## 7th and 8th Grade, October, 2018

C.1201. What is the greatest possible common divisor of 6 different 2-digit positive whole numbers?

C.1202. We built a big cube from identical smaller cubes. An eighth of the small cubes used were red, a fourth were white and the rest were green. We used more than 300 green cubes. How many small cubes did we use if we tried to use as few as possible?

C.1203. In a store you ask for 5 pieces of candy to be put in a paper bag. Then you ask for 10 pieces of candy to be put in another similar paper bag. Each piece of candy has the same mass. The first bag measures 85 grams, and the second bag measures 165 grams. How much does each paper bag cost if the price of a bag of candy weighing 1 kg is \$12?

C.1204. Inscribe an octagon in a circle. Find the sum of four of its inner angles if no two of these angles are next to each other.

C.1205. Cut the perimeter of a convex quadrilateral ABCD at its vertices to get four line segments. Now choose an arbitrary point O, and move these four segments parallel to their original positions until one end of each segment is at point O. Connect the outer ends of these segments to get a new quadrilateral. Suppose you do this to get the quadrilateral XYZV with the greatest possible area. How many times greater is the area of XYZV than ABCD?

C.1206. The parallelogram on the diagram below is cut into three isosceles triangles. (Sides marked similarly are equal in length.) Find the angles of the parallelogram. C.1207. Find the following sum using the most simple calculations: C.1208. We want to cut up a big cube into smaller pieces. First we cut it a few times with vertical planes parallel to two of its opposite faces, then we cut it a few times with vertical planes again but in a perpendicular direction to the previous planes. Finally, we cut it a few times horizontally. All this time the cube was resting on one of its faces on the floor. With every cut we cut through the cube completely. We cut the cube a total of 175 times, at least once in all three of the previously mentioned directions. Could the number of smaller pieces created be:

a) 2145

b) 3006?