Answer
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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.
Alternate solution: try applying the arithmetic-geometric
mean inequality to the numbers 3p and (1-3p) to find the
maximal value of p(1-3p).
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.
It is a curious fact that as long as the side lengths remain the same, the order of the sides does not alter whether or not a polygon can be inscribed in a circle, nor does it alter the size of the circumscribed circle. To see this, verify that if a polygon is inscribed in a circle then switching two adjacent sides produces another polygon which can be inscribed in the same circle. Use this fact to prove our claim. Where did we use this fact above?