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Found: 8

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.

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?

9 straight lines each divide a square into two quadrilaterals, with their areas having a ratio of \(2:3\). Prove that at least three of the nine lines pass through the same point.

Kate and Gina agreed to meet at the underground in the first hour of the afternoon. Kate comes to the meeting place between noon and one o’clock in the afternoon, waits for 10 minutes and then leaves. Gina does the same.

a) What is the probability that they will meet?

b) How will the probability of a meeting change if Gina decides to come earlier than half past twelve, and Kate still decides to come between noon and one o’clock?

c) How will the probability of a meeting change if Gina decides to come at an arbitrary time between 12:00 and 12:50, and Kate still comes between 12:00 and 13:00?

On one island, one tribe has a custom – during the ritual dance, the leader throws up three thin straight rods of the same length, connected in the likeness of the letter capital \(\pi\), \(\Pi\). The adjacent rods are connected by a short thread and therefore freely rotate relative to each other. The bars fall on the sand, forming a random figure. If it turns out that there is self-intersection (the first and third bars cross), then the tribe in the coming year are waiting for crop failures and all sorts of trouble. If there is no self-intersection, then the year will be successful – satisfactory and happy. Find the probability that in 2019, the rods will predict luck.

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.

The population of China is one billion people. It would seem that on a map of China with a scale of 1 : 1,000,000 (1 cm : 10 km), it would be possible to fit a million times fewer people than there is in the whole country. However, in fact, not only 1000, but even 100 people will not be able to be placed on this map. Can you explain this contradiction?