Problems

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Found: 353

The Hatter says that he knows four numbers such that their sum and their product are odd. Is he right? Can you expain why?

The four numbers 1, 1, 1, 2 are written on a piece of paper. Alice and the Hatter are playing a game. They add in turns 1 to any two numbers to make the new four numbers. The winner is the one to make all the four numbers equal. If Alice goes first, who will win, if any?

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.

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.

Is it possible to arrange some group of distinct numbers in a circle so that each number equals the sum of its neighbours?

Sarah is writing down natural numbers starting from 2. She notices that each time she writes the next number the sum of all written numbers is less than their product. She believes she can find such 57 natural numbers (not necessarily different from each other) that their sum will be greater than their product. Do you think it is possible?

(a) Can you find a set of distinct numbers which can be arranged in a circle in such a way that each number equals the product of its neighbours?

(b) Is it true that each solution of Example 1 is determined by the values of two neighbouring numbers?

It’s not that difficult to find a set of \(57\) integers which has a product strictly larger or strictly smaller than their sum. Is it possible to find \(57\) integers (not necessarily distinct) with their sum being equal to their product?