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Problem of the Week

Problem E

Dot Dot Dot

Ponto forms a triangle using dots, and then puts five of these triangles together to make a star. When he does this, the dots in the bottom corners of each adjacent triangle overlap. An example is shown below.

An equilateral triangle is formed using 6 identical dots with 3 dots in the bottom row, 2 dots in the middle row, and 1 dot in the top row. Five of these triangles form a five-pointed star with 25 identical dots showing.

Ponto creates a sequence of such stars as follows.

The first three stars in the sequence are shown. They have 10, 25, and 45 dots showing, respectively.

Which star in the sequence will have 20 020 dots showing?

The first three stars in the sequence made up of identical dots.

In solving the above problem, it may be helpful to use the fact that the sum of the first \(n\) positive integers is equal to \(\tfrac{n(n+1)}{2}\). That is, \[1 + 2 + 3 + … + n = \frac{n(n+1)}{2}\]