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Problem #PRU-79462

Problems Discrete Mathematics Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction Fun Problems Word Problems Tables and tournaments Tables and tournaments (other)

12–14 3.0

The judges of an Olympiad decided to denote each participant with a natural number in such a way that it would be possible to unambiguously reconstruct the number of points received by each participant in each task, and that from each two participants the one with the greater number would be the participant which received a higher score. Help the judges solve this problem!

Problem #PRU-79473

Problems Methods Pigeonhole principle Pigeonhole principle (other) Algebra Algebraic inequalities and systems of inequalities Quadratic inequalites (several variables)

13–14 3.0

The numbers \(a_1, a_2, \dots , a_{1985}\) are the numbers \(1, 2, \dots , 1985\) rearranged in some order. Each number \(a_k\) is multiplied by its number \(k\), and then the largest number is chosen among the resulting 1985 products. Prove that it is not less than \(993^2\).

Problem #PRU-79488

Problems Methods Pigeonhole principle Pigeonhole principle (other) Number Theory Divisibility The fundamental theorm of arithmetic. Prime factorisation.

13–15 4.0

The product of 1986 natural numbers has exactly 1985 different prime factors. Prove that either one of these natural numbers, or the product of several of them, is the square of a natural number.

Problem #PRU-79498

Problems Methods Pigeonhole principle Pigeonhole principle (other) Number Theory Divisibility The fundamental theorm of arithmetic. Prime factorisation.

13–15 4.0

The product of a group of 48 natural numbers has exactly 10 prime factors. Prove that the product of some four of the numbers in the group will always give a square number.

Problem #PRU-79504

Problems Calculus Derivative Increasing and decreasing. Researching functions

15–16 5.0

Find the minimum for all \(\alpha\), \(\beta\) of the maximum of the function \(y (x) = | \cos x + \alpha \cos 2x + \beta \cos 3x |\).

Problem #PRU-79505

Problems Methods Pigeonhole principle Algebraic methods Proof by exhaustion

12–14 3.0

In March 2015 a teacher ran 11 sessions of a maths club. Prove that if no sessions were run on Saturdays or Sundays there must have been three days in a row where a session of the club did not take place. The 1st March 2015 was a Sunday.

Problem #PRU-79506

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

Prove that from any 27 different natural numbers less than 100, two numbers that are not coprime can be chosen.

Problem #PRU-79515

Problems Number Theory Numeral systems Decimal number system Methods Pigeonhole principle Pigeonhole principle (other)

13–15 4.0

7 different digits are given. Prove that for any natural number \(n\) there is a pair of these digits, the sum of which ends in the same digit as the number.

Problem #PRU-79539

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

13–15 3.0

Is there a line on the coordinate plane relative to which the graph of the function \(y = 2^x\) is symmetric?

Problem #PRU-79545

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

In a dark room on a shelf there are 4 pairs of socks of two different sizes and two different colours that are not arranged in pairs. What is the minimum number of socks necessary to move from the drawer to the suitcase, without leaving the room, so that there are two pairs of socks of different sizes and colours in the suitcase?