacci numbers, finding new sequences that were eventually published in the Fibonacci Quarterly. I was (and still am) fascinated by the golden section and its occurrence in natural subdivisions of space (something that was understood by Greek architects that designed buildings such as the Parthenon). My fascination with Euclid's geometry led to my appreciation for Archimedes's and Apollonius's work on circles and, in the case of Archimedes, on approximation.
In the beginning of my second year in the university I proved the generalisation of Riemann's theorem on conditionally convergent series:
In finite dimensional space the set of all sums of all rearrangements of a convergent but not absolutely convergent series is a linear subspace.
Outline:
- It is clear it is enough to do two-dimensional case;
- I rather quickly figured out how to construct a basis with sub-series diverging in norm but converging in direction to basic vectors;
- But then I got kind of stuck on what to do with rest?! Finally, and this was inspiring point, I figured out that I may just add one at a time and then use basis to move back to chosen convergent point.
This was quite a lesson in convergence! This problem was also kind of entry test to be admitted by Prof. Troyanski for training!
Thinking about it, 10 years later I would "do" it in two hours by Google. LOL
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