- Farmer Jones went to the market to sell his farm-grown melons. He sold as many melons as the price of each
melons in cents, i.e., if he sold 15 melons then he got 15¢ for each. With the money he received from selling
his melons he bought a few bags of fertilizers at $19.95 each. When he reached home, he found he had one of the
following amounts of money left. Which one is it?
A) $7.28 B) $8.72 C) $11.94 D) $12.97
- Let N be the product of all the divisors of 1995, i.e., the product of all positive integers which
divide 1995 exactly. What is the number of digits of N ?
A) 6 B) 19 C) 22 D) 27
- If 865 is the smallest positive integer which leaves 7 as remainder when divided by 11 or 13, and 15 as
remainder when divided by 17, what is the sum of the digits of the smallest positive integer which leaves 7 as
remainder when divided by 11, 9 as remainder when divided by 13, and 5 as remainder when divided by 17?
A) 11 B) 21 C) 23 D) 27
- 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}
Note that if a/b and c/d are two consecutive elements of a Farey sequence, then
bc - ad = 1.
The fractions 12/17 and 29/41 are consecutive in F41 and also in F42.
What is the smallest value of n such that 12/17 and 29/41 are not consecutive in Fn?
A) 47 B) 53 C) 58 D) 61
- Mary, Jane, and Ann are three intelligent women who were asked to participate in an experiment. They were
taken to a room and shown five hats, three of which were white and the other two were black. They were told that
the lights in the room would be turned off and three of the hats would be put on their heads and the other two
removed from the room before the lights are turned on again. Each of them would be able to see the hats on her two
friends but not the hat on her own head. This was done. First Mary was asked whether she could tell the color of
her hat. She said no, she couldn't. Then Jane was asked the same question and gave the same answer. Remember that
Jane heard Mary's reply and Ann heard both Mary's And Jane's replies. What was Ann's answer when she was asked
the same question? Assume that the three women answered correctly.
A) I cannot say B) White C) Black
D) Ann's reply will be A), B), or C) depending on what color hat she
sees her friends wearing.
- A and B are two points on a plane and a point P moves on the plane in such a manner that
the ratio of its distances from A and B remains constant (¹ 1).
What is the curve traced out by the point P?
A) a circle B) an ellipse C) a parabola D) a line
- According to the law of sines, a/sin A = b/sin B = c/sin C ,
where a, b, c are the side-lengths and A, B, C are the opposite angles in a triangle.
Which of the following is not always equal to the common ratio above?
A) 2R, where R is the radius of the circum-circle,
B) abc/2D, where D is the area of the triangle,
C) (2/3)(a2 + b2 + c2)1/2,
D) (abc)2/3/(h1h2h3)1/3, where
h1 , h2 , h3 are the three altitudes.
- Let ABC be an equilateral triangle and let D be a point on its circum-circle. If a, b, c
denote the the distances of D from the vertices A, B, C respectively, what is the value of the product
(b + c - a)(c + a - b)(a + b - c) ?
A) 0
B) abc/8
C) (a3 + b3 + c3 - 3abc)/6
D) none of the above.
- Let C be a circle of radius r and let P be a point outside C at a distance d
from its center. If a line through P intersects C at the points Q and R and if
S is the midpoint of QR, then which of the following does not equal the product
(PQ)·(PR), the product of the distances of P from Q and R ?
A) d2 - r2
B) (PT)2 , where T is a point on C and PT is tangent to C,
C) (PS)2 - (QS)·(RS)
D) (PS)2
- Let C be a circle with center at O and P be a point outside C. A line through
P intersects C at Q and R; S is the midpoint of QR. For different
choices of the line through P, what is the curve on which S lies?
A) a straight line,
B) an arc of a circle with P as center,
C) An arc of a circle with OP as diameter,
D) an arc of a parabola with vertex at O and OP as axis.
