We solve problems

Problems Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

4 points $a, b, c, d$ lie on the segment $[0, 1]$ of the number line. Prove that there will be a point $x$ , lying in the segment $[0, 1]$ , that satisfies $\frac{1}{| x - a |} + \frac{1}{| x - b |} + \frac{1}{| x - c |} + \frac{1}{| x - d |} < 40.$

Problem #PRU-98267

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Proof by contradiction

14–16 4.0

Some points with integer co-ordinates are marked on a Cartesian plane. It is known that no four points lie on the same circle. Prove that there will be a circle of radius 1995 in the plane, which does not contain a single marked point.

Problem #PRU-98298

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle

13–15 3.0

Some real numbers $a_{1}, a_{2}, a_{3}, \dots, a_{2022}$ are written in a row. Prove that it is possible to pick one or several adjacent numbers, so that their sum is less than 0.001 away from a whole number.

Problem #PRU-98408

Problems Geometry Plane geometry Circles Central angle. Arc length and circumference Diameter, chords, and secants Chords and secants (other) Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Polygons Regular polygons

12–14 4.0

In a regular shape with 25 vertices, all the diagonals are drawn.

Prove that there are no nine diagonals passing through one interior point of the shape.

Problem #PRU-98413

Problems Discrete Mathematics Algorithm Theory

12–14 3.0

17 squares are marked on an $8 \times 8$ chessboard. In chess a knight can move horizontally or vertically, one space then two or two spaces then one – eg: two down and one across, or one down and two across. Prove that it is always possible to pick two of these squares so that a knight would need no less than three moves to get from one to the other.

Problem #PRU-98415

Problems Methods Examples and counterexamples. Constructive proofs Algebra and arithmetic Mean values

12–14 4.0

A group of psychologists developed a test, after which each person gets a mark, the number $Q$ , which is the index of his or her mental abilities (the greater $Q$ , the greater the ability). For the country’s rating, the arithmetic mean of the $Q$ values of all of the inhabitants of this country is taken.

a) A group of citizens of country $A$ emigrated to country $B$ . Show that both countries could grow in rating.

b) After that, a group of citizens from country $B$ (including former ex-migrants from $A$ ) emigrated to country $A$ . Is it possible that the ratings of both countries have grown again?

c) A group of citizens from country $A$ emigrated to country $B$ , and group of citizens from country $B$ emigrated to country $C$ . As a result, each country’s ratings was higher than the original ones. After that, the direction of migration flows changed to the opposite direction – part of the residents of $C$ moved to $B$ , and part of the residents of $B$ migrated to $A$ . It turned out that as a result, the ratings of all three countries increased again (compared to those that were after the first move, but before the second). (This is, in any case, what the news agencies of these countries say). Can this be so (if so, how, if not, why)?

(It is assumed that during the considered time, the number of citizens $Q$ did not change, no one died and no one was born).

Problem #PRU-98427

Problems Algebra and arithmetic Sequences Recurrent relations Linear recurrent relations Calculus Functions of one variable. Continuity Periodicity and aperiodicity

12–14 3.0

In a row there are 2023 numbers. The first number is 1. It is known that each number, except the first and the last, is equal to the sum of two neighboring ones. Find the last number.

Problem #PRU-102796

Problems Combinatorics Integer lattices Integer lattices (other) Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

Inside a square with side 1 there are several circles, the sum of the radii of which is 0.51. Prove that there is a line that is parallel to one side of the square and that intersects at least 2 circles.

Problem #PRU-102797

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

Solve the equation $⌊ x^{3} ⌋ + ⌊ x^{2} ⌋ + ⌊ x ⌋ = {x} - 1$ .

Problem #PRU-102839

Problems Fun Problems Word Problems

12–14 2.0

In the US, it is customary to record the date as follows: the number of the month, then the number of the day and then the year. In Europe, the number comes first, then the month and then the year. How many days are there in the year, the date of which can be read definitively, without knowing how it was written?