C.1209. During Math class Jim and John were playing 3×3 Tic Tac Toe. (On a 3×3 square grid two players place an X and an O sign alternating, each player having his own sign. The first person having 3 of his signs in a row, column or in a diagonal wins the game.) Upon noticing this activity, their Math teacher gave them the following penalty: They had to calculate how many different grid formations would give a simple victory to one of the players. (A simple victory means that the first 3 moves of a player make him a winner.) Two formations are considered to be the same if one formation can be moved into the other by rotation and/or reflection. If the numbers of X’s and O’s are different, then the formations are considered to be different.)
C.1210. Three different kinds of people live in a city: one who always says the truth, one who always lies, and one who sometimes lies, but other times says the truth at his or her own will. In this city somebody stole the major’s horse. The police picked up 3 men, Jonathan, Jason and James, knowing that one of them is a liar, one of them always says the truth, and with one of them you never know! The police also knows that one of them is the thief, who always says the truth. Thee three men made the following statements:
Jonathan: I’m innocent.
Jason: He really is!
James: I stole the horse.
Who stole the horse and who is the liar?
C.1211. Their boss wants to distribute $3150 among three workers. How much money should each worker get if one of them makes the same piece in 6 minutes, the other makes it in 4 minutes, and the third one makes it in 3 minutes?
C.1212. Find the angles of the right triangle if we could cut it up to three isosceles triangles the following way:
C.1213. How many different ways can you place a king and a rook on the chess board without them hitting each other? (The fields of the chess board are labeled by letters and numbers as usual. Two placements are different if at least one figure is on a different field.)
C.1214. We wrote down all the 2-digit numbers in an increasing order by using a blue, a red, and a green pen, rotating them in this order for each digit. Will every positive digit appear in every color in both the tens and the units places?
C.1215. We inscribe trapezoid ABCD into a circle. Its perpendicular diagonals are the same length as side AB. How big is angle ADB?
C.1216. Substitute a, b, and c with positive whole numbers in the following line of inequalities so that the value of “a” is the least possible. What are the values of b and c then?