Here's one set of math circle problems (on the easier side) and solutions, some of which I adapted from various Moscow State University math circle problem sets. In 2012, I presented these problems at a Tucson middle school in order to recruit students to the UA math circle.
Problems are taken and translated from http://mmmf.msu.ru.
1. (No calculator!) What's larger: 333333*444444, or 222222*666667? By how much?
Solution: 333333*444444 = 3*111111 * 4*111111 = 12 * 111111
222222*666667 = 222222*666666 + 222222 = 2*111111 * 6*111111 + 222222 = 12*111111 + 222222
We therefore get that 222222*666667 is larger, by 222222.
2. Ms. Nisch used a special bar of soap to wash her cat seven times with. After the seventh washing, the bar's length, width and height decreased to one half of the original measurements. How many more times will she be able to wash her cat? [Note: The answer may surprise you, but so would her cat's size.]
Solution: If the each measurement decreased by a factor of 1/2, the total volume decreased to 1/8 of the initial volume; thus seven washes took up 7/8 of the soap, or, equivalently, each washing took up 1/8 of the initial volume. Since 1/8 of the initial volume is all that's left, Rachel will only be able to wash her cat ONCE!
3. Rachel had seven potatoes, Sara had five, and Yusuke from Japan had none. They baked all the potatoes and divided them up amongst themselves equally. Grateful for the healthy meal, Yusuke gave Rachel and Sarah twelve candies. How many candies did Rachel get, if Yusuke was being fair (as usual) (and as you should be)?
Solution: Each person ate four potatoes. It follows that Sara gave Yusuke one potato, and Rachel gave him three; so Rachel should get 3/4th of the candies, which comes out to 9.
3. If this sequences consists of 100 L-shapes, what is the total number of stars in all 100 shapes?[*] [*][*] [*][*][*] ... [*](98 stars)[*] [*] [*] . [*] . . [*]
Solution: If you put all the shapes together, you get a 100x100 square, from which is follows that there are 100^2 = 10 000 stars in all of the shapes combined.
4. Sara and Rachel are playing a game on a white 3x3 board with a black tile in the center: On each person's turn, they can invert all the colors on a given row or column. Is Sara starts first and the object of the game is to color the entire board black, who wins?
Solution: No one ever wins; look at the top left 2x2 "subboard." No matter what the move is, there is always an odd number of black tiles, and an odd number of white tiles!
5. The distance from Tucson to Desert Rock is 90 km; the distance from Desert Rock to Big Cactus is 30 km; the distance from Big Cactus to Rattlesnake Village is 36 km; and the distance from Rattlesnake Village to Tucson is 24 km. What's the distance from Rattlesnake Village to Desert Rock?
Solution: Draw a diagram. A->B is the distance from A to B.
Tucson->DR <= Tucson->RV + RV->DR, so 90 <= 24 + RV->DR, which means 66 <= RV->DR.
On the other hand,
RV->DR <= RV->BC + BC->DR = 36 + 30 = 66
So we get that 66 <= RV->DR <= 66, so RV->DR = 66.
6. In the triangle ABC, B = 90 degrees, AB = BC = 1. A random point is chosen on the side AC, and the sum of the distances from this point to the sides AB and BC is measured. Is it possible to figure out what this sum is?
Solution: Call this point P, and call its projections onto AB, BC X and Y, respectively. Then triangles AXM and MYC are similar to ABC, so they are isoc. triangles, from which we see that the sum is 1.
7. We are given 2012 integers. They have the special property that the sum of any 100 of them is positive. Is the the sum of all 2012 integers positive?
Solution: How many negative integers can we can in total? Not more than 99, since otherwise we could take 100 negative integers and sum them up to a negative number! Suppose we have N negative integers. Sum all of our integers as follows: (N negative integers) + (100 - N positive integers to make the sum of this and the previous term positive) + (rest of the integers, all of which are positive) to get a positive total sum.