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From the morning rounds:
Find an expression for the quadratic function with the following properties using the format ax2+bx+c. It has a y-intercept at 16 and goes through the points (1,30) and (-1,6). Its vertex is at (-3,-2), and its x-intercepts are at -4 and -2. Written in vertex form, its equation is y=2(x+3)2-2. Identify this quadratic whose derivative is 4x+12 and whose antiderivative is (2/3)x3+6x2+16x+C.
Answer: y= OR f(x)= 2x2+12x+16
From the afternoon rounds:This is the smallest counting number which is the radius of a sphere whose volume is an integer multiple of pi. It is also the number of distinct real solutions to the equation x7-19x5=0. This number also gives the ratio between the volumes of a cylinder and a cone with the same heights and radii. Give this number equal to the log base four of sixty-four.
Make sure your answer is completely simplified. This number is equal to the eccentricity for the conic section given by the equation 9x2+5y2=45. It also gives the eccentricity for the conic section with the equation r equals ten divided by the quantity three plus two cosine theta. It also equals the period of the function y equals the cosine of the quantity three pi x. Find this number equal to the slope of the line connecting the points (-3,2) and (12,12).
Give your answer in simple radical form. A cube with a great diagonal of this length will have a surface area of eight-thirds. This value is the positive solution to the equation three x squared minus four equals zero. Give this value equal to the secant of the quantity pi over six radians, which also equals the cosecant of sixty degrees.
Answer: Two Root Three Over Three
This number is the positive geometric mean between 360 and 1440, and it equals the dot product of the vectors <10,40> and <52,5>. It also equals the sum of the degree measures of the interior angles of a hexagon. It also equals six factorial. Identify this number which equals the number of degrees equivalent to four pi radians.
Your answer must be a simplified fraction. This number is the only solution to the equation 338x^2-208x+32=0. In addition, an infinite geometric series that starts with the number one and has a ratio equal to this number will converge to the sum 13/9. This number also equals the probability that a random card pulled from a standard deck is either an ace or a spade. Find this number equal to 27/130 + 1/10.
This number is the only negative solution to the equation x3-x2-4x+4=0. It is also the only negative solution to the equation x2-48x-100=0, and it is the x-coordinate of the hole in the graph y=(x2+8x+12) over (x2+10x+16). Find this number that is the only solution to the equation 36x+35=34x+31.
If you flip a coin six times, this is the probability of getting either exactly zero or exactly two heads. If x is a randomly chosen positive whole number, this number is also the probability that two raised to the x power ends with the digit two. It's also the probability of getting a sum of five or six when you roll two standard dice. If you flip two coins, it's the probability that both of them come up heads. Find this number that equals the probability that a randomly selected card from a standard deck is a club.
Answer: 1/4 (accept .25 or 25%)