We solve problems

Problem #PRU-102844

Problems Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction Algebra and arithmetic Word problems Tables and tournaments Tables and tournaments (other)

12–14 3.0

The cells of a $15 \times 15$ square table are painted red, blue and green. Prove that there are two lines which at least have the same number of cells of one colour.

Problem #PRU-102879

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

What is the maximum number of kings, that cannot capture each other, which can be placed on a chessboard of size $8 \times 8$ cells?

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) Set theory and logic Theory of algotithms 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-104039

Problems Set theory and logic Joke problem

12–14 2.0

We call a natural number “amazing” if it has the form $a^{b} + b^{a}$ (where $a$ and $b$ are natural numbers). For example, the number 57 is amazing, since $57 = 2^{5} + 5^{2}$ . Is the number 2006 amazing?

Problem #PRU-104090

Problems Algebra and arithmetic Rational functions Rational equations

12–14 2.0

Solve the equation: $x + \frac{x}{x} + \frac{x}{x + \frac{x}{x}} = 1$

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-105208

Problems Algebra and arithmetic Word problems Tables and tournaments Number tables and its properties Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 4.0

A teacher filled the squares of a chequered table with $5 \times 5$ different integers and gave one copy of it to Janine and one to Zahara. Janine selects the largest number in the table, then she deletes the row and column containing this number, and then she selects the largest number of the remaining integers, then she deletes the row and column containing this number, etc. Zahara performs similar operations, each time choosing the smallest numbers. Can the teacher fill up the table in such a way that the sum of the five numbers chosen by Zahara is greater than the sum of the five numbers chosen by Janine?

Problem #PRU-107702

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube Combinatorics Dissections, partitions, covers and tilings Features of dissection pieces Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

13–16 3.0

The surface of a $3 \times 3 \times 3$ Rubik’s Cube contains 54 squares. What is the maximum number of squares we can mark, so that no marked squares share a vertex or are directly adjacent to another marked square?

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.