Answer Regional Level Solution ....... 12 1. Let d be the common difference in our arithmetic sequence. Then x=6-d while y=6+d, making x+y = (6-d)+(6+d) = 12. Notice we didn't need to use the fact that the first term equaled 1. (Alternately, one could deduce that d=2.5 since the difference between the first and third terms was 5, then computed x=3.5 and y=8.5 so that x+y=12.) . 2 2. The region in which the two squares overlap with the rectangle is a right triangle, which we shall call T. Because the squares each have area 1, the area of T is also 1. However, upon closer inspection we see that the base and height of the rectangle are the same as the (slanted) base and height of triangle T. Since area(T)=bh/2=1, we find bh=2, so the area of the rectangle is 2. Alternate solution: can the reader figure out how to dissect the two squares into pieces that can be reassembled into the rectangle, thereby showing that they must have the same area? . 12 3. Each of the numbers from 1 to 6 can only have two possible neighbors to avoid sums divisible by 2 or 3. For example, 5 may neighbor only 2 or 6. If we write the numbers 1, 4, 3, 2, 5, and 6 in that order in a ring then each number will be next to its two possible neighbors. (Verify.) Therefore, to order the six numbers successfully we need only start at any point along the ring (6 choices) and list the numbers as they appear around the ring in either direction (2 choices) for a total of 12 possible orderings. . 60 4. Since 30 must divide 72N, there must be factors of 2, 3, and 5 in 72N. The 2 and 3 are already present in 72, so N must be divisible by 5. In the same manner, since 30N is a multiple of 72, N must be a multiple of 12. The smallest positive integer divisible by both 5 and 12 is 60. As a check, we note that 60 divides evenly into (30)(72), so N=60 satisfies all the requirements. . 70 5. In the time it takes the older sister to reach the top, the up escalator has carried her forward 30 steps, because the total length of the escalator is 40 steps and she took 10 steps herself. Therefore in this same time the down escalator pushes the younger sister back 30 steps. To make up this set-back and cover the original 40 steps separating her from the top the younger sister must make a total of 70 steps. . 1/12 6. If I choose a shot that I will make with probability p (where p is between 0 and 1/3), then Michael Jordan will make the same shot with probability 3p. Hence the probability that I make a shot that Jordan subsequently misses is p(1-3p). The graph of this function is a parabola which equals zero when p=0 and p=1/3. By symmetry the vertex (maximum) is midway between the two, at p=1/6. Hence the best chance I have of winning the game is 1/12. . 29pi 7. Notice that a right triangle with legs whose squares are 17 and 99 has the same hypotenuse as a right triangle with legs whose squares are 19 and 97, namely the square root of 116 in both cases. Now place these two right triangles next to one another (not overlapping) so that their hypotenuses coincide. The circle with the common hypotenuse as diameter will neatly circumscribe the resulting quadrilateral since any angle inscribed in a semicircle must be a right angle, and vice versa. To summarize: we have created a quadrilateral with the appropriate side lengths inscribed in a circle with diameter equal to the square root of 116, and hence with area 29pi.