Let a, b, and c be the three roots of x3 - 64x - 14. What is the value of a3 + b3 + c3 ?
Consider the points A(-5 ; -1), B(-1;0), C(1;2), and D(1;3). Let P be a point and let d = PA2 + PB2 + PC2 + PD2 so that d is the sum of the squares of the distances from P to each of A, B, C, and D. What is the least possible value for d?
Let N be the smallest positive number which is the cube of one integer and the fifth power of a different integer. How many digits does N have?
Suppose we draw 100 horizontal lines and 100 vertical lines in the plane. How many "pieces" of the plane are formed by cutting along all of these lines? Note: some of the pieces will have infnite area.
(a) 10000 (b) 10001 (c) 10004 (d) 10201 (e) 10204
Given that 1.000000358112312 = 1.000000xyz2247482444265735361 where x, y, and z denote missing digits, what is the value of x + y + z?
Which one of the following numbers is smallest?
(a) 2600 (b) 3500 (c) 4400 (d) 5300 (e) 6200
A used car dealer sold two cars and received $560 for each car. One of these transactions amounted to a 40% profit for the dealer, whereas the other amounted to a 20% loss. What is the dealer’s net profit on the two transactions?
The area of the square ABCD is equal to 1. Determine the area of the shaded region if the indicated points on the sides of ABCD divide those sides in a 2:1 ratio as shown.
A B C D
What is the remainder when x200 − 2x199 + x50 − 2x49 + x2 + x + 1 is divided by (x−1)(x−2)?
There is a unique positive integer that has less than 11 decimal digits, ends in a 6, and if this 6 is removed and put at the front of the number (e.g., 136 → 613), then the resulting number is exactly four times the original number. How many digits does this number have?
Ten squares of equal size are arranged in the grid below. What is the value of β − α?
How many 9’s are there in the decimal expansion of 999998999992?
Juniors and seniors of a school were polled on the question “Do you believe there is life on Mars?” An equal number of juniors and seniors responded, and every respondent answered either “Yes” or “No.” If 60% of those who said “Yes” were seniors, and 80% of those who said “No” were juniors, then what percentage of the juniors polled said “Yes”?
Let P(x) be a polynomial of degree four such that:
P(2) = P(−2) = P(−3) =−1
and P (1) = P(−1) = 1.
What is P(0)?
There are 6 gallons of pure alcohol in container A and 6 gallons of pure water in container B. An empty bottle is filled with alcohol from A and then emptied into B. After stirring, the bottle is filled with this mixture from B and emptied into A. The ratio of alcohol to water in container A is now 4:1. Assuming there were no spills, what is the size of the bottle in gallons?
The coordinates for A and D are (7,4) and (−5,−3) respectively. What is the shortest possible length of a path ABCD where B is a point on the line y = 2, C is a point on the line y = 0, and the line segment BC is perpendicular to the line y = 0?
A(7;4) D(-5;-3) y = 2 y = 0
Recall that the iterated power \(a^{b^c}\) denotes \(a^{\left(b^c\right)}\). Given that x is a real number which satisfies the equation \(2^{2^x}+4^{2^x}=42\)
what is the value of \(\sqrt{2^{2^x}}\)?
(a) 2 (b) 4 (c) 8 (d) 16 (e) 32
John and Nancy live on the same street and often walk towards each other's home. If they both leave their homes at 8:00 a.m., then they will meet at 8:04 a.m. If Nancy leaves her home at 8:00 a.m. but John does not leave his home until 8:03 a.m., then they will meet at 8:05 a.m. How many minutes does it take for John to walk all the way to Nancy's home? Assume that each person walks at his or her own constant rate.
When written in base 8, the number n is 34112d4357 where d denotes a digit in base 8. If n is divisible by 7, then the value of d is (a) 2 (b) 3 (c) 4 (d) 5 (e) 6
Consider the positive integers n having the property that 2 divides n, 3 divides n + 1, 4 divides n + 2, ..., 10 divides n + 8. The first positive integer with this property is 2. Let N be the 4th positive integer with this property. What is the sum of the digits of N? (a) 12 (b) 14 (c) 16 (d) 18 (e) 20
