On the left there is a circle inscribed in a square of side 1. On the right there are 16 smaller, identical circles, which all together fit inside a square of side 1. Which area is greater, the yellow or the blue one?
In a pentagon \(ABCDE\), diagonal \(AD\) is parallel to the side \(BC\) and the diagonal \(CE\) is parallel to the side \(AB\). Show that the areas of the triangles \(\triangle ABE\) and \(\triangle BCD\) are the same.
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.
Two numbers are given in terms of their prime factorizations: \(a= 2^3 \times 3^2 \times 5 \times 11^2 \times 17^2\) and \(b = 2 \times 5^3 \times 7^2 \times 11 \times 13\).
a) What is the greatest common divisor \(\mathrm{gcd}(a,b)\) of these numbers?
b) What is their least common multiple \(\mathrm{lcm}(a,b)\)?
c) Write down the prime factorization of \(\mathrm{gcd}(a,b) \times \mathrm{lcm}(a,b)\). Then write the prime factorization of \(a \times b\). What do you notice?