## 7th and 8th Grade, September

C.1193. Oliver has a collection of 100 chocolates for the winter. Each of them is either milk or dark, and each of them is either rectangular or circular. 1/5 of all the milk chocolates are circular, and 1/3 of all the circular chocolates are milk chocolates. He has 30 rectangular dark chocolates. How many circular milk chocolates does Oliver have?

C.1194. Kate’s measuring tape has centimeter divisions on both sides but in opposite directions. On the yellow side it starts at one end, and on the green side it starts at the other end. The measuring numbers are written on the centimeter division lines. One night out of curiosity Kate measured the dimensions of her sofa. Its length was the whole tape and an additional 90 cm, its width was 10 cm. From these Kate realized that she used the wrong side while taking each measurement. How long is the measuring tape if the width of the sofa is 2/3 of its length?

C.1195. Every inner angle of the hexagon below is 120 degrees. Prove that AB + AF = CD +DE.

C.1196. In a library 3 children pick from 5 different books in the following way: one copy of each of the 5 different books is placed on the table in a row. The children put their library cards on the book they want to borrow. More than one child may borrow the same book since the library has multiple copies, but each child can borrow only one book. How many different ways can the cards be placed on the books? (Two placements are different if at least one child did not put his card on the same book in the two arrangements.)

C.1197. Fiona has 11 cylinders. If she puts them in order of their heights (all of them standing on their circular bases) then there is a 2 cm height difference between any two consecutive cylinders. The highest cylinder has the same height as the middle one put on top of one of its neighbors. How tall is the column you can build by putting all of these cylinder on top of one another?

C.1198. Olga typed the following addition into her calculator: 1+2+3+4+…+100, starting from the left. Every time she entered a number, the sum up to that number appeared on the display of the calculator. How many times did she see a number divisible by 97 on the display?

C.1199. Adam and Eve are playing Heads-or-Tails with a coin. If it is a head then Eve is the winner and gets 2 pieces of candy from Adam. If it is a tail then Adam is the winner and gets 3 pieces of candy from Eve. After 30 games they each have the same numbers of candies that they started with. How many times did Adam win?

C.1200. We have a subscription for a daily newspaper, but our mailbox is not big enough for the newspaper to fit in without folding it. Our mailman, just for his own entertainment, decided that he will fold the newspaper differently every day. He always folds the paper parallel to its shorter side so that the layers cover each other completely. On the diagram below you can see all possible ‘essentially different’ ways of folding the paper into 3 layers. How many ‘essentially different’ ways can you fold the paper into 5 layers?