We solve problems

Problems Combinatorics Geometry on grid paper

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-104046

Problems Combinatorics

10–13 2.0

A toddler has \(25\) lego pieces in a box:

In how many ways are there to choose three pieces to play with?
In how many ways can he choose three pieces for the foundation, main walls and roof? Note that the order is important.

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-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-107703

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

11–13 2.0

Is it possible to place 12 identical coins along the edges of a square box so that touching each edge there were exactly: a) 2 coins, b) 3 coins, c) 4 coins, d) 5 coins, e) 6 coins, f) 7 coins.

You are allowed to place coins on top of one another. In the cases where it is possible, draw how this could be done. In the other cases, prove that doing so is impossible.

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-108995

Problems Combinatorics Packing problems Methods Pigeonhole principle Pigeonhole principle (area and volume) Systems of points and line segments Systems of points

13–15 3.0

Prove that, in a circle of radius 10, you cannot place 400 points so that the distance between each two points is greater than 1.

Problem #PRU-109018

Problems Combinatorics Dissections, partitions, covers and tilings Covers Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

12–14 3.0

A circle is covered with several arcs. These arcs can overlap one another, but none of them cover the entire circumference. Prove that it is always possible to select several of these arcs so that together they cover the entire circumference and add up to no more than \(720^{\circ}\).

Problem #PRU-109748

Problems Geometry Plane geometry Convex and non-convex figures Convex polygons Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Polygons Polygons (other)

13–15 4.0

We are given a convex 200-sided polygon in which no three diagonals intersect at the same point. Each of the diagonals is coloured in one of 999 colours. Prove that there is some triangle inside the polygon whose sides lie some of the diagonals, so that all 3 sides are the same colour. The vertices of the triangle do not necessarily have to be the vertices of the polygon.