We solve problems

Problem #PRU-104089

Problems Combinatorics Geometry on grid paper Set theory and logic Theory of algotithms Game theory Winning and loosing positions

11–14 3.0

There is a $5 \times 9$ rectangle drawn on squared paper. In the lower left corner of the rectangle is a button. Kevin and Sophie take turns moving the button any number of squares either to the right or up. Kevin goes first. The winner is the one who places the button in upper right corner. Who would win, Kevin or Sophie, by using the right strategy?

Problem #PRU-58080

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

12–13 3.0

A ream of squared paper is shaded in two colours. Prove that there are two horizontal and two vertical lines, the points of intersection of which are shaded in the same colour.

Problem #PRU-60365

Problems Combinatorics Geometry on grid paper Painting problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

An endless board is painted in three colours (each cell is painted in one of the colours). Prove that there are four cells of the same colour, located at the vertices of the rectangle with sides parallel to the side of one cell.

Problem #PRU-86120

Problems Combinatorics Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (other)

13–15 4.0

A board of size $2005 \times 2005$ is divided into square cells with a side length of 1 unit. Some board cells are numbered in some order by numbers 1, 2, ... so that from any non-numbered cell there is a numbered cell within a distance of less than 10. Prove that there can be found two cells with a distance between them of less than 150, which are numbered by numbers that differ by more than 23. (The distance between the cells is the distance between their centres.)

Problem #PRU-104029

Problems Combinatorics Geometry on grid paper

12–13 2.0

a) In the construction in the figure, move two matches so that there are five identical squares created. b) From the new figure, remove 3 matches so that only 3 squares remain.

Problem #PRU-104030

Problems Combinatorics Geometry on grid paper

12–13 2.0

How many squares are shown in the picture?

Problem #PRU-104058

Problems Combinatorics Geometry on grid paper

12–13 2.0

Cut the shape (see the figure) into two identical pieces (coinciding when placed on top of one another).

Problem #PRU-107810

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

13–15 3.0

The centres of all unit squares are marked in a $10 \times 10$ chequered box (100 points in total). What is the smallest number of lines, that are not parallel to the sides of the square, that are needed to be drawn to erase all of the marked points?

Problem #PRU-109808

Problems Combinatorics Geometry on grid paper Integer lattices Integer lattices (other) Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

All of the points with whole number co-ordinates in a plane are plotted in one of three colours; all three colours are present. Prove that there will always be possible to form a right-angle triangle from these points so that its vertices are of three different colours.

Problem #PRU-110072

Problems Combinatorics Dissections, partitions, covers and tilings Dissections with certain properties Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 4.0

A target consists of a triangle divided by three families of parallel lines into 100 equilateral unit triangles. A sniper shoots at the target. He aims at a particular equilateral triangle and either hits it or hits one of the adjacent triangles that share a side with the one he was aiming for. He can see the results of his shots and can choose when to stop shooting. What is the largest number of triangles that the sniper can guarantee he can hit exactly 5 times?