## 7th and 8th Grade, September, 2017

C.1137. Find the second smallest positive whole number that when multiplied by 252 gives you a cube number.

C.1138. Find a 4-digit positive whole number that is 9 times as great as the number you get when you reverse the order of its digits.

C.1139. How many digits of 1 and how many digits of zero are there in the final result of the following additions:

9 + 99 + 999 + 9999 + … + 999…999

(There are 100 digits of 9 in the last addend.)

C.1140. A solid has 6 octagons, 8 hexagons and 12 squares as faces on its surface. Every vertex has 3 edges running from it. How many vertices does this solid have?

C.1141. Some whole numbers have the same value even if you read their digits in the reverse order. How many such numbers are there between 10,000 and 12,000?

C.1142. Newton and Gregory were arguing about the following problem: If you have a sphere, how many spheres of the same radius can you place around it so that all of them would touch the original sphere? The argument was decided only 180 years later. As it turned out in 1874, Newton was right. What do you think, without necessarily proving it, is the maximum number of such spheres?

C.1143. A company decided that it will raise the price of a \$3.00 item by 3 cents. When I learned about it, I immediately bought a few of these items for the old price because I figured out that I could have bought 5 fewer of these items at the new price for the same amount of money. How many of these items did I buy?

C.1144. In the XVII century Lord de Mere, an avid gambler, asked Pascal, one of the great mathematicians of his time: If a player rolls a die 4 times what should I bet on: a) he rolls at least one 6; or b) he does not roll a 6 at all? What would you bet on?