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

B.1201. We do not know the three middle digits of a 7-digit number, but we know that they are identical. The number is the following form: 84aaa48. We also know that you can get this number by multiplying three consecutive even numbers. What is the unknown digit?

B.1202. The circumference of the wheel on Hansel’s bike is twice the circumference of the wheel on Greta’s bike. If Hansel turns his pedal around once, his wheel turns around twice, but when Greta turns her pedal around once, her wheel turns around only once. Greta can bike 2 km in 20 minutes. How long does Hansel take to bike the same distance on his own bike if he makes the same number of turns on his pedal in a minute as Greta did?

B.1203. The dimensions of a wooden block are 2cm x 3cm x 4cm. You paint every face of this block blue, then with parallel cuts to its sides we cut it up to 24 unit cubes. Using these cubes, can you build a 2cm x 3cm x 4cm block that

a) has no blue paint on its surface?

b) has two opposite faces with no blue paint on them but the rest of the faces are blue?

c) has two all-blue opposite faces but the rest of the faces have no blue paint on them?

B.1204. Replaces the letters a, b, c, and d by the digits 2, 3, 4, and 5, so that the sum a/b+c/d is as small as possible.

B.1205. A company wanted a new logo, so they asked a designer to create one for them, who came up with the following design using two kinds of regular hexagons. What is the ratio of the areas of the two kinds of hexagons?

B.1206. a) How many such 4-digit numbers are there from which you can subtract a 2-digit number to get a 3-digit number?

b) How many such 4-digit numbers are there from which you can subtract any 2-digit number and you always get a 3-digit number?

B.1207. Last month my cat gave birth to a few kittens. Each kitten has at least one brother and at least one sister. At least how many kittens were born?

B.1208. In a family the sum of the ages of the four children is 20 years.

a) What is the sum of their ages going to be 4 years from now?

b) The sum of their ages 2 years ago was 13. How old is the youngest child now?