The set of numbers 1,3,8,120 has a remarkable property: the product of any two numbers is a perfect square minus one. Find a fifth number that could be added to the set preserving its property.

Once upon a time, a tsar was holding a reception and the Megamind was among the guests. The tsar decided to test how smart the Megamind was, took him into a dark room, and gave the following task: On the table in this room, there are 50 coins, exactly 10 of them are tails up. In the darkness, it is impossible to determine the sides of these coins. Touching the coins also does not help. The Megamind has to separate these coins into two groups so that the number of tails in both are equal. Can he do it?

You have two incense sticks, that burn unevenly, and a lighter. Each will burn for an hour. How can you time 45 minutes using nothing but these tools?

A Megamind owns a mint with a 100 workers. Each day, he gives 1kg of gold to each worker, and each worker must make 100 coins (10 grams each). The Megamind learnt that one of the workers is a forger, he makes coins that are 1g lighter. How can Megamind determine the forger using only one weighing? The scales can determine the total weight, up to 100 kg.

Using numbers 1,3,4,6, and basic arithmetic operations (addition, subtraction, multiplication, and division) and parentheses, obtain and an expression that evaluates to 24. You may use only these numbers and only these operations. Every number should be used exactly once. Numbers cannot be concatenated, i.e. you cannot use 13 or 146.

A large and merry group of N guys and N girls went to celebrate a birthday in a restaurant. They were randomly seated at a big rotating table by a hurried waiter who quickly took their drink orders. Every girl ordered a wine, and every guys ordered a beer. Bringing the drinks, the waiter brought the right number of beers/wines, but forgot who ordered what and placed the drinks randomly in front of everyone. The party was enraged because more than a half got the wrong drink and demanded to see the manager. The manager appeared and said: "My apologies, but this is not such a big deal. In fact, I can rotate the table without touching the drinks so that more than half of you will get what you ordered." Is the manager's claim correct?

A birthday cake has a form of triangle. Two Megaminds divide as follows: One chooses a point in the triangle, and the other cuts the triangle by a segment passing through this point. The cutter then takes the bigger part. What is the maximal part of the cake guaranteed to be left to the first Megamind? The thickness of the cake is constant.

Two commanders (Caesar and Brutus) conquer a country which forms a connected graph with nodes representing cities and edges representing roads. First, Caesar chooses a city and claims it. Then, Brutus claims any of the remaining cities. Turn by turn, the commanders claim the available cities that are adjacent to the cities they have already claimed. If a commander cannot make a claim, he skips a move. The game continues until all cities have been claimed. Both commanders wish to claim a maximal number of cities. Can Brutus win in this game?

Four toy cars are moving with constant velocities on a flat surface. Their velocities are not parallel, and the cars had started to move a while ago. After a collision, each car continues to move with the same velocity, but it disintegrates after three collisions. Five collisions involving two cars each had already happened, and two toys had disintegrated. What is the fate of the remaining two toys?

Two Megaminds play a game with 100 wooden sticks. The lengths of the sticks are 1,2,3,...,100 inches. Turn by turn, the players choose three of the remaining sticks which form a triangle, and burn them. The player that cannot make a move loses. Which player has a winning strategy (justify your answer)?

A Megamind has two bags and 100 M&Ms. He needs to put all the candies into the bags so that one bag has twice as many M&Ms as the other. He cannot split the M&Ms. What should the Megamind do?

A smart cat named Leopold is hunting a mouse. A mouse is hiding in one of five holes arranged in a row. Leopold can reach into one of the holes and try to catch the mouse. If he is not successful, the scared mouse runs into a right/left neighboring hole. Is it guaranteed that Leopold will catch the mouse? If so, what should he do?

A positive integer is given. How can you determine whether it is a power of 2 without using recursion and floating point operations?

A famous magician takes a standard 52 card deck and gives it to the audience. The spectators choose any 5 cards (they may do it any way they like) and pass these cards to the magician's assistant. The assistant announces 4 of these cards out loud. The magician responds by naming the fifth card. Except for the suit and denomination of each card, the assistant passes no other information to the magician. How does the magician "know" the fifth card?

Using numbers 1,3,4,6, and basic arithmetic operations (addition, subtraction, multiplication, and division) and parentheses, obtain and an expression that evaluates to 24. You may use only these numbers and only these operations. Every number should be used exactly once. Numbers cannot be concatenated, i.e. you cannot use 13 or 146.