20th Annual Lehigh University / AT&T High School Math Contest

  March 4, 2000  1000am

1. Do not open this exam until you are told to do so.

2. Print your name clearly on your answer sheet. On the "Grade" line, write
your grade in school (9,10,11, or 12) not the grade level of your team.

3. Print clearly on the answer sheet your answer to each question. You should
not write decimal approximations to numbers such as  ,and 10!.
You should simplify your answers as much possible. For example
6/4 should be simplified to 3/2, and square roots should not appear in the
denominator. Frequently, several equivalent expressions will be considered
correct. For example, 3/2, 1 1/2, and 1.5 could each be correct.

4. Each correct answer is worth 1 point. A blank and an incorrect answer are each
worth 0 points. An answer which is correct, but not adequately simplified, may
receive half of one point.

5. No books, notes, calculators or headphones may be used.

6. Most students will probably be unable to finish the exam in the 2 hours allotted.
for the most part, the hardest questions are near the end of the exam.

7.  denotes a line segment and  denotes the length of a line segment. The
vertices of a polygon such as ABCD, are indicated in consecutive order as you
move around the polygon in a clockwise manner.

8. You may keep your questions and scratch work. You might want to keep a record
of your responses for comparison with the solutions which will be given out after the

9. The people with the five highest scores of all people taking the test, without regard for
grade level, will receive cash awards of $250, $175, $100, $50, and $25 respectively.
In case of ties, the cash awards will be split. For example if two tie for first, they each
receive ($250+$175)/2 = $212.50.

10. A plaque will be awarded to the top individual and team in each grade level (9-12 for
individuals, 10-12 for teams). In case of a tie, duplicate awards will be presented.

2000 Lehigh University / AT&T High School Mathematics Examination


3. Steve has 5 red poker chips and 7 blue poker chips.  Tom has 9 red poker
chips and 3 blue poker chips.  If Steve's chips are worth a total of $69 and
Tom's are worth a total of $57, determine the value of a blue chip.

4. Determine the slope of the straight line shown in the figure below.

5. Two sides of a triangle are 14 and 20. If the third side of the triangle is an integer,
what is the smallest possible value for the length of the third side?

6. This summer 1 US dollar was worth $1.45 in Canadian money. At that time how much US money would be equivalent to 1 Canadian dollar? Give your answer rounded to the
nearest cent.

7.  What is the equation of a line that passes through the point (1,4) and is parallel
to the line y = - 2x + 3  ?  Put your answer in y = mx + b form.


10. The sides of a triangle are 6,7 and x. What is the largest value that could
 be the area of such a triangle?

11. A grad is a unit of angle measurement where there are 100 grads in a right
angle.  How many regular polygons with side length 1 have angles with an
integer number of grads?

13. Sally travels to work but must only travel north or east. Furthermore due to
construction a section of road is unusable. Determine the number of ways Sally can
get to work?

14. John is given 16 coins that look identical. In this collection there are 15 genuine
coins all of the same weight and one fake coin that is lighter than all of the others.
He has a balance as shown in the figure below that can be used to determine if a stack
of coins on one side is lighter than a stack of coins on the other. Determine the fewest
number of times the scale can be used so that John can guarantee the identification of
the fake coin.


18. In the figure below a square with side length 1 is inscribed in a circle. Find the area
of the shaded region.


21. A class has three teachers, Mr. P, Ms. Q and Mrs. R and six students, Ali, Bob, Cal, Dee, Emy, and Fay. How many ways can they sit in a line of 9 chairs if we have the property that between any two teachers there are exactly two students?

22. From a group of 10 students an executive board is chosen that has one president, one vice-president, one secretary and one treasurer. If each person can fill at most one of the positions, how many different executive boards are possible?



24.  Let P(x) = 0 be a fifth-degree polynomial equation with integer coefficients
that has at least one integral root.  If  P(2) = 13 and P(10) = 5, compute a
value of x that must satisfy P(x) = 0.

25.  Two concentric circles are such that the smaller divides the larger into two
regions of equal area.  If the radius of the smaller circle is 3, compute the
length of a tangent from any point P on the larger circle to the smaller






33. A sphere and a regular octahedron have the same surface area. Determine the square of the ratio between the diameter of the sphere and the distance between opposite vertices of the octahedron.



 36. How many positive integers n are there such that n is an exact divisor of
 at least one of the numbers

1050, 2030?

37. A 2 x 3 rectangle has vertices (0,0), (2,0), (0,3), and (2,3).
Perform the following sequence of steps:
1. Rotate the rectangle 90? degrees clockwise about the point (2,0).
2. Then rotate this new rectangle 90? clockwise about the point (5,0).
(At this point the side of the rectangle that originally was on the x-axis is now parallel to the x-axis).
3. Then rotate the current rectangle 90? clockwise about the point (7,0).

Find the area of the region above the x-axis and below the curve traced out by the point whose initial position is (1,1).

38.   Consider the following carnival game. You toss a penny onto the board and you win
if the entire penny (3/4" in diameter) lies inside one of the gray squares (side length 1").
The board is placed on a square table that has dimensions .
Assuming that the entire penny lies on the table, what is the probability of winning?

39. Consider the following tiles.

A perfect tiling of a 1 x n rectangle is an arrangement of tiles that completely covers the entire rectangle such that no two tiles overlap or extend beyond the boundary of the rectangle. For example when n = 6 we could have the following tilings:

Determine the number of different tilings for a 1 x 12 rectangle.


40. Let S be a set of n distinct real numbers. Let AS  be the set of numbers that occur as the averages of
two distinct elements of S. For a given  what is the smallest possible number of elements in AS?