Each of the integers from 0 to 9, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into a hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10?

Rita and Sam play the following game with n sticks on a table. Each must remove 1, 2, 3, 4 or 5 sticks at a time on altemate tums, and no stick that is removed is put back on the table.The one who removes the last stick(or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays？

The figures above show a hexagonal nut that has a width of 1~$\frac{5}{16}$~ inches and a wrench that, in order to fit the nut. must have a width of at least 1~$\frac{5}{16}$~ inches. Of all the wrenches that fit the nut and have widths that are whole numbers of millimeters, the wrench that fits the nut most closely has a width of how many millimeters?

(Note: 1 inch ≈ 25. 4 millimeters)

The sum of the first 100 positive integers is 5,050. What is the sum of the first 200 positive integers?

When 2/9 of the votes on a certain resolution have been counted, 3/4 of those counted are in favor of the resolution. What fraction of the remaining votes must be against the resolution so that the total count will result in a vote of 2 to 1 against the resolution?

In the figure above, X and Y represent locations in a district of a certain city where the streets form a rectangular grid.In traveling only north or east along the streets from X to Y, how many different paths are possible?

The letters C, I, R, C, L, and E can be used to form 6-letter strings such as CIRCLE or CCIRLE. Using these letters, how many different 6-letter strings can be formed in which the two occurrences of the letter C are separated by at least one other letter?

Pat will walk from intersection X to intersection Y along a route that is confined to the square grid of four streets and three avenues shown in the map above. How many routes from X to Y can Pat take that have the minimum possible length?

The table above shows the average(arithmetic mean) price per dozen eggs sold in a certain store during three successive months. If 2/3 as many dozen were sold in April as in May,and twice as many were sold in June as in April, what was the average price per dozen of the eggs sold over the three-month period?

The map above shows the trails through a wilderness area. If travel is in the direction of the arrows, how many routes along the marked trails are possible from point A to point B?