Math, Set 11, Rounds 13-17

10) Math - Short Answer For the arithmetic sequence with first term 7 and fourth term 19, what is the sum of the first 50 terms?

ANSWER: 5250

20) Math - Short Answer In a particular year, if April 15 is on a Wednesday, on what day of the week is December 25?

ANSWER: FRIDAY

10) Math - Short Answer What is the greatest integer that is less than or equal to the log base 6 of 975?

ANSWER: 3

21) Math - Short Answer Other than 2, what is the smallest positive integer that, when divided by 3, 5, 7, or 9, has a remainder of 2?

ANSWER: 317

23) Math - Short Answer The point (5, 6) is reflected across the y-axis and the resulting point is then translated 4 units down. What are the coordinates of the final image point?

ANSWER: (-5, 2)

22) Math - Short Answer What is the following cross product of vectors: 5i - 3j + k cross 4j?

ANSWER: -4i + 20k

12) Math - Short Answer Evaluate the following expression: -i^33.

ANSWER: -i

10) Math - Short Answer Consider a bag of 7 marbles, 5 red and 2 white. You draw marbles from the bag until you draw the last white marble, at which point you stop. What is the probability you stop drawing after the second marble is selected?

ANSWER: 1/21

21) Math - Short Answer Using the point names A, B, C, D, and E, how many different names are there for pentagon ABCDE?

ANSWER: 10

10) Math - Short Answer What is the remainder when x6 + 3x4 - 2x3 + 7x - 3 is divided by x + 2?

ANSWER: 111

20) Math - Short Answer If exactly 10 diagonals of a particular convex polygon contain the same vertex, how many total sides does the polygon have?

ANSWER: 13

21) Math - Short Answer An octagon is dilated with scale factor 7/4. What is the ratio of the areas of the original octagon to the image?

ANSWER: 16/49

5) Math - Short Answer What is the largest prime factor of the quantity 42014 + 42015 + 42016 + 42017?

ANSWER: 17

10) Math - Short Answer To the nearest degree, what is the measure of each interior angle of a regular 65-sided polygon?

ANSWER: 174

21) Math - Short Answer What is the greatest common divisor of 234 and 2826?

ANSWER: 18

18) Math - Short Answer What is the volume of a circular cylinder formed by rotating a 5 by 6 rectangular region about one of its shorter sides?

ANSWER: 180π

2) Math - Short Answer Consider an infinite geometric series with sum equal to 8/3 and first term 7/5. What is the value of the common ratio r for this series?

ANSWER: 19/40

5) Math - Short Answer Given f inverse of x equals 2 - x, what is f(x)?

ANSWER: 2 - x

18) Math - Short Answer The sums of three whole numbers taken in pairs are 11, 13, and 14. What is the product of these three numbers?

ANSWER: 240

10) Math - Short Answer In a triple-elimination tournament, each game is played by two teams, there are no ties, and teams are eliminated when they have lost 3 times. What is the maximum number of games played if 83 teams enter?

ANSWER: 248

13) Math - Short Answer What is the sum of the squares of the first 20 positive integers?

ANSWER: 2870

16) Math - Short Answer Triangle ABC has all angles with integral degree measures and the measure of angle A equals 5 times the measure of angle B. How many possible degree measures are there for angle C?

ANSWER: 29

10) Math - Short Answer f of x is a twice differentiable function such that f prime of x and f double prime of x are both less than 0 for all real numbers. If f of 3 equals 12 and f of 5 equals 8, what is the largest possible integer value for f of 7?

ANSWER: 3

5) Math - Short Answer If 6 dice are rolled and the numbers on the top faces are added together, how many different sums are possible?

ANSWER: 31

13) Math - Short Answer What is the limit as x approaches 3 of the fraction with numerator x6 - 81 and denominator x3 - 9?

ANSWER: 36

10) Math - Short Answer In the integers mod 12, what is 5 times 8?

ANSWER: 4

23) Math - Short Answer At what x-value does the maximum value of the function f of x = 2x3 - 3x2 - 12x - 6 occur on the closed interval from -4 to 4?

ANSWER: 4

4) Math - Short Answer What is the limit as x approaches 0 of the fraction with numerator x - 1 + e to the power 3x and denominator sine x?

ANSWER: 4

12) Math - Short Answer A triangular right prism has each edge of length 4. What is its surface area?

ANSWER: 48 + 8√3 (ACCEPT: 8√3 + 48)

5) Math - Short Answer A triangle with integer side lengths has longest side that is 4 times as long as a second side, and the third side has length 20. What is the greatest possible perimeter of the triangle?

ANSWER: 50

13) Math - Short Answer If the definite integral from 1 to 6 of f of x dx = 21, then what is the definite integral from 1 to 6 of open parenthesis f of x minus 3 close parenthesis dx?

ANSWER: 6

18) Math - Short Answer A set of test scores is normally distributed with mean 72 and standard deviation 6. What score corresponds to a Z-score of -1.5?

ANSWER: 63

4) Math - Short Answer Consider a triangle with two sides with lengths of 4 and 9. How many different integers could be the length of the third side?

ANSWER: 7

18) Math - Short Answer A 5-digit palindrome has 3 different digits and is divisible by 18. What is its largest possible value?

ANSWER: 89,298

22) Math - Short Answer What base-10 numeral does the following Roman numeral represent: CMXLVI?

ANSWER: 946

5) Math - Multiple Choice Given that the function f of x has a zero at x = 6, at what x-value must 2 + f of x have a zero? W) 3 X) 4 Y) 8 Z) It need not have a zero

ANSWER: Z) IT NEED NOT HAVE A ZERO

5) Math - Short Answer What is the real quadratic polynomial with least possible positive integer leading coefficient that has 7 - 8i as a zero?

ANSWER: x2 - 14x + 113

16) Math - Short Answer The line with equation y = -2x + 5 is reflected across the y-axis. What is the slope-intercept equation of its image?

ANSWER: y = 2x + 5

2) Math - Short Answer What is the distance from the point (1, 3, 5) to the origin?

ANSWER: √35

13) Math - Short Answer What is the absolute value of the complex conjugate of -4 + 5i?

ANSWER: √41