Q1.

Three planets orbit a star circularly in the same plane. Each moves in the same direction and moves at constant speed. Their periods are 60, 84, and 140 years. The three planets and the star are currently collinear. What is the fewest number of years from now that they will all be collinear again?

Q2.

In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let N be the number of shadings with this property. Find the remainder when N is divided by 1000

Q3.

Let the set S = {8, 5, 1, 13, 34, 3, 21, 2}. Susan makes a list as follows: for each two-element subset of S, she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.

Q4.

Let a, b, and c be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation (x-a)(x-b)+(x-b)(x-c)=0?

Q5.

Two sides of a triangle have lengths 10 and 15. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side?