We solve problems

Problem #PRU-100113

Problems Algebra and arithmetic Arithmetics. Mental maths Set theory and logic Mathematical logic Mathematical logic (other)

Jane wrote another number on the board. This time it was a two-digit number and again it did not include digit 5. Jane then decided to include it, but the number was written too close to the edge, so she decided to t the 5 in between the two digits. She noticed that the resulting number is 11 times larger than the original. What is the sum of digits of the new number?

Problem #PRU-100114

Problems Algebra and arithmetic Arithmetics. Mental maths Set theory and logic Mathematical logic Mathematical logic (other)

a) Find the biggest 6-digit integer number such that each digit, except for the two on the left, is equal to the sum of its two left neighbours.

b) Find the biggest integer number such that each digit, except for the rst two, is equal to the sum of its two left neighbours. (Compared to part (a), we removed the 6-digit number restriction.)

Problem #PRU-100337

Problems Set theory and logic Mathematical logic Algebra and arithmetic Number theory. Divisibility Odd and even numbers

Alice marked several points on a line. Then she put more points – one point between each two adjacent points. Show that the total number of points on the line is always odd.

Problem #PRU-100451

Problems Algebra and arithmetic Arithmetic operations. Number identities Set theory and logic Mathematical logic

The product of 22 integers is equal to 1. Show that their sum cannot be zero.

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”.

Problem #PRU-103887

Problems Set theory and logic Mathematical logic Puzzles Algebra and arithmetic Fractions Simple fractions

12–12 2.0

Arrange brackets and arithmetic signs around these numbers so that the correct equality is obtained: \[\frac{1}{2}\quad \frac{1}{6}\quad \frac{1}{6009} \ = \ 2003.\]

Problem #PRU-109423

Problems Set theory and logic Mathematical logic Puzzles Algebra and arithmetic Number theory. Divisibility The fundamental theorm of arithmetic. Prime factorisation.

11–12 2.0

At the end of the term, Billy wrote out his current singing marks in a row and put a multiplication sign between some of them. The product of the resulting numbers turned out to be equal to 2007. What is Billy’s term mark for singing? (The marks that he can get are between 2 and 5, where 5 is the highest mark).

Problem #PRU-109476

Problems Algebra and arithmetic Word problems Solving problems by inequalities. Going through the cases. Set theory and logic Mathematical logic Mathematical logic (other)

11–14 2.0

Sarah believes that two watermelons are heavier than three melons, Anna believes that three watermelons are heavier than four melons. It is known that one of the girls is right, and the other is mistaken. Is it true that 12 watermelons are heavier than 18 melons? (It is believed that all watermelons weigh the same and all melons weigh the same.)

Problem #PRU-35676

Problems Algebra and arithmetic Rational functions Inequalities. Method of Intervals. Set theory and logic Mathematical logic Mathematical logic (other)

12–14 2.0

Let \(x\) be a natural number. Among the statements:

\(2x\) is more than 70;

\(x\) is less than 100;

\(3x\) is greater than 25;

\(x\) is not less than 10;

\(x\) is greater than 5;

three are true and two are false. What is \(x\)?

Problem #PRU-86492

Problems Set theory and logic Mathematical logic Mathematical logic (other) Algebra and arithmetic 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?