We solve problems

Problem #PRU-35710

Problems Probability and statistics Probability theory Probability theory (other) Set theory and logic Algorithm Theory Algorithm Theory

12–13 3.0

Three friends decide, by a coin toss, who goes to get the juice. They have one coin. How do they arrange coin tosses so that all of them have equal chances to not have to go and get the juice?

Problem #PRU-35714

Problems Methods Mathematical induction Mathematical induction (other) Set theory and logic Algorithm Theory Algorithm Theory

12–14 3.0

There are 8 glasses of water on the table. You are allowed to take any two of the glasses and make them have equal volumes of water (by pouring some water from one glass into the other). Prove that, by using such operations, you can eventually get all the glasses to contain equal volumes of water.

Problem #PRU-35719

Problems Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

26 numbers are chosen from the numbers 1, 2, 3, ..., 49, 50. Will there always be two numbers chosen whose difference is 1?

Problem #PRU-35733

Problems Algebra Rational functions Identical transformations Set theory and logic Algorithm Theory Algorithm Theory

12–14 3.0

A broken calculator carries out only one operation “asterisk”: \(a*b = 1 - a/b\). Prove that using this calculator it is possible to carry out all four arithmetic operations (addition, subtraction, multiplication, division).

Problem #PRU-35753

Problems Combinatorics Integer lattices Integer lattices (other) Methods Pigeonhole principle Pigeonhole principle (other)

13–15 3.0

A convex polygon on a plane contains no fewer than \(m^2+1\) points with whole number co-ordinates. Prove that within the polygon there are \(m+1\) points with whole number co-ordinates that lie on a single straight line.

Problem #PRU-35758

Problems Geometry Plane geometry Triangles Incirlce and circumcircle of a triangle Polygons Pentagons Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Regular polygons

13–14 3.0

All the points on the edge of a circle are coloured in two different colours at random. Prove that there will be an equilateral triangle with vertices of the same colour inside the circle – the vertices are points on the circumference of the circle.

Problem #PRU-35791

Problems Algebra Word Problems Solving problems by inequalities. Going through the cases. Methods Extremal principle Algebraic methods Processes and operations

13–14 3.0

Sam and Lena have several chocolates, each weighing not more than 100 grams. No matter how they share these chocolates, one of them will have a total weight of chocolate that does not exceed 100 grams. What is the maximum total weight of all of the chocolates?

Problem #PRU-37549

Problems Geometry Plane geometry Lines, segments, rays, and angles Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Triangles

11–13 3.0

Prove that no straight line can cross all three sides of a triangle, at points away from the vertices.

Problem #PRU-54773

Problems Geometry Plane geometry Lines, segments, rays, and angles

13–14 3.0

Ten straight lines are drawn through a point on a plane cutting the plane into angles.
Prove that at least one of these angles is less than \(20^{\circ}\).

Problem #PRU-55621

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

13–15 3.0

A rectangular billiard with sides 1 and \(\sqrt {2}\) is given. From its angle at an angle of \(45 ^\circ\) to the side a ball is released. Will it ever get into one of the pockets? (The pockets are in the corners of the billiard table).