We solve problems

Problems Set theory and logic Joke problem Algebra and arithmetic Word problems Motion problems

Andrew drives his car at a speed of 60 km/h. He wants to travel every kilometre 1 minute faster. By how much should he increase his speed?

Problem #PRU-36051

Problems Algebra and arithmetic Word problems Motion problems

11–12 2.0

A tourist walked 3.5 hours, and for every period of time, in one hour, he walked exactly 5 km. Does this mean that his average speed is 5 km/h?

Problem #PRU-58313

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

Ten circles are marked on the circle. How many non-closed non-self-intersecting nine-point broken lines exist with vertices at these points?

Problem #PRU-60299

Problems Algebra and arithmetic 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-60336

Problems Algebra and arithmetic Number systems Decimal number system

11–13 2.0

How many distinct seven-digit numbers exist? It is assumed that the first digit cannot be zero.

Problem #PRU-60347

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

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

Problem #PRU-60349

Problems Algebra and arithmetic Number 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-60359

Problems Algebra and arithmetic Number 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 and arithmetic Sequences Progressions Arithmetic progression Methods Pigeonhole principle Pigeonhole principle (other) 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-60366

Problems Algebra and arithmetic Number 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.