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-116573

Problems Geometry Plane geometry Circles Diameter, chords, and secants Chords and secants (other) Polygons Regular polygons

15–17 3.0

What is the maximum number of pairwise non-parallel segments with endpoints at the vertices of a regular \(n\)-gon?

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-102974

Problems Discrete mathematics Algorithm theory

10–12 3.0

There is a 12 litre barrel filled with water, and two empty barrels (one is 5 litres in volume and the other is 8 litres). Using these barrels, try to

a) divide the water into 2 parts with volumes: 3 litres and 9 litres;

b) divide the water into 2 parts with equal volume.

Problem #PRU-102997

Problems Algebra and arithmetic Number systems Binary number system Set theory and logic Theory of algotithms

10–12 3.0

Michael thinks of a number no less than \(1\) and no greater than \(1000\). Victoria is only allowed to ask questions to which Michael can answer “yes” or “no” (Michael always tells the truth). Can Victoria figure out which number Michael thought of by asking \(10\) questions?

Problem #PRU-103791

Problems Algebra and arithmetic Number systems Decimal number system Set theory and logic Mathematical logic Puzzles

12–12 3.0

In the rebus below, replace the letters with numbers such that the same numbers are represented with the same letter. The asterisks can be replaced with any numbers such that the equations hold.

An explanation of the notation used: the unknown numbers in the third and fourth rows are the results of multiplying 1995 by each digit of the number in the second row, respectively. These third and fourth rows are added together to get the total result of the multiplication \(1995 \times ***\), which is the number in the fifth row. This is an example of a “long multiplication table”.