We solve problems

Problem #PRU-60347

Problems Algebra 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

How many nine-digit numbers exist, the sum of the digits of which is even?

Problem #PRU-60349

Problems Number Theory Numeral systems Decimal number system

11–13 2.0

We call a natural number “fancy”, if it is made up only of odd digits. How many four-digit “fancy” numbers are there?

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 \(2k+1\) 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.

Problem #PRU-60362

Problems Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

13–15 4.0

You are given 1002 different integers that are no greater than 2000. Prove that it is always possible to choose three of the given numbers so that the sum of two of them is equal to the third.

Will this still always be possible if we are given 1001 integers rather than 1002?

Problem #PRU-60363

Problems Discrete Mathematics Set theory and logic Mathematical logic Mathematical logic (other)

11–12 2.0

A rectangular table is given, in each cell of which a real number is written, and in each row of the table the numbers are arranged in ascending order. Prove that if you arrange the numbers in each column of the table in ascending order, then in the rows of the resulting table, the numbers will still be in ascending order.

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.