7th and 8th Grade, December, 2018

C.1217. You have 125 identical red cubes. You paint one of them blue and make a 5x5x5 cube using all of them. How many different big cubes can you make? (Two cubes are considered to be the same if one of them can be rotated into the position of the other.)

C.1218. Using all of the digits 2, 3, 4, 6, 7, and 8, exactly once, make two 3-digit numbers with the lowest possible (positive) difference.

C.1219. There are 15 teams in a soccer league. Each team plays against every team once. A team gets 3 points for a victory, 2 points for a tie and 1 point for a lost game. At the end of the season every team had a different number of total points. The last place team had 21 points. Prove that the team in first place had at least one tied game.

C.1220. In quadrilateral ABCD, AB = AC = DB, and its diagonals are perpendicular. Find the sum of angle ACB and angle ADB.

C.1221. How many sets of at least two consecutive numbers are there in which the sum of the numbers is 100?

C.1222. Three girls are practicing calculations with the whole numbers from 1000 to 2018, including these two numbers. Ann added those of them that are divisible by 3, Bea added those that give 1 as a remainder when divided by 3, and Christina added those that give 2 as a remainder when divided by 3. Who got the greatest sum? How much more is this than the smallest sum the girls got?

C.1223. The Eskimos greet each other by rubbing their noses together. They greet the arctic researchers by giving them a “high five”. The researchers say hello to the Eskimos, just as they greet each other. At a party at the North Pole, while everybody was greeting everybody in the proper way, you could hear 444 hello’s and there were 432 “high five’s”. How many Eskimos and how many researchers were present?

C.1224. You have a 2 cm x 3 cm x 4 cm wooden solid. You paint every side of it blue, and cut it up to 24 1 cm x 1 cm x 1 cm cubes with cuts parallel to the sides.

a) Can you build a different sized blue rectangular solid using all 24 of the little unit cubes?

b) How many different sized blue rectangular solids can you build if you do not have to use every unit cube for each?