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

B.1225. The solid below was built by using red and white 1cm x 1cm x 1cm unit cubes. We indicated the lengths of a few edges of the solid. The center region, as shown on the diagram is all red, and the two sides have one layer of all white cubes each. How many red and how many white cubes were used? What is the surface area of the red part?

B.1226. The numerator of a fraction is 50-x, its denominator is 221. For what positive values (less than 50) of x is this fraction reducible, and what are the lowest terms of those fractions?

B.1227. Laszlo is sucking on his favorite candies with a constant speed. When he has only one tenth of a candy in his mouth, he starts sucking on another one, too. When he has one tenth of that new candy in his mouth, he starts sucking on yet another one. Each candy melts in his mouth as if he sucked on only one candy. How long does it take Laszlo to finish 10 candies if it takes him 5 minutes to finish one candy?

B.1228. In how many positive 3-digit integers is the product of the digits 12?

B.1229. Fabienne keeps her red, blue and green pencils in the same pencil holder. Out of all the pencils in her pencil holder, 3 pencils are not red, 4 of them are not blue, and 5 of them are not green. How many pencils of each color does she have in her pencil holder?

B.1230. There are 70 songs in a book, each labeled by the numbers from 1 to 70. Every Sunday the congregation sings 3 different songs from this book. The numbers of the songs are always shown on the bulletin board by little cards with one digit printed on each card. Any 6 can be used as a 9, too. What is the least number of each digit card they have to print so that they could show any combination of the 3 songs on the board?

B.1231. Robert, the robot, is locked into an empty room. It starts walking in a straight line, until it hits the wall. Then it turns right and keeps on walking. If it could not turn right because it would turn into another wall, then it turns left and starts walking. If it cannot turn either left or right, then it turns itself off. Figure 1 is the diagram of a room in which if Robert starts from the point marked by 1 then after walking around the room, it will come back to point 1 and it will turn itself off there. Figure 2 shows a room in which Robert may start either from point 1 or point 2, and it will end up at its starting point, where it turns itself off. Figure 3 shows a room in which if Robert starts at point 1, it will end up in point 2 and shuts down; or if it starts from point 2 then it will end up in point 1 and turns itself off there.

a) Draw the diagram of a room in which there are 3 starting points, so that the robot may start from any one of them and end up at the same point where it will turn itself off.

b) Draw the diagram of a room in which there are 3 starting points, so that if the robot starts at point 1, then it will end up at point 2 and turns itself off there; starting at point 2, the robot ends up at point 3 and turns itself off there; and if it starts at point 3 then it will end up at point 1 and turns itself off there.

Figure 1

Figure 2

Figure 3

B.1232. a) Multiply the first 123 positive integers that are divisible by 3. Is the product odd or even?

b) Add up the first 123 positive integers that are divisible by 3. Is the sum odd or even?