1.a) Prove: if every angle of a polygon is acute, then the polygon is a triangle. That is, it can't be a quadrilateral, or a pentagon, or any n-gon with n > 3.
b) If a polygon has one right angle and otherwise acute angles, how many sides can it have? Prove your answer.
c) If a polygon has at most one non-acute angle, how many sides can it have? In your proofs, you may use standard facts about angles in polygons.
2. Consider the three numbers
651000 - 82014 - 31773, 791212 - 92412 + 22001, and 244493 - 58192 + 71777.
Prove that the product of some two of these numbers is nonnegative.
3. Consider the integers from 0 to 999, written in the normal way. We will ask several questions about these numbers, and then ask you some questions about the questions.
a) How many of these numbers contain exactly one digit 0?
b) How many of these numbers contain at least one digit 0?
c) What is the total number of digits 0 that appear in all these numbers?
4. Now, sometimes the numbers from 0 to 999 have to be written with three digits no matter what, for instance, in marking answers on a computer scan sheet for the AIME exam. In such cases, 0 is written as 000 and 23 is written as 023. Answer a)-c) again for the numbers from 000 to 999 written this way. That is, answer
a) How many of these specially written numbers contain exactly one digit 0?
b) How many of these specially written numbers contain at least one digit 0?
c) Which set of questions, a)-c) or a′)-c′), did you prefer? Why?
d) Of your six solutions, which did you like best? Why?
5. Addition of real numbers is associative: (a + b) + c = a + (b + c). So is multiplication: (ab)c = a(bc). But not every operation is associative.
a) Subtraction is not associative; in general (a - b) - c = a - (b - c). When is it associative?
That is ?nd all solutions to (a - b) - c = a - (b - c).
b) Find all solutions to (a/b)/c = a/(b/c).
c) What about exponentiation? Find all solutions in positive real numbers to (a∧b)∧c =a∧(b∧c). If stuck, consider more restricted questions, such as what are all the solutions to this equation if a, b, c are positive integers .
6. Let I1, I2, . . ., In be a ?nite set of closed intervals on the number line. So for instance, I1 might be [1, π], the set of all real numbers x such that 1 ≤ x ≤ π. The intervals are called closed because they contain their endpoints. Throughout this problem, suppose that every pair of intervals intersects.
a) Prove or disprove: all the intervals share a common point, that is, the intersection of all the intervals is nonempty.
b) Suppose the intervals need not be closed. They may be open, as in 1 < x < π, or half open, as in 1 ≤ x < π or 1 < x ≤ π. Prove or disprove: all the intervals share a common point.
c) Suppose the intervals are now line segments in the plane. Prove or disprove: all the intervals share a common point. (In this and the next part, the answer may depend on whether you restrict to closed intervals or not.)
d) Return to the number line, but now suppose there can be in?nitely many intervals. Again investigate: must all the intervals share a common point?