Prove that, with central symmetry, a circle transforms into a circle.
The opposite sides of a convex hexagon are pairwise equal and parallel. Prove that it has a centre of symmetry.
A parallelogram \(ABCD\) and a point \(E\) are given. Through the points \(A, B, C, D\), lines parallel to the straight lines \(EC, ED, EA,EB\), respectively, are drawn. Prove that they intersect at one point.
Prove that a circle under the axial symmetry transforms into a circle.
A quadrilateral has an axis of symmetry. Prove that this quadrilateral is either an isosceles trapezoid or is symmetric with respect to its diagonal.
Prove that if a shape has two perpendicular axes of symmetry, then it has a centre of symmetry.
40% of adherents of some political party are women. 70% of the adherents of this party are townspeople. At the same time, 60% of the townspeople who support the party are men. Are the events “the adherent of the party is a townsperson” and “the adherent of party is a woman” independent?
The tower in the castle of King Arthur is crowned with a roof, which is a triangular pyramid, in which all flat angles at the top are straight. Three roof slopes are painted in different colours. The red roof slope is inclined to the horizontal at an angle \(\alpha\), and the blue one at an angle \(\beta\). Find the probability that a raindrop that fell vertically on the roof in a random place fell on the green area.
Investigating one case, the investigator John Smith discovered that the key witness is the one from the Richardson family who, on that fateful day, came home before the others. The investigation revealed the following facts.
1. The neighbour Maria Ramsden, wanting to borrow some salt from the Richardson’s, rang their doorbell, but no one opened the door. At what time though? Who knows? It was already dark...
2. Jill Richardson came home in the evening and found both children in the kitchen, and her husband was on the sofa – he had a headache.
3. The husband, Anthony Richardson, declared that, when he came home, immediately sat down on the sofa and had a nap. He did not see anyone, nor did he hear anything, and the neighbour definitely did not come – the doorbell would have woken him up.
4. The daughter, Sophie, said that when she returned home, she immediately went to her room, and she does not know anything about her father, however, in the hallway, as always, she stumbled on Dan’s shoes.
5. Dan does not remember when he came home. He also did not see his father, but he did hear how Sophie got angry about his shoes.
“Aha,” thought John Smith. “What is the likelihood that Dan returned home before his father?”.
Out of the given numbers 1, 2, 3, ..., 1000, find the largest number \(m\) that has this property: no matter which \(m\) of these numbers you delete, among the remaining \(1000 - m\) numbers there are two, of which one is divisible by the other.