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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?

Problem #PRU-79570

Problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction

13–14 3.0

In a group of seven boys, everyone has at least three brothers who are in that group. Prove that all seven are brothers.

Problem #PRU-79595

Problems Algebra Functional equations

14–16 4.0

The function \(f (x)\) for each real value of \(x\in (-\infty, + \infty)\) satisfies the equality \(f (x) + (x + 1/2) \times f (1 - x) = 1\).

a) Find \(f (0)\) and \(f (1)\). b) Find all such functions \(f (x)\).

Problem #PRU-79603

Problems Fun Problems Word Problems Percentages and ratios Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

A council of 2,000 deputies decided to approve a state budget containing 200 items of expenditure. Each deputy prepared his draft budget, which indicated for each item the maximum allowable, in his opinion, amount of expenditure, ensuring that the total amount of expenditure did not exceed the set value of \(S\). For each item, the board approves the largest amount of expenditure that is agreed to be allocated by no fewer than \(k\) deputies. What is the smallest value of \(k\) for which we can ensure that the total amount of approved expenditures does not exceed \(S\)?