There are fewer than 30 people in a class. The probability that at random a selected girl is an excellent student is \(3/13\), and the probability that at random a chosen boy is an excellent pupil is \(4/11\). How many excellent students are there in the class?
In a box of 2009 socks there are blue and red socks. Can there be some number of blue socks that the probability of pulling out two socks of the same colour at random is equal to 0.5?
On a Christmas tree, 100 light bulbs hang in a row. Then the light bulbs begin to switch according to the following algorithm: all are lit up, then after a second, every second light goes out, after another second, every third light bulb changes: if it was on, it goes out and vice versa. After another second, every fourth bulb switches, a second later – every fifth and so on. After 100 seconds the sequence ends. Find the probability that a light bulb straight after a randomly selected light bulb is on (bulbs do not burn out and do not break).
In the set \(-5\), \(-4\), \(-3\), \(-2\), \(-1\), \(0\), \(1\), \(2\), \(3\), \(4\), \(5\), replace one number with two other integers so that the set variance and its mean remain unchanged.
Valerie wrote the number 1 on the board, and then several more numbers. As soon as Valerie writes the next number, Mike calculates the median of the already available set of numbers and writes it in his notebook. At some point, in Mike’s notebook, the numbers: 1; 2; 3; 2.5; 3; 2.5; 2; 2; 2; 2.5 are written.
a) What is the fourth number written on the board?
b) What is the eighth number written on the board?
An abstract artist took a wooden \(5\times 5\times 5\) cube and divided each face into unit squares. He painted each square in one of three colours – black, white, and red – so that there were no horizontally or vertically adjacent squares of the same colour. What is the smallest possible number of squares the artist could have painted black following this rule? Unit squares which share a side are considered adjacent both when the squares lie on the same face and when they lie on adjacent faces.
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 \(n\)?
Prove that for any odd natural number, \(a\), there exists a natural number, \(b\), such that \(2^b - 1\) is divisible by \(a\).
Prove that for any positive integer \(n\), it is always possible to find a number, consisting of the digits \(1\) and \(2,\) that is divisible by \(2^n\). (For example, \(2\) is divisible by \(2\), \(12\) is divisible by \(4,\) \(112\) is divisible by \(8,\) \(2112\) is divisible by \(16\) and so on...).
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.