## 7th and 8th Grade, November, 2017

C.1153. Six soccer teams are in a league. (Every team has to play against every other team.) The season is not over yet, and the diagram below shows the games played already, and their outcomes. (An arrow points from the winner to the loser of that game. A dot on a line indicates a tie.) A team gets 3 points for a win, 1 point for a tie, and 0 points for a loss. Teams with the same number of points finish in the same place.

a) How many games are left to play this season?

b) What is the current order of the teams?

c) Which teams won the remaining games if we know that the team that is the current leader, did not win the overall championship; and that the team that is in last place currently, did not finish last after all?

C.1154. The diagonals AC and BE of a regular octagon ABCDEFGH intersect each other at point P. Find the angles of triangle APE.

C.1155. On the diagram below you have to get from the marked field containing 2 to the marked field containing 8. You can step on each square field no more than once. From each field you can step only horizontally or vertically. (No diagonal steps are allowed.) Add up the numbers on the fields you step on. What is the greatest sum you can get?

C.1156. If 2.017171717… = m/n, where m and n are relatively prime, find the value of m+n.

C.1157. A quadrilateral in a coordinate system is given by the coordinates of its vertexes: A(0,0), B(5,0), C(3,2), and D(0,1). Prove that the angle between its diagonals is 45 degrees.

C.1158. Find two consecutive prime numbers with exactly 2017 consecutive composite numbers in between them.

C.1159. How many different ways can a King on a chess board get from c5 to g2 using the least number of steps possible? (A King may take a step to the neighboring fields horizontally, vertically or diagonally.)

C.1160. The sum of the ages of Tamara and her mother is 49 years. 16 years from now Tamara is going to be the same age as her mother was when Tamara was born. 20 years from now Tamara is going to be the same age as her father was when Tamara was born. How old are the family members now?