24 = 2^3 * 3, and 70 = 2 * 5 * 7.
We have 24x divisible by 70, so 2, 5, and 7 all need to be factors of 24x. 2 is already a factor of 24, but 5 and 7 are not. So 5 and 7 need to be factors of x.
We have 70x divisible by 24, so 2^3 and 3 need to be factors of 70x. 2 is already a factor of 70, so we need an additional 2^2 and a 3 in x.
So x needs to contain factors of 2^2, 3, 5, and 7. To minimize x, we use just these factors. Thus x = 2^2 * 3 * 5 * 7 = 420.
Wednesday, June 13, 2012
Problem #3
[This problem is worth 5 points.]
ABCD is a square with side length 13. E and F are points exterior to the square, with AE = CF = 5 and BE = DF = 12. Find EF.
ABCD is a square with side length 13. E and F are points exterior to the square, with AE = CF = 5 and BE = DF = 12. Find EF.
Tuesday, June 12, 2012
Problem #2
[This problem is worth 4 points]
Given that x+y = 1, and xy = 1, what is the sum of the cubes of x and y?
Given that x+y = 1, and xy = 1, what is the sum of the cubes of x and y?
Monday, June 11, 2012
Problem #1
[This problem is worth 2 points.]
The three positive integers 24, 70, and x have the property that the product of any two is divisible by the third. What is the minimum possible value of x?
The three positive integers 24, 70, and x have the property that the product of any two is divisible by the third. What is the minimum possible value of x?
Saturday, June 9, 2012
2012 Summer Math Problem Solving Ultramarathon
I'll be running a ten-week, fifty-problem math competition for math team members (and anyone else interested in participating) this summer, starting Monday June 11 and ending Friday August 17. The structure:
1. A new problem will be posted on this blog each weekday during the competition. Answers to questions should be e-mailed to me. I'll only accept one answer per person (to avoid scatter-shot problem solving methods), so be sure you've got your final answer before sending it.
2. You will have 48 hours from the time of posting of the question to send me an answer. After 48 hours, a solution to the problem will be posted, and I won't accept further answers.
3. Each problem will be worth between 1 and 10 points, based on my assessment of its difficulty. Anyone who sends a correct answer will get that number of points added to their running score. The first person to send a correct answer will get a bonus point. [Note: last year I gave double points for the first correct answer. I've changed that this year to a single bonus point, because the bonus points were overwhelming other factors, making the competition into a speed event.]
4. Comments will not be open on the question posts, but they will be on the solution posts. You're encouraged to post alternative solutions in the comments, or to ask questions about the solution if it doesn't make sense to you.
5. At the end of each week, I'll post a list of the standings (using just first initials, to preserve anonymity). The top three finishers from math team (either middle school or high school) will receive some nominal prize when math team starts back in the fall.
6. Calculators are permitted on questions unless I specify otherwise. Many of the questions posted will be drawn from old math competitions, and you could probably find most of them with a bit of googling. I don't have any good mechanism to stop people from doing that, so you're just on the honor system -- your answers should be entirely your own.
The first question will appear here Monday. Good luck to everyone!
1. A new problem will be posted on this blog each weekday during the competition. Answers to questions should be e-mailed to me. I'll only accept one answer per person (to avoid scatter-shot problem solving methods), so be sure you've got your final answer before sending it.
2. You will have 48 hours from the time of posting of the question to send me an answer. After 48 hours, a solution to the problem will be posted, and I won't accept further answers.
3. Each problem will be worth between 1 and 10 points, based on my assessment of its difficulty. Anyone who sends a correct answer will get that number of points added to their running score. The first person to send a correct answer will get a bonus point. [Note: last year I gave double points for the first correct answer. I've changed that this year to a single bonus point, because the bonus points were overwhelming other factors, making the competition into a speed event.]
4. Comments will not be open on the question posts, but they will be on the solution posts. You're encouraged to post alternative solutions in the comments, or to ask questions about the solution if it doesn't make sense to you.
5. At the end of each week, I'll post a list of the standings (using just first initials, to preserve anonymity). The top three finishers from math team (either middle school or high school) will receive some nominal prize when math team starts back in the fall.
6. Calculators are permitted on questions unless I specify otherwise. Many of the questions posted will be drawn from old math competitions, and you could probably find most of them with a bit of googling. I don't have any good mechanism to stop people from doing that, so you're just on the honor system -- your answers should be entirely your own.
The first question will appear here Monday. Good luck to everyone!
Tuesday, August 2, 2011
Summer Problem Solving Marathon Question #47
[Value = 3 points]
A certain polynomial has a remainder of 3 when divided by x-1, and a remainder of 5 when divided by x-3. What is its remainder when divided by x2 - 4x + 3?
A certain polynomial has a remainder of 3 when divided by x-1, and a remainder of 5 when divided by x-3. What is its remainder when divided by x2 - 4x + 3?
Wrapping Up the Summer Problem Solving Marathon
We're going to end the Summer Problem Solving Marathon at the end of this week, which will take us through Problem #50. Math team meetings will resume on Tuesday September 6.
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