If a positive two-digit integer is c times the sum of its digits, the number formed by interchanging the digits is the sum of the digits multiplied by what expression that involves c?

On Sunday, John drove from his house to his uncle’s house for a visit. If his average speed had been 10 miles per hour slower, the trip would have taken 2 hours longer. If his average speed had been 20 miles per hour faster, the trip would have taken 2 hours less. How many miles is it from John’s house to his uncle’s house?

Of the first 2011 natural numbers, how many have exactly three digits of 1 when written in base 2 form?

Base BCD of right tetrahedron ABCD is an equilateral triangle with sides of length 6 mm. Each of the lateral sides of the tetrahedron is an isosceles right triangle. What is the volume of the tetrahedron? Express your answer in simplest radical form. 

The number 20 can be expressed as a sum of three natural numbers in many ways. Three distinct examples are 1 + 12 + 7, 3 + 14 + 3 and 14 + 3 + 3. Including the examples shown, in how many distinct ways can 20 be expressed as a sum of three natural numbers? 

 Two red, two yellow and two green faces, all unit squares, are available for building a cube. How many distinct cubes can be built? 

 Ben came up with a new idea for a clock and telling time. He wants to measure a day as two 10-"hour" cycles rather than two 12-hour cycles. However, Ben does not want to change the duration of a minute, so a day would still have the same number of minutes. How many minutes are there in a “Ben-clock hour”?

Three rectangles have the same area. Their dimensions are m by n, (m + 2) by (n – 2) and (m - 2) by (n + 10). What is the area of each rectangle?  

A small square is inscribed in a circle that has another, larger square circumscribed about it. What is the ratio of the area of the small square to the area of the large square? Express your answer as a common fraction.

The mean, median and range are all 6 for a collection of five positive integers. What is the sum of all possible distinct values that could be the greatest number in all the possible collections? 

 For each positive two-digit integer, John adds the two digits. For example, 34 gives 3 + 4 = 7. What is the sum of all of his results?

What is the product of x and y if x\(^2\) + y\(^2\) = 36 – 2xy and x\(^2\) – y\(^2\) = 12? 

One hundred red balls are lined up in a row. Starting from the left end, every fourth ball is replaced with a green ball. Then, starting from the right end, every fifth ball is replaced with a white ball. Finally, starting from the left end, every sixth ball is replaced with a yellow ball. How many red balls remain in the row?

 At 3:00, the hour hand and minute hand of a 12-hour clock are perpendicular. What is the least number of minutes that must elapse for this to be true again? Express your answer as a mixed number.

 In triangle RST, J is on line segment RT, with RJ:JT = 2:1. Also, K is on line segment ST, with TK:KS = 2:1. Line segments SJ and RK intersect at point P. If the area of triangle SPK is 7 units2, what is the area of triangle RPS? 

 Given that n > 1, what is the smallest positive integer n whose divisors have a product of n\(^8\)? 

Two natural numbers have a greatest common factor of 3 and a least common multiple of 216. If the difference between the two numbers is 3, what is the sum of the two numbers?

Eight identical marbles are to be placed in five boxes numbered 1 through 5 so that each box contains at least one marble. In how many ways can this be done? 

 For every day his homework is late, Sam’s grade is reduced by 10%. For example, if Sam's homework is 2 days late, his score will be multiplied by 0.8. Sam will get 4 more problems correct for each extra day he works on his homework past the due date. If Sam has 14 out of 30 problems correct the day his assignment is due, how many days late should he turn it in to get the maximum grade?

 In convex pentagon JKLMN, KL = LM = MN = 4 units and JK = JN = 8 units. If m∠J = 60° and m∠L = m∠M, what is the area of the pentagon, expressed in simplest radical form?