We solve problems

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 $π k$ .

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-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-58299

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

13–15 3.0

Suppose that $n \geq 3$ . Are there n points that do not lie on one line, whose pairwise distances are irrational, and the areas of all of the triangles with vertices in them are rational?

Problem #PRU-58300

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

13–15 3.0

Do there exist three points $A$ , $B$ and $C$ on the plane such that for any point $X$ the length of at least one of the segments $X A$ , $X B$ and $X C$ is irrational?

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 $2 n$ , $n + 1$ numbers are chosen. Prove that among the chosen numbers there are two, one of which is divisible by another.

Problem #PRU-60352

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 2.0

A sack contains 70 marbles, 20 red, 20 blue, 20 yellow, and the rest black or white. What is the smallest number of marbles that need to be removed from the sack, without looking, in order for there to be no less than 10 marbles of the same colour among the removed marbles.

Problem #PRU-60353

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–13 2.0

Some points from a finite set are connected by line segments. Prove that two points can be found which have the same number of line segments connected to them.

Problem #PRU-60354

Problems Methods Pigeonhole principle Pigeonhole principle (other)

13–15 4.0

There are $2 k + 1$ cards numbered with the numbers $1$ to $2 k + 1$ . 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.