GCS 2014-15 November 7th and 8th Grade

7th and 8th graders

November, 2014

C.985. Pete has the same number of sisters as brothers. However, each of her sisters has half as many sisters as brothers. How many boys and how many girls are there in this family?

C.986. Connect an inner point of an equilateral triangle to all three vertexes. Prove that for any inner point you can construct a triangle using these three segments.

C.987. Two students were bored in Science class and they invented the following math game: taking turns, they draw the diagonals of a regular 7-sided polygon. You are not allowed to draw a diagonal that would intersect another diagonal already drawn. The person who cannot draw such a new diagonal loses the game. Who has a winning strategy?

C.988. One of the outer angles of a triangle is 135 degrees. The difference of two of its inner angles is 29 degrees. What could the inner angles of this triangle be?

C.989. Place 2015 pieces on a 2015x2015 chess board, so that they are symmetrical to the center square. Prove that there is a piece in the 1008th row.

C.990. In a store you can buy coats, shoes for women, and shoes for men. One time the inventory showed that 25% of all the items are coats, and 60% of all the shoes are for men. What percent of all the items in the store are shoes for women?

C.991. a) In the sum of the first 2014 positive whole numbers we change the sign of a few numbers. Could the sum be 2015?

b) In the sum of the first 2015 positive whole numbers we change the sign of a few numbers. Could the sum be 2016?

C.992. Start a number sequence with 14. Calculate every next element by adding the cubes of the digits of the previous element. What is the 2014th element? What would the 2014th element be if the first element is 13?

Please, send your solutions here.

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