We solve problems

Problem #PRU-103964

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

12–14 3.0

There are \(n\) integers. Prove that among them either there are several numbers whose sum is divisible by \(n\) or there is one number divisible by \(n\) itself.

Problem #PRU-30375

Problems Algebra and arithmetic Number theory. Divisibility

12–14 2.0

Prove that \(n^2 + 1\) is not divisible by \(3\) for any natural \(n\).

Problem #PRU-31302

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

12–14 3.0

Recall that a natural number \(x\) is called prime if \(x\) has no divisors except \(1\) and itself. Solve the equation with prime numbers \(pqr = 7(p + q + r)\).

Problem #PRU-31304

Problems Number Theory Number theory. Divisibility Equations in integer numbers

12–14 3.0

Solve the equation with natural numbers \(1 + x + x^2 + x^3 = 2y\).

Problem #PRU-60514

Problems Algebra and arithmetic Algebraic equations and systems of equations Diophantine equations

12–15 3.0

Numbers \(a, b, c\) are integers with \(a\) and \(b\) being coprime. Let us assume that integers \(x_0\) and \(y_0\) are a solution for the equation \(ax + by = c\).

Prove that every solution for this equation has the same form \(x = x_0 + kb\), \(y = y_0 - ka\), with \(k\) being a random integer.

Problem #PRU-78751

Problems Set theory and logic Theory of algotithms Game theory Game theory (other)

13–14 3.0

A monkey escaped from it’s cage in the zoo. Two guards are trying to catch it. The monkey and the guards run along the zoo lanes. There are six straight lanes in the zoo: three long ones form an equilateral triangle and three short ones connect the middles of the triangle sides. Every moment of the time the monkey and the guards can see each other. Will the guards be able to catch the monkey, if it runs three times faster than the guards? (In the beginning of the chase the guards are in one of the triangle vertices and the monkey is in another one.)

Problem #PRU-102818

Problems Set theory and logic Theory of algotithms Game theory Symmetric strategies

12–13 3.0

A two-player game with matches. There are 37 matches on the table. In each turn, a player is allowed to take no more than 5 matches. The winner of the game is the player who takes the final match. Which player wins, if the right strategy is used?

Problem #PRU-102841

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

11–12 2.0

Two weighings. There are 7 coins which are identical on the surface, including 5 real ones (all of the same weight) and 2 counterfeit coins (both of the same weight, but lighter than the real ones). How can you find the 3 real coins with the help of two weighings on scales without weights?

Problem #PRU-102862

Problems Set theory and logic Theory of algotithms Theory of algorithms (other)

10–12 2.0

Multiplication of numbers. Restore the following example of the multiplication of natural numbers if it is known that the sum of the digits of both factors is the same.

Problem #PRU-102863

Problems Set theory and logic Theory of algotithms Theory of algorithms (other)

10–12 2.0

Restore the example of the multiplication.