## 7th and 8th grade, January, 2018

C.1169. In which set, the set of two-digit numbers, or the set of four-digit numbers, is the ratio of the count of those numbers containing the digit 7 to those that do not, greater?

C.1170. A and B are positive whole numbers. Neither of them is divisible by 10, and their product is 10,000. What is their sum?

C.1171. Find the last two digits of: C.1172. There are many different roads between Town A, Town K, and Town F. We know that between any two of these towns the number of direct roads is at least 3 but no more than 10. You can get from Town A to Town F directly or through Town K in a total of 33 different ways. Similarly, you can get from Town K to Town F directly or through Town A in a total of 23 different ways. In how many different ways can you get from Town K to Town A?

C.1173. Angle A of the convex quadrilateral ABCD is 100 degrees. We know that diagonal AC breaks up the quadrilateral into an equilateral and an isosceles triangle. How big are the inner angles in ABCD?

C.1174. We glue together 27 regular dice into a 3x3x3 cube. What is the least amount of dots you can see on this cube? (On a regular dice the number of dots are 1 to 6, and the dots on opposite facing sides add up to 7.)

C.1175. The sides of a rectangle are 14 cm and 24 cm. We draw the diagonal from one vertex, and we draw straight segments from this vertex to the mid-, third-, and quarter-points of the longer side facing this vertex, a total of 6 segments including the diagonal). What are the areas of the triangles we created?

C.1176. One year a monthly calendar looked like the diagram below. The sum of the numbers in one of the 3×3 segments of this calendar is 162. What is the smallest number in that 3×3 section? 