- If 19941994 is divided by 7, what is the remainder?
A) 1 B) 3 C) 4 D) 6
- Consider the number 61994. It is easy to see that its last digit is 6, since 6 raised to
any power ends in 6. What is the next to last digit (the 10's place digit) in 61994 ?
A) 1 B) 3 C) 5 D) 9
- The n-th Farey sequence Fn consists of all rational numbers x = p/q , such that
0 £ x £ 1 and q £ n
in ascending order.
e.g., F4 = {0, 1/4, 1/3, 1/2, 2/3, 3/4, 1}
and F5 = {0, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1}
What would you get if you subtract the sum of the even numbered elements from the sum of the odd numbered elements
in the 100-th Farey sequence F100 ?
A) 0 B) 0.5 C) 1 D) -1
- Let ABC be a scalene triangle with three acute angles.
What is the location of the point P such that the sum of its distances from the three vertices AP + BP + CP is
minimum?
A) centroid of ABC, i.e., the point of concurrence of the three medians,
B) in-center of ABC, i.e., the center of the circle inscribed in the triangle ABC,
C) circum-center of ABC, i.e., the center of the circle passing through A, B, and C,
D) a point in the triangle such that the three lines AP, BP, CP make 120° angles with each other.
- Let ABC be a scalene triangle with three acute angles.
What is the location of the point P such that the sum of the squares of its distances from the three vertices
AP2 + BP2 + CP2 is minimum?
A) centroid of ABC, i.e., the point of concurrence of the three medians,
B) in-center of ABC, i.e., the center of the circle inscribed in the triangle ABC,
C) circum-center of ABC, i.e., the center of the circle passing through A, B, and C,
D) a point in the triangle such that the three lines AP, BP, CP make 120° angles with each other.
- Let N be the largest number such that the difference between the maximum number of coins and the minimum
number of coins in pennies, nickels, dimes, and quarters to make up the amount N cents is 109. The sum of the digits
of N is
A) 10 B) 11 C) 12 D) 14
- Four players A, B, C, D are playing a game similar to poker. Each of them gets 100 chips in
the beginning. They play 4 rounds. In each round each player bets a few chips and at the end of the round
the winner takes all.
In the first round, A bets 20 chips, B bets 53 chips, C bets 48 chips, and D
bets 51 chips.
In the second round, A bets 23 chips, B bets 56 chips, C bets 25 chips, and D
bets 48 chips.
In the third round, A bets 35 chips, B bets 61 chips, C bets 27 chips, and D
bets 56 chips.
In the fourth round, A bets 21 chips, B bets 43 chips, C bets 83 chips, and D
bets 84 chips.
Who wins the third round?
A) A B) B C) C D) D
- If the sum of squares of the ages of Larry, Peter, and, Martin is 926 and the sum of squares of the
pairwise differences of their ages is 74, what is the sum of their ages?
A) 52 B) 62 C) 78 D) 87
- Given that 32 + 42 = 52 and 52 + 122 =
132 , how many points with integer coordinates lie on the sphere of radius 13 centered at
the origin?
Or, in other words, how many different ordered triplets (x, y, z) of integers satisfy
the equation x2 + y2 + z2 = 132?
A) 52 B) 62 C) 78 D) 87
- Let ABC be a triangle. Pick points D, E and F outside the triangle
such that the triangles DBC, ECA, and FAB are isosceles triangles with
ÐBDC = ÐCEA = Ð
AFB = 120°. Then the triangle DEF is
A) a right triangle,
B) an equilateral triangle,
C) A triangle similar to DABC,
D) has the same area as that of DABC.
