Mathcounts

What is the greatest integer k such that 2^k is a factor of 67! ?

The units digit of a three-digit number, ABC, is moved to the left of the remaining two digits to make a new three-digit number, CAB. If CAB − ABC = 162, what is the sum of the least and greatest possible values of ABC?

If the six-digit number 3D6,D92 is divisible by 11, what is the value of D? 

Drew purchased a used 50-page book at the book fair. Drew later realized that the book, in which left-hand pages contained even page numbers and right-hand pages contained odd page numbers, did not contain all 50 pages. The sum of the page numbers on the pages that Drew’s book did contain was 1242. What is the greatest page number that could be on a page missing from Drew’s book?

Three identical boxes contain tennis balls, baseballs or both. A label is affixed to each box. The three labels correctly describe the three boxes, but none of the labels is on the correct box. Box 1 is labeled “Tennis Balls.” Box 2 is labeled “Baseballs.” Box 3 is labeled “Tennis Balls & Baseballs.” Devon reaches into Box 3 and pulls out a baseball. Which box contains only tennis balls? 

In a survey, 30 people reported that they enjoy some combination of walking, hiking and jogging. The number who enjoy only walking, the number who enjoy only hiking and the number who enjoy only jogging are all equal. Likewise, the number who enjoy only walking and hiking, the number who enjoy only walking and jogging and the number who enjoy only hiking and jogging are equal. In addition, the survey showed that half as many people enjoy exactly two of these activities as those who enjoy only one activity. If three people enjoy all three activities, how many people enjoy jogging?  

Celia, Desi and Everett are each wearing a hat that displays a different whole number from 1 to 9, inclusive. Each number cannot be seen by the person wearing it, but that number is visible to the other two individuals. Everett says, “The sum of the numbers I see is 6.” Celia says, “The product of the numbers I see is 10.” What is the sum of the numbers that Everett could possibly have on his hat? 

Each digit 0 through 9 is used exactly once to create two five-digit numbers. What is the sum of the digits of the greatest product of two such numbers?

Sam, Taylor and Pat counted the number of fish in each of their fish tanks. They noticed that Sam’s tank had exactly 25% more fish than Taylor’s tank, and Pat’s tank had exactly 24% more fish than Sam’s tank. If each tank had at least one fish, what is the minimum combined number of fish that could have been in the three tanks?

One circular base of a cylinder with radius 2 inches and height 4 inches is glued flush to the center of each face of a 4-inch cube. What is the surface area of the resulting solid? Express your answer as a decimal to the nearest tenth.