During the year, the price for a strudel were twice raised by 50%, and before the New Year they were sold at half price. How much does one strudel cost now, if at the beginning of the year it cost 80 pence?
Harry and Matt came down from a mountain. Harry walked on foot, and Matt went skiing, which was seven times faster than Harry. Halfway down, Matt fell, broke his skis and his leg, and hence travelled twice as slow as Harry. Who will descend first from the mountain?
A country is called a Fiver if, in it, each city is connected by airlines with exactly with five other cities (there are no international flights).
a) Draw a scheme of airlines for a country that is made up of 10 cities.
b) How many airlines are there in a country of 50 cities?
c) Can there be a Fiver country, in which there are exactly 46 airlines?
Prove that if the irreducible rational fraction \(p/q\) is a root of the polynomial \(P (x)\) with integer coefficients, then \(P (x) = (qx - p) Q (x)\), where the polynomial \(Q (x)\) also has integer coefficients.
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 \(3x^2 + 2 = y^2\).
Can seven phones be connected with wires so that each phone is connected to exactly three others?
Write out in a row the numbers from \(1\) to \(9\) (every number once) so that every two consecutive numbers give a two-digit number that is divisible by \(7\) or by \(13\).
Prove that the sum of
a) any number of even numbers is even;
b) an even number of odd numbers is even;
c) an odd number of odd numbers is odd.
Prove that the product of
a) two odd numbers is odd;
b) an even number with any integer is even.