5th and 6th Grade, October, 2019

B.1257. Two apricots and a pear are lighter than two apples and a pear, but this second group is lighter than two pears and an apricot. The same fruits have the same weight. Which fruit is the lightest?

B.1258. Find the smallest possible 2-digit number with the most possible divisors.

B.1259. How many different ways can you pick two different positive whole numbers, so that none of them is greater than 20 and their product is divisible by 10?

B.1260. 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. One of the discs has 4 lemons, 4 bananas and 4 oranges; another disc has 4 lemons, 4 oranges, and 4 bananas; and one has 2 lemons, 2 bananas, 2 oranges, 2 lemons, 2 bananas, and 2 oranges, 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 a lemon on the disc behind that window. If you pull the arm once to rotate the discs, then from left to right orange, lemon, orange can be seen in the windows. If you pull the arm one more time, then banana, banana, lemon can be seen in the windows from left to right. Which disc is in which place in the toy?

B.1261. Write the following numbers as the sum of two square numbers:

a) 10

b) 100

c) 1000

B.1262. We made a big cube out of small, unit cubes. We painted every sides of the big cube blue, and then we took the big cube apart. There we 72 small cubes with exactly 2 sides painted blue. How many small cubes have exactly 3 blue sides?

B.1263. In some countries the date at Christmas is written as 12/25, but in other countries they write it as 25/12. Life takes you to a foreign country where you do not know which way they write the dates.

a) How many dates are there in a year about which you cannot decide for sure which day they represent?

b) How many dates are there in a year that do not allow you to know which way that country uses to write the dates?

B.1264. Peter has to vacuum the apartment. It takes twice as long to clean the living room than his own room, and he finishes the kitchen in 3 less minutes than the hallway. He cleans his own room for 5 minutes longer than the hallway, and he finishes the whole job in 37 minutes. How long does Peter vacuum in the living room and his own room together?