The Hatter is obsessed with odd numbers. He is very determined to represent 1 as \[1 = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d},\] where \(a\), \(b\), \(c\), and \(d\) are all odd.
Alice is very sceptical about it. Do you think you can help Alice to persuade the Hatter that it is impossible?
Prove that the equation \(x^2 + 4034 = y^2\) does not have solutions in integer numbers.
Find a solution of the equation \(x^2 + 2017 = y^2\) in integer 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.
Express the number 111 as a sum of 51 natural numbers so that each of the terms has the same sum of digits.
a) Express the number 221 as a sum of 52 natural numbers so that each of the terms has the same sum of digits.
(b) Express the number 226 as a sum of 52 natural numbers so that all terms have the same sum of digits.
The Queen has introduced a new currency in the world of Wonderland. This currency consists of three golden coins with values \(3\), \(5\) and \(15\). Is it possible for Alice to change an old note with value \(100\) using \(11\) new coins?
One sunny day Alice met the White Rabbit. The Rabbit told her that he owns a pocket watch which has 11 gears arranged in a chain loop. The rabbit asked Alice if it was possible for all the gears to rotate simultaneously. What is your opinion on this matter? Can all the gears rotate simultaneously?
After the Mad Tea-Party, the Hatter was so excited that he decided to cool down by going on a short walk across the chessboard. He started at position a1, then walked around in steps taking each step as if he was a knight, and eventually returned back to a1. Show that he made an even number of steps.
Is it possible that odd integers \(a\), \(b\), \(c\), \(d\) satisfy \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=1\)?