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### Pyramidal Math Tossups

Posted: Sun Dec 16, 2007 3:30 pm
Inspired by a previous thread, we put some pyramidal math in our tournament yesterday.

Anybody who was there have any reaction? Anybody not there have anything to say about it?

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
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.
Answer: 3
From the afternoon rounds:
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).
Answer: 2/3
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.
Answer: 720
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.
Answer: 4/13
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.
Answer: -2
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%)

Posted: Sun Dec 16, 2007 9:23 pm
I really like the style of these questions; the idea of putting lots of pyramidally ordered small computation problems has a lot of promise for making pyramidal computational math both easier to write and more accessable to the average player.

I fear, however, that having the early clues be too difficult might actually hurt stronger math players. It seems that the early clues need to to be immediately solvable by players with better knowledge, as if they take too long to solve, it becomes advantageous to ignore that clue and wait for an easier one. As an example of a tossup I really like:
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.
Answer: 720
Each clue is more accessible than the previous one, but most importantly, none of the clues require that much time to solve. Someone who is going to get the answer off a given clue will get it immediately, so the pyramidality works out. Another good one is this:
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.
Answer: 3
Again, each clue becomes easier and easier, and each clue is very quick to solve, so taking a shot at each clue is the best strategy. An example of a tossup where the first clue might be too time-consuming is this one:
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.
Answer: 4/13
That quadratic is solvable, but by the time anybody will get close, the far easier geometric series clue will have been read, and thus it is to your disadvantage to try and do anything with the first clue. Similarly,
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
I found this question had the same problem; working from the first clue would be tremendously difficult; once the x-intercepts are given, it is possible to find the scaling factor on the entire quadratic from previous clues, but I worry that ignoring the first sentence might have been optimal.

Overall, I think this brand of computational math might be the way to go in the future; it seems to be a merger of computation and theory that eliminates many of the problems with traditional pencil-and-paper math tossups.

EDIT: Added answer to one of the quoted questions.

DOUBLE EDIT: I think the phrasing of my post didn't accurately convey my overall thoughts; I really liked these math tossups, and I really like this style of pyramidal math computation. I just had a couple of thoughts on the overall "lots of small problems" format, which I thought these sample questions allowed me to express most easily.

Posted: Sun Dec 16, 2007 10:55 pm
Dan Schafer: The Only Person In Quizbowl Who Can Write Good Math Questions

Posted: Thu Dec 20, 2007 1:03 am
The one thing I liked about these questions is that the answer choice isn't limited to "small numbers" or anything. I found it difficult to write about large numbers like the 720 one or whatever. I haven't really tried it too much, but now that I have seen some examples, I will try it out.

Another good thing I thought was that none of the questions were too wordy; they asked for something, and got on to the next clue pretty quickly. for the 4/13 one, like Dan Schafer said, the quadratic is solvable, but the numbers seem pretty ugly to me.

I really liked the clues for the 2/3 question. they were pretty creative.

The ones I am currently writing are not entirely computational - some reward more advanced knowledge, such as things that would be useful in low-difficulty olympiads. I am pretty sure, though, that they will be fairly accessible to the intended audience.

Posted: Thu Dec 20, 2007 7:33 am
While the quadratic example seems ugly, you are told that there is only one solution, so a really good math player will know that it is a perfect square trinomial. This can help you solve the problem quickly in a few different ways. The one root will be x=-b/2a, or you can divide everything by 2 and quickly recognize how it factors since you know it is a perfect square.

Matt Beeken
Auburn High School
Riner, VA

I also like these math questions better than the math questions that you typically hear in pyramidal sets. What I have a hard time getting my really good math player to do, is to start ignoring the rest of the question once he has enough information to solve the problem. In NAQT it seems like 2/3 of the question is devoted to telling you how to solve the problem which can distract a player from actually solving it.

Posted: Thu Dec 20, 2007 11:35 am
What I have a hard time getting my really good math player to do, is to start ignoring the rest of the question once he has enough information to solve the problem.
Understood... it's like taking the Math Olympiad test at a Lollapalooza concert. :)

Posted: Thu Dec 20, 2007 12:29 pm
AHS Eagles wrote:While the quadratic example seems ugly, you are told that there is only one solution, so a really good math player will know that it is a perfect square trinomial. This can help you solve the problem quickly in a few different ways. The one root will be x=-b/2a, or you can divide everything by 2 and quickly recognize how it factors since you know it is a perfect square.
Ah, I completely missed the word "only" in the line describing the quadratic; knowing that it is a perfect square does help immensely.

Posted: Thu Dec 27, 2007 1:59 am
Having additional clues on the NAQT math questions sometimes helps because it allows enough time to actually do the calculation. Aegis has done this on occasion, since it gives people who know what they are doing time to do it but it also gives people who are fast but not clever a chance at the problem too.
Tossup 3: Math Algebra- Computational- 30 seconds
Calculate 259 squared minus 241 squared. You can do this by squaring each number and then subtracting your two results, though it would be much easier to realize that the final result is equal to a difference of squares, and can be separated into the product of a sum and difference which is much easier to compute. Either way, what is 259 squared minus 241 squared?
The first clue is difficult to solve on its own- you'd have to be pretty clever to see the easy way to do the problem and start working on it right from the start. But if you didn't see that method, you could sit there puzzled until the "easy" way was given. At that point, the cleverer player has a head start of sorts, but a very fast player could still beat the cleverer one to the buzzer.