Problems

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A rectangle is cut into several smaller rectangles, the perimeter of each of which is an integer number of meters. Is it true that the perimeter of the original rectangle is also an integer number of meters?

Arrange brackets and arithmetic signs around these numbers so that the correct equality is obtained: \[\frac{1}{2}\quad \frac{1}{6}\quad \frac{1}{6009} \ = \ 2003.\]

The number \(A\) is positive, \(B\) is negative, and \(C\) is zero. What is the sign of the number \(AB + AC + BC\)?

In Mongolia there are in circulation coins of 3 and 5 tugriks. An entrance ticket to the central park costs 4 tugriks. One day before the opening of the park, a line of 200 visitors queued up in front of the ticket booth. Each of them, as well as the cashier, has exactly 22 tugriks. Prove that all of the visitors will be able to buy a ticket in the order of the queue.

a) A 1 or a 0 is placed on each vertex of a cube. The sum of the 4 adjacent vertices is written on each face of the cube. Is it possible for each of the numbers written on the faces to be different?

b) The same question, but if 1 and \(-1\) are used instead.

Gerard says: the day before yesterday I was 10 years old, and next year I will turn 13. Can this be?

Doctor Smith gave out 2006 miracle tablets to four sick animals. The rhinoceros received one more tablet than the crocodile. The hippopotamus got one more tablet than the rhino. The elephant got one more tablet than the hippo. How many tablets did the elephant have to eat?

There are two numbers \(x\) and \(y\) being added together. The number \(x\) is less than the sum \(x+y\) by 2000. The sum \(x+y\) is bigger than \(y\) by 6. What are the values of \(x\) and \(y\)?

The digits of a 3 digit number \(A\) were written in reverse order and this is the number \(B\). Is it possible to find a value of \(A\) such that the sum of \(A\) and \(B\) has only odd numbers as its digits?