We solve problems

Problem #PRU-35599

Problems Algebra and arithmetic Algebraic geometry Cryptography Number theory. Divisibility Division with remainders. Arithmetic of remainders Division with remainder Methods Pigeonhole principle Pigeonhole principle (other)

14–15 3.0

Upon the installation of a keypad lock, each of the 26 letters located on the lock’s keypad is assigned an arbitrary natural number known only to the owner of the lock. Different letters do not necessarily have different numbers assigned to them. After a combination of different letters, where each letter is typed once at most, is entered into the lock a summation is carried out of the corresponding numbers to the letters typed in. The lock opens only if the result of the summation is divisible by 26. Prove that for any set of numbers assigned to the 26 letters, there exists a combination that will open the lock.

Problem #PRU-35606

Problems Methods Pigeonhole principle Pigeonhole principle (other)

13–14 3.0

In an $n$ by $n$ grid, $2 n$ of the squares are marked. Prove that there will always be a parallelogram whose vertices are the centres of four of the squares somewhere in the grid.

Problem #PRU-35627

Problems Methods Pigeonhole principle Pigeonhole principle (other) Set theory and logic Set theory Set theory (other) Algebra and arithmetic Fractions Simple fractions

12–15 3.0

A hostess bakes a cake for some guests. Either 10 or 11 people can come to her house. What is the smallest number of pieces she needs to cut the cake into (in advance) so that it can be divided equally between 10 and 11 guests?

Problem #PRU-35710

Problems Probability and statistics Probability theory Probability theory (other) Set theory and logic Theory of algotithms Theory of algorithms (other)

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-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-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 Incircle 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 and arithmetic 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 $20^{\circ}$ .