MAΘ Problem Set 8
November 3, 2003
1. What parity must n3 − n + 1 have, for whole numbers n? (Parity refers
to whether a number is even or odd). Prove your result.
√
2. The equation 13 − 17x2 = k has exactly one solution x for certain values
of k. Find all such values.
√
3. Define a sequence {ai } by aj+1 = aj or aj+1 = 2aj . The first term
satisfies a0 > 100, 000. (For example, if a0 = 160, 000, then{ai } might√be
{160,
400, 200, 100, 50, 25...}, or it might be {160,000, 80,000, 200 2,
√ 000, √
100 2, 10 4 2,...} Find the least possible a0 such that 1268 appears as a
term somewhere in the sequence.
4. The length of the hypotenuse of a certain right triangle is an integer and
is one unit greater than the length of the longer leg. How many of the
following numbers are possibilities√for the number of units in the measure
of the shorter leg: {3, 27, 64, 121, 210, 216, 410 , 510 }?
√
5. Find the greatest integer that does not exceed 9x2 + 6x, where x =
400, 001.
6. Find the minimum possible area of a trapezoid in which two externally
tangent circles of radius 1 and 2 may be inscribed.