Prove that rational numbers from \([0; 1]\) can be covered by a system of intervals of total length no greater than \(1/1000\).
The positive irrational numbers \(a\) and \(b\) are such that \(1/a + 1/b = 1\). Prove that among the numbers \(\lfloor ma\rfloor , \lfloor nb\rfloor\) each natural number occurs exactly once.
A rectangular billiard with sides 1 and \(\sqrt {2}\) is given. From its angle at an angle of \(45 ^\circ\) to the side a ball is released. Will it ever get into one of the pockets? (The pockets are in the corners of the billiard table).
Suppose that \(n \geq 3\). Are there n points that do not lie on one line, whose pairwise distances are irrational, and the areas of all of the triangles with vertices in them are rational?
Do there exist three points \(A\), \(B\) and \(C\) on the plane such that for any point \(X\) the length of at least one of the segments \(XA\), \(XB\) and \(XC\) is irrational?
Ten circles are marked on the circle. How many non-closed non-self-intersecting nine-point broken lines exist with vertices at these points?
How many nine-digit numbers exist, the sum of the digits of which is even?
Calculate the following sums:
a) \(\binom{5}{0} + 2\binom{5}{1} + 2^2\binom{5}{2} + \dots +2^5\binom{5}{5}\);
b) \(\binom{n}{0} - \binom{n}{1} + \dots + (-1)^n\binom{n}{n}\);
c) \(\binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n}\).
In the expansion of \((x + y)^n\), using the Newton binomial formula, the second term was 240, the third – 720, and the fourth – 1080. Find \(x\), \(y\) and \(n\).
Here is a fragment of the table, which is called the Leibniz triangle. Its properties are “analogous in the sense of the opposite” to the properties of Pascal’s triangle. The numbers on the boundary of the triangle are the inverses of consecutive natural numbers. Each number is equal to the sum of two numbers below it. Find the formula that connects the numbers from Pascal’s and Leibniz triangles.