We solve problems

Problems Algebra and arithmetic Number theory. Divisibility Odd and even numbers

Let $m$ and $n$ be integers. Prove that $m n (m + n)$ is an even number.

Problems Methods Pigeonhole principle Pigeonhole principle (other)

The numbers $1, 2, \dots, 99$ are written on 99 cards. Then the cards are shuffled and placed with the number facing down. On the blank side of the cards, the numbers $1, 2, \dots, 99$ are once again written.

The sum of the two numbers on each card are calculated, and the product of these 99 summations is worked out. Prove that the end result will be an even number.

Problem #PRU-60732

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

12–15 3.0

Prove that any $n$ numbers $x_{1}, \dots, x_{n}$ that are not pairwise congruent modulo $n$ , represent a complete system of residues, modulo $n$ .

Problem #PRU-60739

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

13–16 3.0

Prove that for any natural number there is a multiple of it, the decimal notation of which consists of only 0 and 1.

Problem #PRU-60806

Problems Algebra and arithmetic Number systems Decimal number system Number theory. Divisibility Divisibility rules Divisibility tests for 3 and 9 Methods Pigeonhole principle Pigeonhole principle (other)

13–15 3.0

Without calculating the answer to $2^{30}$ , prove that it contains at least two identical digits.

Problem #PRU-60839

Problems Algebra

11–13 2.0

Write the following rational numbers in the form of decimal fractions: a) $\frac{1}{7}$ ; b) $\frac{2}{7}$ ; c) $\frac{1}{14}$ ; d) $\frac{1}{17}$ .

Problem #PRU-60845

Problems Methods Proof by contradiction Calculus Real numbers Rational and irrational numbers

12–14 3.0

Let the number $α$ be given by the decimal:

a) $0.101001000100001000001 \dots$ ;

b) $0.123456789101112131415 \dots$ .

Will this number be rational?

Problem #PRU-60877

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

13–15 3.0

Prove that if $(m, 10) = 1$ , then there is a repeated unit $E_{n}$ that is divisible by $m$ . Will there be infinitely many repeated units?

Problem #PRU-60897

Problems Combinatorics

12–14 2.0

There are 4 weights and scales. How many loads that are different by weight can be accurately weighed using these weights, if

a) weights can be placed only on one side of the scales;

b) weights can be placed on both sides of the scales?

Problem #PRU-60906

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

12–13 2.0

Will thought of a number: 1, 2 or 3. You can ask him only one question, to which he can answer “yes”, “no” or “I do not know”. Can you guess the number by asking just one question?