7th and 8th graders

March 2015

C.1017. Bob and John form a team together. Bob is as old as John will be when Bob is twice as old as John was when Bob was half as old as the sum of their current ages. John is as old as Bob was when John was half as old as he will become over ten years. How old are Bob and John?

C.1018. There are red and blue balls in a box. At least 90% of the balls are red. Kate is taking the balls out of the box one by one. Only one of the first 50 balls was blue. While she was taking out the rest of them, every 8th ball was blue. At most, how many balls could there be in the box?

C.1019. At a corporation the members of the Board of Directors and the workers had to vote on an important issue. 19% of the members of the Board of Directors and 91% of the workers voted yes. We know that every member of the Board of Directors and every worker voted, and that 90% of all the votes were yes. What is the ratio of the number of members of the Board of Directors and the number of workers?

C.1020. Pick any point P inside of rectangle ABCD. The bisecting perpendiculars of segments PA, PB, PC, and PD create a quadrilateral. Prove that the diagonals of this quadrilateral intercept each other in the midpoint of rectangle ABCD.

C.1021. Tie up a 6 cm-diameter and an 18 cm-diameter cylinder with a rope as shown on the diagram. How long is the rope if we used 15 cm of rope for the knot to tie the two ends of the rope together?
C.1022. Two cities (A and B) are connected by a straight road. There are 3 other cities on this road between cities A and B. The distance between any two out of these 5 cities is a whole number of kilometers. If somebody walks from one city to any other city, from the number of kilometers walked you can tell exactly between which two cities the person walked. What is the shortest possible distance between A and B?

C.1023. A red line is going upwards in a spiral from the bottom of a cylinder to the top, rising by the same rate all along, making 6 complete rotations around the cylinder. The radius of the cylinder is 12 centimeter, and its height is 1 meter. How long is the line. (Ignore the width of the line.)

C.1024. We wrote down the 3-digit numbers one after another in a row, so that the digits of the even numbers are written in red, and the digits of the odd numbers are written in blue. What is the 2015th digit, and what is its color? 

Please, send your solutions here.

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