## 5th and 6th Grade, November, 2018

B.1209. Find the last three digits of the sum of the first 17 members of the following number sequence:

7, 77, 777, 7777, …

B.1210. Fill in the blank fields of the following chart using X’s and O’s, so that no four of the same kind of marks are side-by-side in a row or column or diagonally.

B.1211. Find three integers so that their product is a positive prime number, and if you put them in an increasing order, the difference between any two consecutive numbers is the same. (A number is prime if it has exactly two positive divisors.)

B.1212. How many vertices of a regular 7-sided polygon can you pick so that the distance between any two selected vertices is different?

B.1213. How many year numbers are there in this millennium with two zeroes in them?

B.1214. Olga started to build a corner fence as you can see on the diagram below. She is using identical cubic building blocks. If she continued building by the same pattern, how many blocks would she need to build a 21 story high fence?

B.1215. In Calendaria, a hidden country of the world, a calendar- and bookstore owner came up with the following idea: the 2019 calendar costs 365 darias before the year begins and then it costs one daria less every day that has gone by during the year. However, he will raise the price of every book and every calendar by 10 darias on the first day of April, July and October. On which day of the year can little Darius buy the current year’s calendar for 100 darias from this store owner?

B.1216. The actual length of an inching worm is 2 cm. When he walks he takes up the following two extreme positions: both his head part and tail part touch the ground on a 2 mm section, so his length varies between 4 mm and 16 mm. The inching worm progresses by pushing his body forward while hanging on to the ground by his tail, and then he pulls his tail up while standing on his head. He switches from position B to position A periodically. It takes a half a second for him to extend his body and then a half a second to pull up his tail. How long does it take his head to get to the end of a 10 cm long branch if it started from standing at the very beginning of that branch in position A?