We solve problems

Problem #PRU-58085

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Covers Geometry Plane geometry Polygons Inscribed and circumscribed polygons Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

13–15 4.0

A unit square contains 51 points. Prove that it is always possible to cover three of them with a circle of radius \(\frac{1}{7}\).

Problem #PRU-58107

Problems Geometry Plane geometry Area Finding the area by rearrangment Discrete Mathematics Set theory and logic Set theory Inclusion-exclusion principle Methods Pigeonhole principle Pigeonhole principle (area and volume) Proof by contradiction

14–15 3.0

a) A square of area 6 contains three polygons, each of area 3. Prove that among them there are two polygons that have an overlap of area no less than 1.

b) A square of area 5 contains nine polygons of area 1. Prove that among them there are two polygons that have an overlap of area no less than \(\frac{1}{9}\).

Problem #PRU-60299

Problems Number Theory Divisibility Divisibility of a number. General properties Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

13–15 4.0

From the set of numbers 1 to \(2n\), \(n + 1\) numbers are chosen. Prove that among the chosen numbers there are two, one of which is divisible by another.

Problem #PRU-60359

Problems Number Theory Numeral systems Decimal number system Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

14–16 4.0

We are given 51 two-digit numbers – we will count one-digit numbers as two-digit numbers with a leading 0. Prove that it is possible to choose 6 of these so that no two of them have the same digit in the same column.

Problem #PRU-60364

Problems Algebra Sequences Progressions Arithmetic progression Methods Pigeonhole principle Pigeonhole principle (other) Fun Problems Word Problems Tables and tournaments Tournament tables

12–14 3.0

In a volleyball tournament teams play each other once. A win gives the team 1 point, a loss 0 points. It is known that at one point in the tournament all of the teams had different numbers of points. How many points did the team in second last place have at the end of the tournament, and what was the result of its match against the eventually winning team?

Problem #PRU-60365

Problems Discrete Mathematics 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-60366

Problems Number Theory Numeral systems Decimal number system Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

13–14 4.0

Prove that amongst any 11 different decimal fractions of infinite length, there will be two whose digits in the same column – 10ths, 100s, 1000s, etc – coincide (are the same) an infinite number of times.

Problem #PRU-60489

Problems Number Theory Divisibility Equations in integer numbers Methods Extremal principle Extremal principle (other) Methods from geometry Geometric interpretations in algebra Pigeonhole principle Pigeonhole principle (other) The greatest common divisor (GCD) and the least common multiplier (LCM). Mutually prime numbers

13–15 4.0

Let \(a\), \(b\), \(c\) be integers; where \(a\) and \(b\) are not equal to zero.

Prove that the equation \(ax + by = c\) has integer solutions if and only if \(c\) is divisible by \(d = \mathrm{GCD} (a, b)\).

Problem #PRU-60732

Problems Number Theory Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Methods Pigeonhole principle Pigeonhole principle (other)

12–15 3.0

Prove that any \(n\) numbers \(x_1,\dots , x_n\) that are not pairwise congruent modulo \(n\), represent a complete system of residues, modulo \(n\).

Problem #PRU-60739

Problems Number Theory Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Numeral systems Decimal number system Methods Pigeonhole principle Pigeonhole principle (other)

13–16 3.0

Prove that for any natural number there is a multiple of it, the decimal notation of which consists of only 0 and 1.