## 7th and 8th Grade, February, 2020

C.1289. Color a few more squares, so that the diagram becomes symmetrical about a straight line. You may color a square if its symmetrical partner is colored already. What are the possible new designs?

C.1290. How many different ways can you pick two different positive whole numbers so that none of them is greater than 20, and their product is not divisible by 10? The order of picking the two numbers is irrelevant.

C.1291. Andrew, Ben, Charlie, and Daniel are playing cards. In each round, only three of the four boys are playing, and depending on the result of that round they may win points from each other: either one player from the other two, or two players from the third one. (If there are two winners or two losers, they may not win or lose the same amount.) At the beginning of the game everybody has 100 points. The boys played 4 rounds, and in each round a different person did not participate. After every round, if the number of a person’s points changed, they wrote the new number of his points right under the last number in his column. So they got the following chart. Which boy was left out in which round?

C.1292. There are three kinds of boxes in a room: small, medium and big. (Same size boxes cannot be placed in one another.) We put 11 big boxes on the table. We left a few of them empty, and put 8 medium boxes in each of the rest. Then we left a few medium boxes empty, and put 8 small (empty) boxes in each of the rest of the medium boxes. There are 102 empty boxes on the table now. How many boxes are there on the table all together?

C.1293. There are 700 songs in a Hymnal book, numbered from 1 to 700. Every Sunday people sing one song from this book, and it is announced on a board using little wooden cubes, each cube showing one digit on each side. The face with the digit 6 can also be used to show a 9. The one- and two-digit numbers are always shown starting with zeros. For example: the number 3 is shown as 003, the number 26 is shown as 026.

a) How many wooden cubes are needed so that every song in this book can be shown on the board? Design one such set of cubes.

b) How many more songs could be added to the book without increasing the number of wooden cubes?

C.1294. Snow White made dumplings for the seven dwarfs. She did not make more than 29 dumplings for each of them. She put all the dumplings on a plate and went to the forest to pick some berries. One of dwarfs left his lamp behind, so he came back home for it. When he saw the dumplings, he could not control his appetite, ate one fifth of the dumplings and went back to the mine. A short time later, another dwarf came back home because he forgot his shovel. He could not resist the dumplings either, so he ate one third of the dumplings on the plate, and went back to work. By the time they all returned home in the evening, Snow White divided the dumplings evenly among the dwarfs without noticing that some of them were missing. During the day none of the two dwarfs used a knife to cut any of the dumplings in order to carry out their plans. How many dumplings did Snow White make?

C.1295. On the Math final exam there are 150 questions. For a correct answer you get 2 points, but you lose a point for not answering a question or for an incorrect answer. How can you have a total of:

a) 273

b) 163 points?

C.1296. The two parallel sides of a trapezoid are 64 cm and 70 cm long. The third side is 60 cm. How big are the angles on this third side if the area of the trapezoid is 2010 square centimeters?