An airplane started 8 miles south and 4 miles west of a radar station. The airplane travels due northeast at a speed of 3 miles per minute. In how many minutes is it again the same distance from the radar station as it started? Express your answer as a decimal to the nearest hundredth.  

  In trapezoid ABCD, bases AB and CD are 13 and 39 units, respectively. Legs BC and DA are 24 and 10 units, respectively, and sides BC and DA lie on lines that are perpendicular to each other. What is the area of ABCD? 

Kendra starts counting with a and counts by d, where a and d are both positive integers. For example, for a = 5 and d = 3 the sequence would be 5, 8, 11, …. The sum of two terms in Kendra’s sequence is 10. How many different possible pairs (a, d) are there?

Equilateral triangle ABC has a side length of 6 units. Point D lies on segment BC such that DC = 2(BD). What is the length of the altitude of triangle ADC from point C? Express your answer as a common fraction in simplest radical form.

When 97, 151 and 241 are each divided by a positive integer K, the remainder is the same. What is the largest possible value of K?

A circle with radius 1 unit lies in the first quadrant and is tangent to both the x- and y-axes. A second larger circle lies in the first quadrant, is tangent to both axes and is externally tangent to the first circle. What is the radius of the second circle? Express your answer in simplest radical form.

On a 5 by 5 grid of unit squares, one unit square is colored blue, one unit square is colored red, and the rest of the unit squares are white. Grids are considered different if no rotation could turn one into the other. How many different grids are there?  

The cities of Smallville and Largeville are 300 miles apart. Jim left from Smallville to go to Largeville at 10 a.m. Mickey left Largeville to go to Smallville at 10:30 a.m. on the same day. Jim traveled at a constant speed that was twice Mickey’s constant speed, and they both arrived at a point 90 miles from Largeville at the same time. What was Mickey’s constant speed, in miles per hour?

A large tank contains a 400-kg mixture of water and alcohol. The mixture is 64% alcohol by weight. At each step, 100 kg of the mixture will be drained from the tank, replaced with 100 kg of water, and then stirred. After three steps, what percent of the final solution will be alcohol?

A “deletable prime” is a positive integer that (1) is prime and (2) is either a onedigit integer or, after removing one digit, results in another deletable prime. For example, 439 is deletable because 439 is prime and deleting the 9 results in another deletable prime, 43, which is deletable because removing the 4 results in the prime 3. What is the smallest deletable prime larger than 443?  

Let T be a positive integer whose only digits are 0s and 1s. If X = T ÷ 12, and X is an integer, what is the smallest possible value of X?

Consider all of the positive five-digit integers that can be formed using each of the digits 3, 4, 5, 6 and 7 exactly once . What is the sum of these integers? 

If &a represents πa⁴, then the volume of a sphere of radius 3 units could be represented by &x cu units for some positive value of x. What is the value of x? Express your answer in simplest radical form.

The gravitational force on an asteroid varies inversely with the square of its distance from the sun. By what percent must the distance decrease in order that the gravitational force be multiplied by 3? Express your answer to the nearest tenth.

How many different right triangles with integer side lengths have one leg 15 units long?

There is a method traditionally used in some Russian villages to see which of the young women in the village are to be married the next year. Three blades of grass are folded in half and held in such a way that the six ends of the blades are visible but the rest of the blades are hidden. A young woman ties the ends together in pairs at random such that there are three knots and each end is tied to exactly one other end. If, on release, a three-blade loop is formed, the woman will be married the next year. What is the probability of getting a three-blade loop? Express your answer as a common fraction.

Suppose a regular hexagon has a perimeter equal to the circumference of a circle. What is the ratio of a side of the hexagon to the radius of the circle? Express your answer as a common fraction in terms of π

The side lengths of a right triangle are each an integral number of units. If one of the legs is 13 units, what is the perimeter of the triangle?

Six pipes each having a radius of 0.5 feet are stacked in a triangular pile with three pipes on the ground tangent to each other, two in the next row and then one on top. What is the height of the pile? Express your answer in simplest radical form.  

What is the sum of the 13 smallest positive palindromes that have a tens digit of 3 and a ones digit of 7?