We solve problems

Problem #PRU-115384

Problems Discrete Mathematics Combinatorics Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 4.0

A game of ’Battleships’ has a fleet consisting of one \(1\times 4\) square, two \(1\times 3\) squares, three \(1\times 2\) squares, and four \(1\times 1\) squares. It is easy to distribute the fleet of ships on a \(10\times 10\) board, see the example below. What is the smallest square board on which this fleet can be placed? Note that by the rules of the game, no two ships can be placed on horizontally, vertically, or diagonally adjacent squares.

Problem #PRU-115397

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

The sequence \(a_1, a_2, \dots\) is such that \(a_1 \in (1,2)\) and \(a_{k + 1} = a_k + \frac{k}{a_k}\) for any positive integer \(k\). Prove that it cannot contain more than one pair of terms with an integer sum.The sequence \(a_1, a_2, \dots\) is such that \(a_1 \in (1,2)\) and \(a_{k + 1} = a_k + \frac{k}{a_k}\) for any positive integer \(k\). Prove that it cannot contain more than one pair of terms with an integer sum.

Problem #PRU-115447

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

Prove that if the expression

takes a rational value, then the expression

also takes on a rational value.

Problem #PRU-115449

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

13–16 4.0

What is the smallest number of ‘L’ shaped ‘corners’ out of 3 squares that can be marked on an \(8\times 8\) square grid, so that no more ’corners’ would fit?

Problem #PRU-115454

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

13–15 4.0

An airline flew exactly 10 flights each day over the course of 92 days. Each day, each plane flew no more than one flight. It is known that for any two days in this period there will be exactly one plane which flew on both those days. Prove that there is a plane that flew every day in this period.

Problem #PRU-115467

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–15 4.0

10 children, including Billy, attended Billy’s birthday party. It turns out that any two children picked from those at the party share a grandfather. Prove that 7 of the children share a grandfather.

Problem #PRU-115470

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

A class has 25 pupils. It is known that for any two girls in the class, the number of male friends they have in the class is different. What is the maximum number of girls that it is possible for there to be in the class?

Problem #PRU-116278

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

13–14 4.0

Two ants crawled along their own closed route on a \(7\times7\) board. Each ant crawled only on the sides of the cells of the board and visited each of the 64 vertices of the cells exactly once. What is the smallest possible number of cell edges, along which both the first and second ants crawled?

Problem #PRU-116301

Problems Geometry Plane geometry Geometrical inequalities Area inequalities Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Area

13–14 4.0

101 random points are chosen inside a unit square, including on the edges of the square, so that no three points lie on the same straight line. Prove that there exist some triangles with vertices on these points, whose area does not exceed 0.01.

Problem #PRU-116435

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

14–16 3.0

On a chessboard, \(n\) white and \(n\) black rooks are arranged so that the rooks of different colours cannot capture one another. Find the greatest possible value of \(n\).