GCS 2014-15 October 5th and 6th Grade

5th and 6th graders

October, 2014

B.977. You are saving money in a box. You are collecting $1, $2, and $5 bills. Yesterday you had an equal number of each of these bills in your box. Today you put one more bill in the box, and the total now is $1517. What kind of bill did you put in the box today?

B.978. The pages in a book are numbers from the 3rd page till the 1500th page. Which kind of page numbers are there more of in the book: the ones that contain the digit 1, or the ones that do not?

B.979. Andrew, Ben, and Carl are classmates. They visit each other very often and they always take the shortest road. One day Andrew visited Ben, and then from there he went to see Carl, so he walked a total of 1353 meters. Another day Carl went over to Andrew and then from there he went to see Ben, so he walked a total of 1334 meters. Next day Ben went to Carl and then from there he went to see Andrew, so he walked a total of 1315 meters. How far do the boys live from each other?

B.980. The first three elements of a number series are 3, 1, 1. Starting with the second element, the product of the two neighbors of any element is the same. What is the sum of the first 2014 elements?

B.981. You show 3 pens (a red, a blue and a yellow) to Kate, Pete and Frank, and then you hide one in each of the three children's school bag. Then you ask them to guess the color of the pen you put in their bags. Kate says red, Pete says yellow and Frank says that you did not put the yellow pen into his bag. When they find the pens it turns out that only one of the guesses was incorrect. Who had what color of pen put in their bags?

B.982. Olga wrote five not necessarily different integers on a piece of paper. She noticed that the square of each number is written on the paper, also. What numbers could Olga write on the paper, and how many different options did she have?

B.983. There are 100 trees in a forest arranged in a 10x10 grid with lines in the North-South and East-West directions, 10 trees in each line. There are no trees outside of this forest. A bird may fly from any tree to the nearest tree in the North, the South-east, or the South-west directions. Prove that the bird can get from any tree to any other tree in this forest.

B.984. The math teacher cut a polygon into a triangle, a quadrilateral, and a pentagon by drawing two diagonals starting from the same vertex of the original polygon. How many sides does the original polygon have?

Please, send your solutions here.

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