1. S is a set of positive integers, none greater than 21. No two pairs of the elements of S have the same sum. Find the largest possible sum of the elements of S.
Answer:___________________
2. Look at the set of integers {1000,1001,1002,...1998,1999,2000}. There are times when a pair of consecutive integers can be added without "carrying": 1213 + 1214 requires no carrying, whereas 1217 + 1218 does require carrying. For how many pairs of consecutive integers is no carrying required when the two numbers are added?
Answer:______________________
3. You are given the four simultaneous equations:
a + 2b + 3c + 4d = 262
4a + b + 2c + 3d = 123
3a + 4b + c + 2d = 108
2a + 3b + 4c + d = 137.
Evaluate 27a+ 28b + 29c + 30d.
Answer:_______________________
4. If the integer k is added to each of the numbers 36, 300, and 596, one obtains the squares of three consecutive terms of an arithmetic progression. Find k.
Answer:_______________________
5. A glass cylinder 8 inches high and 4 inches in diameter contains water to some depth.
(a) Suppose the depth is 6 inches. If the cylinder is tilted at an angle of 45 degrees to the horizontal, the edge of the surface of the water forms an ellipse. Find the equation of the ellipse in the form
b) Suppose the depth of water is 7 inches. The cylinder is tilted so that the water reaches the rim of the cylinder (and tilting any more would spill water). Find the equation of the ellipse that is the edge of the surface (again in the form
).
Answer (a):________________ Answer (b):_____________
6. In triangle ABC, line segments AE, BD, and CF intersect at point P and lengths AP=PE=6, BP=9, DP=3, and CF=20.<
a) Find the length of CP.
b) Find the area of triangle ABC.
Answer (a):__________________Answer (b):____________
7. Ashley, Bob and Calvin will toss a die in turn and will continue to toss the die until the die shows either 1 or 6. What is the probability that Calvin is the first to toss a 1 or 6?
Answer:_________________
8. Find the sum of the terms
Answer:___________________