Problems

Which triangle has the largest area? The dots form a regular grid.

What is the ratio between the red and blue area? All shapes are semicircles.

In a parallelogram $ABCD$, point $E$ belongs to the side $CD$ and point $F$ belongs to the side $BC$. Show that the total red area is the same as the total blue area:

The figure below is a regular pentagram. What is larger, the black area or the blue area?

A circle was inscribed in a square, and another square was inscribed in the circle. Which area is larger, the blue or the orange one?

In a square, the midpoints of its sides were marked and some segments were drawn. There is another square formed in the centre. Find its area, if the side of the square has length $10$.

In a parallelogram $ABCD$, point $E$ belongs to the side $AB$, point $F$ belongs to the side $CD$ and point $G$ belongs to the side $AD$. What is more, the marked red segments $AE$ and $CF$ have equal lengths. Prove that the total grey area is equal to the total black area.

Jane wrote another number on the board. This time it was a two-digit number and again it did not include digit 5. Jane then decided to include it, but the number was written too close to the edge, so she decided to t the 5 in between the two digits. She noticed that the resulting number is 11 times larger than the original. What is the sum of digits of the new number?

a) Find the biggest 6-digit integer number such that each digit, except for the two on the left, is equal to the sum of its two left neighbours.

b) Find the biggest integer number such that each digit, except for the rst two, is equal to the sum of its two left neighbours. (Compared to part (a), we removed the 6-digit number restriction.)

Matt built a simple wooden hut to protect himself from the rain. From the side the hut looks like a right triangle with the right angle at the top. The longer part of the roof has $20$ ft and the shorter one has $15$ ft. What is the height of the hut in feet?