Problems

There are eight points inside a circle of radius 1. Show that there are at least two points with distance between them less then 1.

Leo’s dad was making a pizza for lunch. He decided to place 7 pieces of pineapple on it. Assuming the pizza is a circle of $20$ cm radius, show that some two pieces of pineapple were placed closer than $20$ cm apart.

Each point on a circle was painted either red or green. Show that there is an isosceles triangle whose vertices are on the circumference of the circle, such that all three vertices are red or all three are green.

Anna has a garden shaped like an equilateral triangle of side $8$ metres. She wants to plant $17$ plants, but they need space – they need to be at least $2$ metres apart in order for their roots to have access to all the microelements in the ground. Show that Anna’s garden is unfortunately too small.

Consider the powers of the number five: 1, 5, 25, 125, 625, ... We form the sequence of their first digits: 1, 5, 2, 1, 6, ...

Prove that any part of this sequence, written in reverse order, will occur in the sequence of the first digits of the powers of the number two (1, 2, 4, 8, 1, 3, 6, 1, ...).

Three functions are written on the board: $f_1(x)=x+1/x$, $f_2(x)=x^2,f_3(x)=(x-1{)}^2$. You can add, subtract and multiply these functions (and you can square, cube, etc. them). You can also multiply them by an arbitrary number, add an arbitrary number to them, and also do these operations with the resulting expressions. Therefore, try to get the function $1/x$.

Prove that if you erase any of the functions $f_1,f_2,f_3$ from the board, it is impossible to get $1/x$.

A continuous function $f$ has the following properties:

1. $f$ is defined on the entire number line;

2. $f$ at each point has a derivative (and thus the graph of f at each point has a unique tangent);

3. the graph of the function $f$ does not contain points for which one of the coordinates is rational and the other is irrational.

Does it follow that the graph of $f$ is a straight line?

Prove that any convex polygon contains not more than $35$ vertices with an angle of less than ${170}^{\circ }$.

A circle is covered with several arcs. These arcs can overlap one another, but none of them cover the entire circumference. Prove that it is always possible to select several of these arcs so that together they cover the entire circumference and add up to no more than ${720}^{\circ }$.

Solve the equation: $(x^3-2)(2^{\sin x}-1)+(2^{x^3}-4)\sin x=0$.