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

C.1281. Sara, Kate, and Sophie want to buy the same book.  However, none of them have enough money to buy it. Sara is missing 1/3 of the price of the book, Kate is missing 1/4 of the price of the book, and Sophie is missing 1/5 of the price of the book. If the book were \$4.70 cheaper then they could buy 3 copies of the book if they spent all their money together.  How much does a copy of this book cost?

C.1282. Every competitor in a building competition has to build a continuous structure using 8 identical cubes so that each cube in the structure shares at least one full side with each of its vertical and/or horizontal neighbor. There can be no hole in the structure, cubes cannot be glued together, and you have to be able to climb up from the floor level to the top of the building one level at a time for each step. The value of a building is the product of two numbers: the area of its base and its height. At the end of the competition the judges were surprised to see that buildings by different competitors received different scores. How many competitors could there be at most in this competition? What was the value of the winning structure?

C.1283. There are 7 houses on Dragon Street, numbered from 1 to 7. Mailman, Deli Were, brought mail to exactly two of these houses each day last week. Two of these houses never received mail; all the others received mail twice last week. The sums of the house numbers where Deli went each day are: Monday 5, Tuesday 8, Wednesday 9, Thursday 13 and Friday 7. (There is no mail service on Saturdays and Sundays.) Which two houses did not receive mail?

C.1284. One year January 1st fell on a weekday, while December 31st fell on the weekend (Saturday or Sunday). The school year started on September 1st. What day of the week was that?

C.1285. There are 3 kinds of chocolates, a total of 16 pieces in a box, as shown on the diagram. The total mass of the 4 chocolates in the top row is 14 dkg,  and it is 11 dkg in the row below that. The total mass of the 4 chocolates in the left column is 10 dkg. How many dkg of chocolate is there in the box?

C.1286. Place all the whole numbers from 1 to 9  in the circles of the house on the following diagram so that the sum of the three numbers in any 3 circles falling on a straight line is 13. How many different solutions are there? (Two solutions are different if there is at least one circle with different numbers in it.)

C.1287. If you take the 2020 digit number with every digit equal to 2 and divide this 2020 digit number by 3, how many of the digits in the quotient will be a 7?

C.1288. In triangle ABC, angle B is the smallest, which is 1/5 of the sum of the two other angles. C is 4 cm from line AB. How long is side BC?