Problems

Let’s compute the infinite sum: \[1+2 + 4 + 8 + 16 + ... + 2^n + ... = c\] Observe that \(1+2+4+8+... = 1 + 2(1+2+4+8+16+...)\), namely \(c = 1+2c\), then it follows that \[c = 1+2+4+8+... = -1.\]

Let’s prove that any \(90^{\circ}\) angle is equal to any angle larger than \(90^{\circ}\). On the diagram

We have the angle \(\angle ABC = 90^{\circ}\) and angle \(\angle BCD> 90^{\circ}\). We can choose a point \(D\) in such a way that the segments \(AB\) and \(CD\) are equal. Now find middles \(E\) and \(G\) of the segments \(BC\) and \(AD\) respectively and draw lines \(EF\) and \(FG\) perpendicular to \(BC\) and \(AD\).
Since \(EF\) is the middle perpendicular to \(BC\) the triangles \(BEF\) and \(CEF\) are equal which implies the equality of segments \(BF\) and \(CF\) and of angles \(\angle EBF = \angle ECF\), the same about the segments \(AF=FD\). By condition we have \(AB=CD\), thus the triangles \(ABF\) and \(CDF\) are equal, thus \(\angle ABF = \angle DCF\). But then we have \[\angle ABE = \angle ABF + \angle FBE = \angle DCF + \angle FCE = \angle DCE.\]

Let’s prove that \(1\) is the smallest positive real number: Assume the contrary and let \(x\) be the smallest positive real number. If \(x>1\) then \(1\) is smaller, thus \(x\) is not the smallest. If \(x<1,\) then \(\frac{x}{2}<x\) so \(x\) can not be the smallest either. Then \(x\) can only be equal to \(1\).

There are \(100\) people in a room. Each person knows at least \(67\) others. Show that there is a group of four people in this room that all know each other. We assume that if person \(A\) knows person \(B\) then person \(B\) also knows person \(A\).

The numbers \(a\) and \(b\) are integers and the number \(p \ge 3\) is prime. Suppose that \(a+b\) and \(a^2 +b^2\) are divisible by \(p\). Show that \(a^2 + b^2\) is divisible by \(p^2\).

There are \(33\) cities in the Republic of Farfarawayland. The delegation of senators wants to pick a new capital city. They want this city to be connected by roads to every other city in the Republic. They know for a fact that given any set of \(16\) cities, there will always be some city that is connected by roads to all those selected cities. Show that there exists a suitable candidate for the capital.

The diagram shows a \(3 \times 3\) square with one corner removed. Cut it into three pieces, not necessarily identical, which can be reassembled to make a square:

Find all possible non-zero digits \(A\) for which the following holds \((AA+AA+1) \times A = AAA\). (Recall \(AA\) means the two-digit number whose first and second digits are \(A\))

A square has been divided into \(4\) rectangles and a square. If the rectangle in the bottom left corner has dimensions \(1 \times 4\) and the one in the top right is \(2 \times 5\), what is the area of the small square in the middle?

There are \(25\) bugs sitting on the squares of a \(5 \times 5\) board, \(1\) at each square. When I clap my hands, each bug jumps to a square diagonally from where it was before. Show that after I clap my hands, at least \(5\) squares will be empty.