Reference no: EM13588325

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?