If each side of parallelogram ~$P$~ has length 1, what is the area of ~$P$~ ?

1. One angle of ~$P$~ measures 45 degrees.

2. The altitude of ~$P$~ is  ~$\frac{\sqrt{2}}{2}$~.

If ~$a$~ and ~$b$~ are integers, is ~$a^5 < 4^b$~ ?

1. ~$a^3 = -27$~

2. ~$b^2 = 16$~

The total price of 5 pounds of regular coffee and 3 pounds of decaffeinated coffee was $21.50. What was the price of the 5 pounds of regular coffee?

1. If the price of the 5 pounds of regular coffee had been reduced 10 percent and the price of the 3 pounds of decaffeinated coffee had been reduced 20 percent, the total price would have been 18.45.

2. The price of the 5 pounds of regular coffee was S3.50 more than the price of the 3 pounds of decaffeinated coffee.

What number is 6 more than ~$x + y$~?

1. ~$y$~ is 3 less than ~$x$~.

2. ~$y$~ is twice  ~$x$~.

Jack picked 76 apples. Of these, he sold ~$4y$~ apples to Juanita and ~$3t$~ apples to Sylvia. If he kept the remaining apples, how many apples did he keep?(~$t$~ and ~$y$~ are positive integers.)

1. ~$y ≥ 15$~ and ~$t$~ = 2

2. ~$y = 17$~

In any sequence of ~$n$~ nonzero numbers, a pair of consecutive terms with opposite signs represents a sign change. For example, the sequence -2, 3,-4, 5 has three sign changes. Does the sequence of nonzero numbers ~$s_1,s_2,\ldots, s_n$~ have an even number of sign changes?

1. ~$s_k = (-1)^k$~ for all positive integers ~$k$~ from 1 to ~$n$~.

2. ~$n$~ is odd.

If ~$r$~ and ~$s$~ are positive numbers and \theta is one of the operations, ＋,－, ×, or ÷, which operation is \theta?

1. If ~$r = s$~, then ~$r \theta s$~ = 0.

2. If ~$r ≠ s$~, then ~$r \theta s ≠ s \theta r$~.

What is the remainder when the positive integer ~$n$~ is divided by 5?

1. When ~$n$~ is divided by 3, the quotient is 4 and the remainder is 1.

2. When ~$n$~ is divided by 4, the remainder is 1.

On a map, ~$\frac{1}{2}$~ inch represents 100 miles. According to this map, how many miles is City ~$X$~ from City ~$Y$~?

1. City ~$X$~ is 3 inches from City ~$Y$~ on the map.

2. Cities ~$X$~ and ~$Y$~ are each 300 miles from City ~$Z$~.

Is ~$n$~ equal to zero?

1. The product of ~$n$~ and some nonzero number is 0.

2. The sum of ~$n$~ and 0 is 0.