Sam and Lena have several chocolates, each weighing not more than 100 grams. No matter how they share these chocolates, one of them will have a total weight of chocolate that does not exceed 100 grams. What is the maximum total weight of all of the chocolates?
Around a table sit boys and girls. Prove that the number of pairs of neighbours of different sexes is even.
Could the difference of two integers multiplied by their product be equal to the number 1999?
a) There are 21 coins on a table with the tails side facing upwards. In one operation, you are allowed to turn over any 20 coins. Is it possible to achieve the arrangement were all coins are facing with the heads side upwards in a few operations?
b) The same question, if there are 20 coins, but you are allowed to turn over 19.
Two friends went simultaneously from A to B. The first went by bicycle, the second – by car at a speed five times faster than the first. Halfway along the route, the car was in an accident, and the rest of the way the motorist walked on foot at a speed half of the speed of the cyclist. Which of them arrived at B first?
Andrew drives his car at a speed of 60 km/h. He wants to travel every kilometre 1 minute faster. By how much should he increase his speed?
A tourist walked 3.5 hours, and for every period of time, in one hour, he walked exactly 5 km. Does this mean that his average speed is 5 km/h?
Solve the equations \(x^2 = 14 + y^2\) in integers.
Solve the equations in integers:
a) \(3x^2 + 5y^2 = 345\);
b) \(1 + x + x^2 + x^3 = 2^y\).
Find all the prime numbers \(p\) such that there exist natural numbers \(x\) and \(y\) for which \(p^x = y^3 + 1\).