## 5th and 6th Grade, February, 2019

B.1233. I thought of an integer. I multiplied it by 10 and I added the original number to the product. The first three digits of the resulting number are 350. What was the smallest positive integer I could have started with?

B.1234. Jane is playing a number game with her friends. Somebody starts the game by saying “One”, and then the other players one-by-one say the number 1 higher than the previously heard number, or say “Boo” if their number would contain any of the digits: 3, 6, or 9. If Jane said 45, how many boo’s should be heard by then?

B.1235. Leslie’s calculator does not work properly. She realized that two number keys’ functions got switched: if she presses one of these two numbers, the calculator reacts as if she pressed on the other number, and vice versa. If she punches in 8×45, the result is 360. If she punches in 239×10, the result is 2390. If she punches in 6×87 then the result is 602. Which two number keys got switched?

B.1236. Teresa wants to divide the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 into two groups so that one of the groups would contain four consecutive numbers whose sum is half as much as the sum of the numbers in the other group. What four consecutive numbers are in that group?

B.1237. Fiona went to a lighting store and saw chandeliers with 3 or 4 arms, and one light bulb on each arm. By the time she got home, she could not remember exactly how many chandeliers she saw from each kind, but recalled that she saw either 6 or 8 of the 3-arm chandeliers and 5 or 7 of the 4-arm chandeliers. How many light bulbs could possible be all together on all the chandeliers Fiona saw in that store?

B.1238. In a fairy tale, the citizens of Clocktown divided the day into 60 hours (instead of 24), and divided an hour into 24 minutes (instead of 60). (Therefore, one minute is the same length of time in Clocktown as in our world.) They even made a digital clock that shows the time every minute from 0 hour 0 minute to 59 hours 23 minutes, and then switches to 0 hour 0 minute again. When our clock shows 0 hour 0 minute, so does the clock in Clocktown.

a) What time is shown in Clocktown when our clock shows 15 hours 30 minutes?

b) What is the time in our world when the clock in Clocktown shows 48 hours 15 minutes?

B.1239. Inge is playing a number game with her friends. Somebody starts the game by saying “One”, and then the other players one-by-one say the number 1 higher than the previously heard number, or say “Boo” if their number is divisible by 5 or the sum of the digits of their number is divisible by 5. Inge said the 21st “Boo”. What number did this “Boo” replace?

B.1240. Leslie’s calculator does not work properly. She realized that two number keys’ functions got switched: if she presses one of these two numbers, the calculator reacts as if she pressed on the other number, and vice versa. If she punches in 18×45, the result is 972. If she punches in 15×84, the result is 1190. Which two number keys got switched if we know that both of them are used in both of these two calculations?