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-58089

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

12–15 4.0

What is the minimum number of points necessary to mark inside a convex \(n\)-gon, so that at least one marked point always lies inside any triangle whose vertices are the vertices of the polygon?

Problem #PRU-58094

Problems Geometry Plane geometry Circles Central angle. Arc length and circumference Connection between the arc length, the angle, and the chord Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 4.0

Several chords are drawn through a unit circle. Prove that if each diameter intersects with no more than \(k\) chords, then the total length of all the chords is less than \(\pi k\).

Problem #PRU-58097

Problems Geometry Plane geometry Circles Central angle. Arc length and circumference Plane transformations Projection onto a line Orthogonal projection Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–16 3.0

Several circles, whose total length of circumferences is 10, are placed inside a square of side 1. Prove that there will always be some straight line that crosses at least four of the circles.

Problem #PRU-58101

Problems Geometry Plane geometry Area Area of a circle, sector, segment Quadrilateral Parallelogram Special cases Rectangles and squares. Criteria and properties

14–15 4.0

A square of side 15 contains 20 non-overlapping unit squares. Prove that it is possible to place a circle of radius 1 inside the large square, so that it does not overlap with any of the unit squares.

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-60341

Problems Probability and statistics Probability theory Probability theory (other)

13–15 3.0

How many are there six-digit numbers that are divisible by \(5\)?

Problem #PRU-60354

Problems Methods Pigeonhole principle Pigeonhole principle (other)

13–15 4.0

There are \(2k+1\) cards numbered with the numbers \(1\) to \(2k+1\). What is the largest number of cards that can be chosen so that no number on a chosen card is equal to the sum of two numbers from two other chosen cards?

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.