Are there such integer numbers \(a,b,c,d,e,f\) that the numbers \[a-b,\ b-c,\ c-d,\ d-e,\ e-f,\ f-a\] are consecutive integers? (Note: they obviously can be negative.)
There are numbers \(1,2,3,4,5,6,7,8,9\) and \(10\) written on a board. Each time you make a “move” you can erase three of the remaining numbers, \(a,b\) and \(c\), and replace them with the numbers \(2a+b\), \(2b+c\) and \(2c+b\). The goal is to make all the \(10\) numbers left on the board equal. Is it possible?
On a certain island there are 17 grey, 15 brown and 13 crimson chameleons. If two chameleons of different colours meet, both of them change to the third colour. No other colour changes are allowed. Is it possible that after a few such colour transitions all the chameleons have the same colour?
Sixteen lightbulbs are arranged in a \(4 \times 4\) grid. Some are on, some are off. You are allowed to change the state of all the bulbs in a column, in a row, or along any diagonal (note: there are 14 diagonals in total!). Is it possible to go from the arrangement in the left to the one on the right by repeating this operation?
There are numbers from \(1\) to \(2018\) written on a board. In one go, we can erase two numbers and replace them with an absolute value of their difference. Can we achieve a sequence consisting only of several numbers \(0\) after some number of moves?
The distance from school to the monument in the town centre is \(4.2\) km and the distance from Anna’s house to school is \(0.7\) km. Given that the distance from Anna’s house to the monument is an integer number of kilometres, what is this distance?
Tom and his grandma live on the same side of a straight river. Tom wants to visit his grandma, but also wants to stop by the river and fill his bottle with water. What is the shortest path that starts at his house, touches the river and ends at his grandma’s house?
A point \(P\) is somewhere inside the triangle \(\triangle ABC\). Show that \(AP + BP < AC + BC\).
The distance between London and Warsaw equals \(1450\) km, between Warsaw and Kyiv is \(680\) km. The distance from London to New Delhi, is \(6700\) km and the distance from Kyiv to New Delhi is \(4570\) km. What is the distance from London to Kyiv?
Show that for any three points on the plane \(A,B\) and \(C\), \(AB \ge |BC - AC|\).