- Tom bought three television sets for his new home, a large one for his living room,
a medium one for his bedroom, and a small one for his den. In total he spent $2001 for the three
TV's. Each TV set cost him more than $500. The price of the large set was at least $100 more than
that of the medium set which in turn cost at least $100 more than the small set. How many
different price choices were possible?
A price choice is a triplet of integers, e.g., (790, 660, 551), the prices in dollars of the
large, medium, and small TV sets.
A) 2001 B) 3367 C) 5407 D) 10005
- 40! is a 48-digit number ending in 9 zeroes. What is the last non-zero (10th from the
right) digit?
A) 2 B) 4 C) 6 D) 8
- What are the last 5 digits of 72001?
A) 00001 B) 00007 C) 04907 D) 14357
- The "binomial coefficient" mCk is defined by
, where m and k are positive integers.
What is the smallest positive integer n such that 2nCn
is divisible by 1000?
A) 13 B) 25 C) 63 D) 100
- ABCD is a fixed quadrilateral (which is not a parallelogram). A point P
is moving in such a way that the difference of the sum of squares of its distances from A
and B and the sum of squares of its distances from C and D remains
constant. What is the path traced by P?
A) a line B) a circle C) a parabola D) a hyperbola
- Two men in motorboats are crossing a lake from opposite shores travelling perpendicular to the
shores which are parallel. They start at the same time and travel at different but constant
speeds. They meet when they are 900 meters from the nearer shore. Each of them continues on
until he reaches the opposite shore and immediately begins the return journey. This time they
meet when they are 500 meters from the nearer shore. What is the width of the lake (in meters)?
A) 2000 B) 2100 C) 2200 D) 2400
- a, b, c, d are four positive real numbers with
a2 + b2 + c2 + d2 = 100.
What is the ratio of the maximum and minimum possible values of the sum
a + b + c + d ?
A) 2 B) 3 C) 4 D) 5
- A said, "B is always truthful and C always lies."
B said, "What A said is not true."
C said, "A is telling the truth."
Who is really telling the truth?
A) A B) B C) C D) none of them
- It can be shown easily that the maximum area of a triangle inscribed inside a square is
half the area of the square. We get a similar result if a triangle is inscribed in a regular
hexagon. What is the maximum area of a triangle inscribed in a regular pentagon of unit area?
A) 1/3 B) Ö5/6 C) 2/5 D) 1/Ö5
- A friend of mine, who likes to watch movies on his DVD player, learned that the local
video store is having a sale on all movie videos; all DVD's for $9 each. He collected some cash
in $1, $5, $10, and $20 bills. The total amount in dollars, he noticed, was the sum of digits of
n6, where n was his ATM PIN. He bought as many DVD's he could with his money and found a little
was left over with which he bought a bag of popcorn. How much did he spend on popcorn?
A) $1 B) $2 C) $3 D) $4
