Shmerlin managed to enter the cave and explore it. On his way back, he was once again stopped by Drago. He learns that the door out of the cave is locked again, this time with a more powerful lock. The key required to open it now includes four positive integers, which are no longer digits – they can be much larger. Shmerlin once again can choose four integer numbers: \(x, y, z\) and \(w\), and the dragon will tell him the value of \(A \times x + B \times y + C \times z + D \times w\), where \(A, B, C\) and \(D\) are the four secret integer numbers that open the lock. Because the lock is much more difficult to crack now, Drago agrees to let Shmerlin try twice. He can choose his four integer numbers and then, basing on what he learns from the dragon, choose again. Will he be able to leave the cave or is he doomed to stay inside forever?
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?
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.
Using five sixes, arithmetic operations and exponentiation, form the numbers from 1 to 14.
Using five sevens, arithmetic operations and exponentiation, form the numbers from 1 to 22.