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Let’s prove that 1=2. Take a number a and suppose b=a. After multiplying both sides we have a2=ab. Subtract b2 from both sides to get a2b2=abb2. The left hand side is a difference of two squares so (ab)(a+b)=b(ab). We can cancel out ab and obtain that a+b=b. But remember from the start that a=b, so substituting a for b we see that 2b=b, dividing by b we see that 2=1.

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 x2<x so x can not be the smallest either. Then x can only be equal to 1.

In how many ways can you read the word TRAIN from the picture below, starting from T and going either down or right at each step?

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 p3 is prime. Suppose that a+b and a2+b2 are divisible by p. Show that a2+b2 is divisible by p2.

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.

John drunk a 16 of a full cup of black coffee and then filled the cup back up with milk. Then he drunk a third of what he had in the cup. Then, he refilled it back to full with milk again, and after that, drunk a half of the cup. Finally, he once again refilled the cup with milk and drunk everything he had. What did he drink more of - coffee or milk?

A round necklace contains 45 beads of two different colours: red and blue. Show that it is possible to find two beads of the same colour next to each other.

Cut this figure into 4 identical shapes. (Note: you have to use the entire shape. Rotations and reflections count as identical shapes)

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