In some country 89 roads emerge from the capital, from the city of Dalny – one road, from the remaining 1988 cities – 20 roads (in each).
Prove that from the capital you can drive to Dalny.
In a graph there are 100 vertices, and the degree of each of them is not less than 50. Prove that the graph is connected.
The faces of a polyhedron are coloured in two colours so that the neighbouring faces are of different colours. It is known that all of the faces except for one have a number of edges that is a multiple of 3. Prove that this one face has a multiple of 3 edges.
Solve the equation in integers \(2x + 5y = xy - 1\).
Prove there are no integer solutions for the equation \(4^k - 4^l = 10^n\).
Recall that a natural number \(x\) is called prime if \(x\) has no divisors except \(1\) and itself. Solve the equation with prime numbers \(pqr = 7(p + q + r)\).
Can you find
a) in the 100th line of Pascal’s triangle, the number \(1 + 2 + 3 + \dots + 98 + 99\)?
b) in the 200th line the sum of the squares of the numbers in the 100th line?
Prove there are no integer solutions for the equation \(3x^2 + 2 = y^2\).
On the dining room table, there is a choice of six dishes. Every day Valentina takes a certain set of dishes (perhaps, she does not take a single dish), and this set of dishes should be different from all of the sets that she took in the previous days. What is the maximum number of days that Valentina will be able to eat according to such rules and how many meals will she eat on average during the day?
Three people play table tennis, and the player who lost the game gives way to the player who did not participate in it. As a result, it turned out that the first player played 10 games and the second played 21 games. How many games did the third player play?