1. Social optimism.
In a survey people are asked ``Do you think the new president will be better than the last one?'' Suppose a people say ``better'', b say ``the same'' and c say ``worse''. Sociologists calculate two measures of ``social optimism'':  SO₁=a + (b/2) and SO₂=a-c. If 100 people respond to the survey and SO₁=40, find SO₂.
 

2. Sums of cubes.
Prove that the sum of the cubes of three consecutive integers is always a multiple of 9.
 

3. A nonlinear equation.

 

4. Max and min. (Reminder: no calculators.)
Find the maximum and minimum values of

for   |x+1| <= 3.  Justify your answer.
 
 

5. Polynomial evaluation.
(No calculators) If

6. An isosceles triangle.
In the rectangle ABCD, sides AD and CD have lengths 10 and 15, respectively. The point P lies inside the rectangle, and the lengths of AP and BP are, respectively, 12 and 9. Prove that triangle APD is isosceles.
 
 

7. Pick an integer.
Let k be a given positive integer. Suppose that each of k+1 persons independently chooses an integer (positive, negative or zero). Prove that there are some two distinct persons whose chosen integers differ by a multiple of k.
 
 

8. f(x)=f(x+1)?
Let d be a positive integer, and   f:[0,d] -> R  continuous, with f(0)=f(d). Prove that:  f(x)=f(x+1) for some x in [0,d-1].
 
 

9. No real roots.
In the equation

a is a real number and i is a square root of -1. Find a necessary  and sufficient condition on a that the equation (1) have NO real roots. (Remember that you must justify your answer.)
 
 

10. A term divisible by 1997?
The sequence (a_n) is defined recursively by  a₁=a₂=a₃=1
and for   n >= 1,

Prove that for every positive integer r there is a term a_s divisible by r.