We solve problems

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

A carpet of size 4 m by 4 m has had 15 holes made in it by a moth. Is it always possible to cut out a 1 m \(\times\) 1 m area of carpet that doesn’t contain any holes? The holes are considered to be points.

Problem #PRU-79650

Problems Number Theory Divisibility Division with remainders. Arithmetic of remainders Division with remainder Methods Pigeonhole principle Pigeonhole principle (other)

12–13 2.0

Prove that in any group of 2001 whole numbers there will be two whose difference is divisible by 2000.

Problem #PRU-79661

Problems Algebra and arithmetic Arithmetics. Mental maths

11–12 2.0

The natural number \(a\) was increased by 1, and its square increased by 1001. What is \(a\)?

Problem #PRU-79664

Problems Algebra and arithmetic Arithmetics. Mental maths

11–12 2.0

In a basket, there are 30 red and green apples. Among any 12 apples there is at least one red one, and among any 20 apples there is at least one green one. How many red apples and how many green apples are there in the basket?

Problem #PRU-86091

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

11–12 2.0

In the numbers of MEXAILO and LOMONOSOV, each letter denotes a number (different letters correspond to different numbers). It is known that the products of the numbers of these two words are equal. Can both numbers be odd?

Problem #PRU-86492

Problems Discrete Mathematics Set theory and logic Mathematical logic Mathematical logic (other) Number Theory Divisibility Odd and even numbers

12–14 2.0

On an island there are 1,234 residents, each of whom is either a knight (who always tells the truth) or a liar (who always lies). One day, all of the inhabitants of the island were broken up into pairs, and each one said: “He is a knight!" or “He is a liar!" about his partner. Could it eventually turn out to be that the number of “He is a knight!" and “He is a liar!" phrases is the same?

Problem #PRU-86501

Problems Algebra Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other)

12–15 3.0

Determine all natural numbers \(m\) and \(n\) such as \(m! + 12 = n^2\).

Problem #PRU-86553

Problems Discrete Mathematics Algorithm Theory Theory of algorithms (other)

10–12 2.0

A game with 25 coins. In a row there are 25 coins. For a turn it is allowed to take one or two neighbouring coins. The player who has nothing to take loses.

Problem #PRU-86556

Problems Discrete Mathematics Algorithm Theory Game theory Game theory (other) Methods Invariants and semi-invariants Invariants

11–13 3.0

There are three piles of rocks: in the first pile there are 10 rocks, 15 in the second pile and 20 in the third pile. In this game (with two players), in one turn a player is allowed to divide one of the piles into two smaller piles. The loser is the one who cannot make a move. Which player would be the winner?

Problem #PRU-86557

Problems Discrete Mathematics Algorithm Theory Game theory Symmetric strategies

11–13 2.0

In the first pile there are 100 sweets and in the second there are 200. Consider the game with two players where: in one turn a player can take any amount of sweets from one of the piles. The winner is the one who takes the last sweet. Which player would win by using the correct strategy?