## 5th and 6th Grade, March, 2020

B.1297. How many positive square numbers are there up to 2020?

B.1298. How many such rectangles are there for which the lengths of the sides are whole numbers and the area is 2020 units?

B.1299. I moved all of an encyclopedia’s many (more than 2) volumes from one shelf to another. Later I realized that the volumes are in a totally wrong order:

1) none of the volumes are in the place where they should be,

2) none of them are followed by the right volume.

How many volumes could this encyclopedia have?

B.1300. Find the missing digits of the 523abc six-digit number, so that it would be divisible by 7, 8, and 9, also.

B.1301. Is it true that if I pick 30 numbers out of the first 50 positive whole numbers, then there is going to be two such numbers among them that one of these two numbers is twice the other one?

B.1302. Pete chose two consecutive positive integers, and wrote them down on two separate sheets of paper. He gave one of the numbers to Andrew and gave the other number to Tom, who both knew that the numbers they were given are consecutive positive integers. Then the following highly intellectual conversation took place between Andrew and Tom:

Andrew: ” I do not know what number you got.”

Tom: “I don’t know what number you got either.”

Andrew: ” I do not know what number you got.”

Tom: “I don’t know what number you got either.” ….

Andrew and Tom both said the same sentence 10 times. Then Andrew’s 11th sentence sounded like:

“Now I know what number you got.”

What number did Tom get?

B.1303. Find the smallest square number starting with the digit-series 2468.

B.1304. What is the sum of all the digits of the following numbers:

1, 2, 3, …, 1000?