## 5th and 6th Grade, December, 2019

B.1273. Tom Sawyer and his friends are painting marks on the planks of a wooden fence. Tom paints a vertical line on every other plank. Huckleberry Finn paints an X on every third plank, and Joe Harper puts a circle on every seventh plank, as shown on the diagram.

a) Which plank will have all three marks on it first?

b) What is painted on the 40th plank?

c) If the fence contains 128 planks, how many planks will have no mark on them?

B.1274. Since 75 = 3x5x5, the sum of the prime numbers in its prime factorization is 13. Find all such positive whole numbers.

B.1275. You divide the 2019-digit number containing all digit 2’s by 3. How many digit 7’s are there in the quotient?

B.1276. Using positive whole numbers, fill out the empty fields of the following grid so that if you pick 3 numbers from the grid so that no two of them are from the same column and or from the same row either, then the product of the three numbers is always the same.

B.1277. Place two Queens on a 5×5 chess board so that there are only two fields that cannot be hit by any one of the Queens. (A Queen can hit the fields in the same row or the same column or the same diagonal lines with her.)

B.1278. In a warehouse they covered the floor using 50 cm x 50 cm square tiles. In one of the rooms they placed five 1 m x 1.5 m rectangular based boxes, as shown on the diagram. A young mouse wanted to get from one corner of the room, marked by an X, to the opposite corner of the room, marked by an X, also. As he was young and restless, instead of going around the boxes, he chewed himself through them, so that the route he traveled on was always vertical or horizontal on the diagram, traveling 1 meter on the bottom of each box. Show a possible route for this mouse!

B.1279. His wife asked the baker how many strawberry cheese danish he sold that day. The baker said: “I wanted to sell 70 of them, but if I sold twice as many as I did, I still would have sold two less than what I was hoping to.” How much money did the baker collect that day for the danishes if each danish costs 80 cents?

B.1280. Three rabbits are playing cards. At the beginning of the game each rabbit had a different number of carrots, but in the first round the first rabbit lost so much that the other two rabbits doubled their numbers of carrots each. In the second round, the second rabbit lost so much that the other two rabbits doubled their numbers of carrots in that round each. In the third round, the third rabbit lost so much that the other two rabbits doubled their numbers of carrots in that round each. How many carrots did each rabbit start the game with if they each had 24 carrots after these three rounds?