## 7th and 8th Grade, February, 2019

C.1233. Find the two smallest positive integers that give a remainder of 2 when divided by either 3 or 7, and give a remainder of 3 when divided by 5.

C.1234. The first three elements of a number sequence are 1, 2, and 3. (As you can see, the middle number is one less than the product of its neighbors. This is true for the rest of the elements of this sequence. Now add the elements of the sequence up to a certain element.

a) Could this sum be 2019?

b) Could this sum be 2020?

C.1235. A spiral walkway leads to the center of a 20 x 10 meters rectangular garden of a castle. The owner of the castle walks this maze every day to the middle of the garden, waters the 1 x 1 meter flower bed and walks back. He always walks in the middle of the path and makes a right angle turn every time. How long of a walk does the owner of the castle have to make every day to do this? C.1236. In Sophia’s logic board game you have to place 9 red, 9 yellow and 9 green rings on 9 different colored sticks glued onto a square shaped board. The diagram below shows the arrangement of the sticks as seen from above. An arrangement is considered ‘good if there are 3 different colored rings on each stick, and there are 3 different colored rings at the same level on the three sticks in each direction indicated on the diagram. How many different good arrangements of the rings are there? (Two arrangements are considered to be different if the rings are in a different order in the two arrangements on at least one of the sticks.) C.1237. In triangle ABC, X is the midpoint of side AB, Y is the midpoint of side AC, and the intersecting point of BY and CX is S. Prove that

a) the areas of triangles SBX and SCY are the same.

b) the areas of triangle SBC and quadrilateral AXSY are the same.

C.1238. A number magician tells you: “Pick any positive integer and multiply it by 18. Delete one of the non-zero digits of the product. Tell me the sum of the remaining digits of the product and I will tell you what digit you deleted.” Explain this trick.

C.1239. From the decimal form of 1/13 we pick 99 consecutive digits and treat it as a 99-digit number. Can this number be divisible by 3?

C.1240. We cut off a 2 cm wide piece from a cube-shaped piece of margarine in a parallel direction to one of its sides. The volume of the remaining margarine is 384 cubic centimeters. How long are the edges of the original margarine if we know that they are whole numbers when measured in centimeters?