Download Ithaca College Math Day Competition April 18, 2007 Part I

Ithaca College Math Day Competition
April 18, 2007
Individual Question Sheets
Part I
1. What is the value of i1 + i2 + i3 + · · · + i2007 ? (Recall that i =
√
−1.)
2. Write the equation of the line (in form y = mx+b) containing the points of intersection
of the parabolas
y = 2x2 + x + 3 and y = x2 − x + 18.
3. The centers of adjacent faces of a unit cube are joined to form an octahedron. What
is the volume of the octahedron?
q
√
4. For how many real numbers x is 144 − x an integer?
5. How many of the three digit numbers that can be formed from all of the digits 3, 5,
and 7 (used only once each) are prime?
6. Two boats on the opposite shores of a river start moving towards each other. When
they pass each other they are 500 meters from one shoreline. They each continue to
the opposite shore, immediately turn around and start back. When they meet again
they are 150 meters from the other shoreline. Each boat maintains a constant speed
throughout. How wide is the river?
7. Call a number “prime-looking” if it is composite but not divisible by 2, 3, or 5. The
three smallest prime-looking numbers are 49, 77, and 91. There are 168 prime numbers
less than 1000. How many prime-looking numbers are there less than 1000? (Recall
that a composite number is any integer greater than 1 and not prime.)
8. For real numbers a and b, define the operation a ? b =
a
. What is the value of
a+b
(2 ? 3) ? 4?
9. If you put a square on each side of a unit square and connect the outside vertices,
then you get an octagon. But, this octagon is not regular. Find the dimensions of
the rectangles that you would have to put on the edges of the unit square so that the
resulting figure is a regular octagon.
10. Pascal has six yellow marbles and four blue marbles, each distinct from the others. He
arranges his ten marbles in a row. What is the probability that no blue marble lies
next to another?
11. The odometer in Jane’s car reads 15,951 miles. Law-abiding Jane notices that this
is a palindrome and says, “It will be a long time before that happens again.” But 2
hours later, the odometer shows a different palindromic number. How fast was the car
traveling in those 2 hours, assuming that Jane traveled at a constant speed during the
2 hours?
12. The zeros of the polynomial x3 − 24x2 + 191x + C are in arithmetic progression. What
is the value of C?
13. Define a sequence x1 , x2 , . . . by
x1 = 1,
x2 = 2,
and
xn =
xn−1
for each integer n ≥ 3.
xn−2
What is x2007 ?
14. If 9x + 9−x = 14, then what is 81x + 81−x ?
15. Which is greater, the probability of getting at least one six with four rolls of a single
die or the probability of getting at least one double 6 in 24 rolls of a pair of dice?
Part II
16. Given that −1 < q < 1, express the following infinite product
(1 + q)(1 + q 2 )(1 + q 4 )(1 + q 8 )(1 + q 16 ) · · ·
in the simplest algebraic form.
17. Given that p(x) is a polynomial of degree six such that p(n) = 1/n for n = 1, 2, 3, . . . , 7,
find p(8).
18. Find two different numbers, each of which is the square of the other.
√
5
1
+
, show (without using a calculator) that tan2 (54◦ ) tan2 (18◦ )
19. Given that cos(36◦ ) =
4
is a rational number.
20. Let P be a point inside a square such that the distances from P to the four vertices
(in order) are 6, 30, 42, and x. What is the value of x?