For each pair of real numbers \(a\) and \(b\), consider the sequence of numbers \(p_n = \lfloor 2 \{an + b\}\rfloor\). Any \(k\) consecutive terms of this sequence will be called a word. Is it true that any ordered set of zeros and ones of length \(k\) is a word of the sequence given by some \(a\) and \(b\) for \(k = 4\); when \(k = 5\)?
Note: \(\lfloor c\rfloor\) is the integer part, \(\{c\}\) is the fractional part of the number \(c\).
Is there a sequence of natural numbers in which every natural number occurs exactly once, and for any \(k = 1, 2, 3, \dots\) the sum of the first \(k\) terms of the sequence is divisible by \(k\)?
Ten pairwise distinct non-zero numbers are such that for each two of them either the sum of these numbers or their product is a rational number.
Prove that the squares of all numbers are rational.
A convex figure and point \(A\) inside it are given. Prove that there is a chord (that is, a segment joining two boundary points of a convex figure) passing through point \(A\) and dividing it in half at point \(A\).
Author: A.K. Tolpygo
An irrational number \(\alpha\), where \(0 <\alpha <\frac 12\), is given. It defines a new number \(\alpha_1\) as the smaller of the two numbers \(2\alpha\) and \(1 - 2\alpha\). For this number, \(\alpha_2\) is determined similarly, and so on.
a) Prove that for some \(n\) the inequality \(\alpha_n <3/16\) holds.
b) Can it be that \(\alpha_n> 7/40\) for all positive integers \(n\)?
Numbers \(1,2,\dots,20\) are written on a whiteboard. In one go Louise is allowed to wipe out any two numbers \(a\) and \(b\), and write their sum \(a+b\) instead. Louise enjoys erasing the numbers, and continues the procedure until only one number is left on the whiteboard. What number is it?
Three tablespoons of milk from a glass of milk are poured into a glass of tea, and the liquid is thoroughly mixed. Then three tablespoons of this mixture are poured back into the glass of milk. Which is greater now: the percentage of milk in the tea or the percentage of tea in the milk?
Louise has a chessboard \(8\times8\) without two opposite corners (see the picture), and 31 dominoes \(2\times1\). Can she tile the crippled chessboard with dominoes she got?
Numbers \(1,2,\dots,20\) are written on a whiteboard. In one go Louise is allowed to wipe out any two numbers \(a\) and \(b\), and write instead
(a) \(a+b-1\); (b) \(a\times b\).
As you already know, Louise enjoys erasing the numbers, and has fun until only one number is left on the whiteboard. What number is it?
There is a \(3 \times 3\) grid filled with zeros. Louise is allowed to add 1 to each small square inside any \(2\times2\) grid. Can she ever get the following table as a result of her actions?