Two people play the following game. Each player in turn rubs out 9 numbers (at his choice) from the sequence \(1, 2, \dots , 100, 101\). After eleven such deletions, 2 numbers will remain. The first player is awarded so many points, as is the difference between these remaining numbers. Prove that the first player can always score at least 55 points, no matter how played the second.
A six-digit phone number is given. How many seven-digit numbers are there from which one can obtain this six-digit number by deleting one digit?
The city plan is a rectangle of \(5 \times 10\) cells. On the streets, a one-way traffic system is introduced: it is allowed to go only to the right and upwards. How many different routes lead from the bottom left corner to the upper right?
27 coins are given, of which one is a fake, and it is known that a counterfeit coin is lighter than a real one. How can the counterfeit coin be found from 3 weighings on the scales without weights?
In a group of friends, each two people have exactly five common acquaintances. Prove that the number of pairs of friends is divisible by 3.
There are two purses and one coin. Inside the first purse is one coin, and inside the second purse is one coin. How can this be?
Write in terms of prime factors the numbers 111, 1111, 11111, 111111, 1111111.
Specify any solution of the puzzle: \(2014 + YES =BEAR\).
You are mixing four magic potions, and you choose how much of each one to use. Let \(a\), \(b\), \(c\), and \(d\) be the amounts of the four potions you pour in, each chosen between \(0\) and \(1\) liter. The wizard tells you that the magic power of your mix is given by the formula \[a + b + c + d - ab - bc - cd - da.\] What is the largest magic power you can create?
To a certain number, we add the sum of its digits and the answer we get is 2014. Give an example of such a number.