## 7th and 8th Grade, March, 2020

C.1297. How many six-digit cube numbers are there?

C.1298. The sum of two whole numbers (both greater than 10) is 1000. Prove that the last three digits of the squares of these two numbers are identical?

C.1299. Stick another digit into the six-digit number 975 312 (even in front or at the end) so that the new number would be divisible by 468.

C.1300. How big could the greatest prime-divisor of the ababab type 6-digit numbers in base 10 be?

C.1301. Find the greatest whole number which is not the sum of 100 composite numbers.

C.1302. A computer responds to the following six commands:

1) Let the starting value of X be 3, and the starting value of S be 0.

2) Increase the value of X by 2.

3) Increase the value of S by X.

4) If S is at least 10 000, then complete the 5th command, otherwise go back to the 2nd command.

5) Print the value of X.

6) Stop

What will the computer print out?

C.1303. How many 10-digit numbers are there in which only the digits 2 and 5 appear, and there are no digit 2s next to each other?

C.1304. Is it possible to write the first six positive whole numbers on the perimeter of a circle so that for any three a, b, c consecutive numbers on the circle

is divisible by 7?