The numbers \(x\) and \(y\) satisfy \(x+3 = y+5\). Prove that \(x>y\).
The numbers \(x\) and \(y\) satisfy \(x+7 \geq y+8\). Prove that \(x>y\).
Prove that there are infinitely many natural numbers \(\{1,2,3,4,...\}\).
Is it possible to colour the cells of a \(3\times 3\) board red and yellow such that there are the same number of red cells and yellow cells?
A coin is tossed six times. How many different sequences of heads and tails can you get?
Eleven people were waiting in line in the rain, each holding an umbrella. They stood closely together, so that the umbrellas of the neighbouring people were touching (see fig.)
The rain stopped and all people closed their umbrellas. They now stood keeping a distance of \(50\) cm between neighbours. By how many times has the queue length decreased? People can be considered points, and umbrellas are circles with a radius of \(50\) cm.
Cut a square into five triangles in such a way that the area of one of these triangles is equal to the sum of the area of other four triangles.
The prime factorization of the number \(b\) is \(2 \times 5^2 \times 7 \times 13^2 \times 17\). The prime factorization of the number \(c\) is \(2^2 \times 5 \times 7^2 \times 13\). Is the first number divisible by the second one? Is the product of these two numbers, \(b \times c\), divisible by \(49000\)?
A new customer comes to the hotel and wants a room. It happened today that all the rooms are occupied. What should you do?
Now imagine you got \(10\) new guests arriving to the completely full hotel. What should you do now?