2010 High School Team Math
Read each problem carefully. All answers must be expressed in simplest form. Units of measure are not required. Do not make
approximations unless prescribed by the problem. Each problem is worth one point (all or nothing). Each team shall submit a
single answer sheet. Only legible answers on that sheet will be considered for scoring.
1. The equivalent resistance of multiple resistors wired in series is the sum of the individual resistances. 30Ω wired in
series with 40Ω is equivalent to a single 30Ω + 40Ω = 70Ω resistor.
30Ω
40Ω
The equivalent resistance of multiple resistors wired in parallel (i.e. all resistors are electrically
connected at each end) is the reciprocal of the sum of the reciprocals of the individual resistances.
1
Req = 1 + 1 1+... 1 . 90Ω wired in parallel with 30Ω is the equivalent of a single 1 +
=
1
R1
R2
Rn
30Ω
90Ω
22.5Ω resistor.
70Ω
You have a box with ten 60Ω resistors. You need to construct a circuit whose equivalent
resistance is exactly 100Ω.
90Ω
What is the fewest number of resistors you can use to produce a circuit whose equivalent
resistance is exactly 100Ω?
30Ω
2. Evaluate this product:
∞
Y
(−i)
102
22.5Ω
i=0
Note: The −i is the exponent of the 2, which is the exponent of the 10.
3. You fold a rectangular sheet of paper so that two diagonally opposite corners come together. If the length of the crease
you just formed is exactly equal to the length of the longer side of the paper, what is the exact ratio of the longer side
to the shorter side of the sheet of paper?
4. The units digit of each number in the Fibonacci sequence forms a repeating sequence of its own. How many terms are
in the sequence before it repeats?
5. You have five colors of paint with which you’d like to paint a number of regular tetrahedrons. You decide that, for any
given tetrahedron, you will paint each of its faces a different color than any other face. You also decide that no two
tetrahedrons will be exactly alike. (Two tetrahedrons are considered exactly alike if you can rotate one of them so that
the colors of corresponding faces match.)
How many different tetrahedrons will you paint?
6. How many prime numbers exist between 300 and 400?
7. How many degrees are there in A + B + C + D + E + F? Those letters represent the interior angles of the points of
the star.
C
B
D
A
E
F
8. A certain third-degree polynomial has three rational roots. The sum of the roots is 0. The product of the roots is
−2592. If you multiply the first root and the second root, add that to the product of the first root and the third root,
and add that to the product of the second and third roots, the sum is −468. What are the roots of this polynomial?
9. What is the largest prime factor of 15, 475, 420, 832, 256?
10. Calculate the area of the triangle with these vertices: (8,-3), (12,20), and (30,5).
Tie Breakers
T1. You have been captured by a tribe of mostly truth tellers and mostly liars. Mostly truth tellers tell the truth
exactly 80% of the time, while mostly liars lie 80% of the time. This tribe sets before you two doors, one
that leads to certain death and the other that leads to freedom. The doors are guarded by exactly one mostly
truth teller and one mostly liar (though you cannot tell them apart), both of whom will answer questions. You
try the standard question, “Which door would the other guy say leads to certain death?” but realize that you
can’t be certain the answer is false. You decide to ask the question again, realizing that you may get a different answer.
How many times in a row must you get the same answer in order to be at least 99.9% sure that the answer is false?
Another quirk of the members of that tribe is that each of them can predict future events, like how another member
of the tribe would answer some arbitrary question, regardless of whether or not the answer is true or false.
T2. Find the smallest pair of consecutive, positive integers whose product is a multiple of 2010. What is the sum of the
two numbers?
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