We solve problems

Problems Methods Pigeonhole principle Pigeonhole principle (other) Algebra and arithmetic Sequences

A sequence of natural numbers $a_{1} < a_{2} < a_{3} < \dots < a_{n} < \dots$ is such that each natural number is either a term in the sequence, can be expressed as the sum of two terms in the sequence, or perhaps the same term twice. Prove that $a_{n} \leq n^{2}$ for any $n = 1, 2, 3, \dots$

Problem #PRU-73727

Problems Algebra and arithmetic Number theory. Divisibility Divisibility of a number. General properties Geometry Solid geometry Lines and planes in space Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

Out of the given numbers 1, 2, 3, ..., 1000, find the largest number $m$ that has this property: no matter which $m$ of these numbers you delete, among the remaining $1000 - m$ numbers there are two, of which one is divisible by the other.

Problem #PRU-73741

Problems Algebra and arithmetic Number systems Decimal number system Methods Pigeonhole principle Pigeonhole principle (other)

14–16 4.0

An infinite sequence of digits is given. Prove that for any natural number $n$ that is relatively prime with a number 10, you can choose a group of consecutive digits, which when written as a sequence of digits, gives a resulting number written by these digits which is divisible by $n$ .

Problem #PRU-76543

Problems Algebra and arithmetic Number theory. Divisibility Division with remainders. Arithmetic of remainders Division with remainder Methods Pigeonhole principle Pigeonhole principle (other)

12–15 3.0

Prove that, for any integer $n$ , among the numbers $n, n + 1, n + 2, \dots, n + 9$ there is at least one number that is mutually prime with the other nine numbers.

Problem #PRU-77894

Problems Algebra and arithmetic Number theory. Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Methods Pigeonhole principle Pigeonhole principle (other)

13–14 4.0

If we are given any 100 whole numbers then amongst them it is always possible to choose one, or several of them, so that their sum gives a number divisible by 100. Prove that this is the case.

Problem #PRU-78043

Problems Methods Methods of calculus Monotonicity and boundedness Pigeonhole principle Pigeonhole principle (other) Algebra Polynomials Quadratic polynomials

13–15 4.0

The expression $a x^{2} + b x + c$ is an exact fourth power for all integers $x$ . Prove that $a = b = 0$ .

Problem #PRU-78051

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

13–15 4.0

The numbers $⌊ a ⌋, ⌊ 2 a ⌋, \dots, ⌊ N a ⌋$ are all different, and the numbers $⌊ 1 / a ⌋, ⌊ 2 / a ⌋, \dots, ⌊ M / a ⌋$ are also all different. Find all such $a$ .

Problem #PRU-78086

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

13–15 4.0

2022 points are selected from a cube, whose edge is equal to 13 units. Is it possible to place a cube with edge of 1 unit in this cube so that there is not one selected point inside it?

Problem #PRU-78206

Problems Methods Algebraic methods Counting in two ways Pigeonhole principle Pigeonhole principle (other)

13–14 3.0

30 pupils in years 7 to 11 took part in the creation of 40 maths problems. Every possible pair of pupils in the same year created the same number of problems. Every possible pair of pupils in different years created a different number of problems. How many pupils created exactly one problem?

Problem #PRU-78234

Problems Methods Pigeonhole principle Algebraic methods Processes and operations

13–15 4.0

The number $A$ is divisible by $1, 2, 3, \dots, 9$ . Prove that if $2 A$ is presented in the form of a sum of some natural numbers smaller than 10, $2 A = a_{1} + a_{2} + \dots + a_{k}$ , then we can always choose some of the numbers $a_{1}, a_{2}, \dots, a_{k}$ so that the sum of the chosen numbers is equal to $A$ .