We solve problems

Problem #PRU-103787

Problems Methods Algebraic methods Counting in two ways Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

One term a school ran 20 sessions of an after-school Astronomy Club. Exactly five pupils attended each session and no two students encountered one another over all of the sessions more than once. Prove that no fewer than 20 pupils attended the Astronomy Club at some point during the term.

Problem #PRU-103819

Problems Methods Pigeonhole principle Pigeonhole principle (other) Discrete Mathematics Algorithm Theory Theory of algorithms (other)

12–14 3.0

In Mexico, environmentalists have succeeded in enacting a law whereby every car should not be driven at least one day a week (the owner informs the police about their car registration number and the day of the week when this car will not be driven). In a certain family, all adults want to travel daily (each for their own business!). How many cars (at least) should the family have, if the family has a) 5 adults? b) 8 adults?

Problem #PRU-103869

Problems Discrete Mathematics Set theory and logic Order relations Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

Solve the puzzle: $B A O \times B A \times B = 2002$ .

Problem #PRU-103960

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

Prove that if 21 people collected 200 nuts between them, there are two people in the group who collected the same number of nuts.

Problem #PRU-103964

Problems Number Theory Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

There are $n$ integers. Prove that among them either there are several numbers whose sum is divisible by $n$ or there is one number divisible by $n$ itself.

Problem #PRU-103987

Problems Methods Pigeonhole principle Pigeonhole principle (other)

10–12 2.0

a) Prove that in any football team there are two players who were born on the same day of the week.

b) Prove that in the population of London, which is almost 9 million, there will be ten thousand people who celebrate their birthday on the same day.

Problem #PRU-104021

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube Number Theory Divisibility Odd and even numbers Methods Pigeonhole principle Pigeonhole principle (other)

12–13 2.0

a) A 1 or a 0 is placed on each vertex of a cube. The sum of the 4 adjacent vertices is written on each face of the cube. Is it possible for each of the numbers written on the faces to be different?

b) The same question, but if 1 and $- 1$ are used instead.

Problem #PRU-105052

Problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction Fun Problems Word Problems Tables and tournaments Tables and tournaments (other)

11–13 3.0

In a chess tournament, each participant played two games with each of the other participants: one with white pieces, the other with black. At the end of the tournament, it turned out that all of the participants scored the same number of points (1 point for a victory, $\frac{1}{2}$ a point for a draw and 0 points for a loss). Prove that there are two participants who have won the same number of games using white pieces.

Problem #PRU-105204

Problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction

12–14 3.0

In a mathematical olympiad, $m > 1$ candidates solved $n > 1$ problems. Each candidate solved a different number of problems to all the others. Each problem was solved by a different number of candidates to all the others. Prove that one of the candidates solved exactly one problem.

Problem #PRU-107762

Problems Methods Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (other)

13–15 4.0

Each of the 1994 deputies in parliament slapped exactly one of his colleagues. Prove that it is possible to draw up a parliamentary commission of 665 people whose members did not clarify the relationship between themselves in the manner indicated above.