5th and 6th Grade, September, 2019

B.1249. What is the last digit of the following product:

2015 x 2017 x 2019 x 2021

B.1250. The first three elements of a number sequence are 3, 1, 1. Starting with the second element, the product of the two neighbors of any element is the same. What is the sum of its first 2019 elements?

B.1251. Using the positive whole numbers, we created the following series of fractions. What number goes in place of the question mark?

1/1, 2/2, 3/1, 4/3, 5/2, 6/1, 7/4, 8/3, 9/2, 10/1, …, 2019/?

B.1252. Mr. Shee Lee Hoi is the owner of the most famous restaurant in Hong Kong. Once he decided that he will write his name on the window by using straight neon line-segments, so that you could figure out the phone number of his restaurant by looking at the sign. What is that 10-digit phone number?

B.1253. To a whole number add the number you get by reversing the digits of the original number. Can the sum be 2019?

B.1254. How many rectangles with whole number long sides have perimeter equal to 2019?

B.1255. How many rectangles with whole number long sides have area equal to 2019?

B.1256. There are 3 discs rotating around the same axle in a toy. There are 12 pictures of fruits on each disc, and every time the discs stop spinning, only one picture can be seen on each disc through a small window for each disc. The first disc has 4 lemons, 4 bananas and 4 oranges; the second disc has 2 lemons, 2 bananas, 2 oranges, 2 lemons, 2 bananas, and 2 oranges; the third disc has a lemon, a banana, an orange, a lemon, a banana, an orange, and so on, in the order of the rotation of the discs. (It means that if the discs rotate slowly, these are the orders of the pictures appearing in each window.) When you pull the arm to rotate the discs, they will rotate so fast that you cannot see the pictures on them until they all stop at the same time. The rotation of the discs is set so that they always stop 7 places ahead of the previously shown pictures. At the beginning of the game each window showed the first lemon of an all-lemon segment on the disc behind that window.

a) How many times do we have to pull the arm in order to see 3 oranges for the first time?

b) Can we ever see 3 bananas?