Problems

Integer numbers \(a,b\) and \(c\) are such that the sum of digits of a number \(a+b\) is less than \(5\), the sum of digits of a number \(b+c\) is less than \(5\), the sum of digits of a number \(a+c\) is less than \(5\), but the sum of digits of a number \(a+b+c\) is greater than \(50\). Can you find such three numbers \(a,b\) and \(c\)?

Catherine asked Jennifer to multiply a certain number by 4 and then add 15 to the result. However, Jennifer multiplied the number by 15 and then added 4 to the result, but the answer was still correct. What was the original number?

Alex laid out an example of an addition of numbers from cards with numbers on them and then swapped two cards. As you can see, the equality has been violated. Which cards did Alex rearrange?

The number \(A\) is positive, \(B\) is negative, and \(C\) is zero. What is the sign of the number \(AB + AC + BC\)?

Pinocchio correctly solved a problem, but stained his notebook. \[(\bullet \bullet + \bullet \bullet+1)\times \bullet= \bullet \bullet \bullet\]

Under each blot lies the same number, which is not equal to zero. Find this number.

Compare the numbers: \(A=2011\times 20122012\times 201320132013\) and \(B= 2013\times 20112011 \times 201220122012\).

Using five twos, arithmetic operations and exponentiation, form the numbers from 1 to 26.

Using five threes, arithmetic operations and exponentiation, form the numbers from 1 to 39.

Using five fours, arithmetic operations and exponentiation, form the numbers from 1 to 22.

Using five fives, arithmetic operations and exponentiation, form the numbers from 1 to 17.