Some algebra for today.
Difficulty:Medium if the AHA ! moment is found. Hard if you don't see the path.
Difficulty: Medium, but the kids will find it Hard.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 2013
answer 37, answer 38. Published 5/5.
Showing posts with label Algebra. Show all posts
Showing posts with label Algebra. Show all posts
Saturday, January 19, 2013
Sunday, January 13, 2013
2013 ... 25 & 26
Today, we have a nice addition to the questions about synthetic division and complex numbers.
Difficulty: Easy, but the "three solutions must be positive" is a nice twist on this.
Difficulty: Medium, because of the time pressure ... determining the patterns is straightforward.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 2013
answer 25, answer 26. Published 4/29.
Difficulty: Easy, but the "three solutions must be positive" is a nice twist on this.
Difficulty: Medium, because of the time pressure ... determining the patterns is straightforward.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 2013
answer 25, answer 26. Published 4/29.
Saturday, January 12, 2013
2013 ... 23 & 24
Two questions today include a very cool number theory/algebra reasoning question and a graph of a type to make them think.
Difficulty: Medium. My absolute favorite question on the whole test. So cool how this works out.
Difficulty: Easy-Medium. Graphing piecewise functions with absolute value mixed in is not something that most students have had experience with. Easy enough to figure out from principles, but time pressure makes this harder than it really is.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 2013
answer 23, answer 24. Published 4/29.
Difficulty: Medium. My absolute favorite question on the whole test. So cool how this works out.
Difficulty: Easy-Medium. Graphing piecewise functions with absolute value mixed in is not something that most students have had experience with. Easy enough to figure out from principles, but time pressure makes this harder than it really is.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 2013
answer 23, answer 24. Published 4/29.
Sunday, January 6, 2013
2013 ... 11 & 12
Classic trigonometry problem that demonstrates the spirit of the UVM Math Contest. Do you remember your angle formulas?
Difficulty: medium, if only because of the time pressure.
Difficulty: Pretty easy, once you see where you're going.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 2013
answer 11, answer 12.
Difficulty: medium, if only because of the time pressure.
Difficulty: Pretty easy, once you see where you're going.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 2013
answer 11, answer 12.
Saturday, January 5, 2013
2013 ... 9 & 10
We've got some real, rootin'-tootin' algebra to contend with today!.
In Ms. Direction's class? Really?
Difficulty: easy.
Here it is!
Difficulty: easy-medium, depending on whether they have that flexibility that allows them to follow a dead-end idea to a point and recognize it's a dead end, and then re-focus on a better tack.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 2013
answer 09, answer 10.
In Ms. Direction's class? Really?
Difficulty: easy.
Here it is!
Difficulty: easy-medium, depending on whether they have that flexibility that allows them to follow a dead-end idea to a point and recognize it's a dead end, and then re-focus on a better tack.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 2013
answer 09, answer 10.
Monday, January 22, 2007
2007:. 41
Since there are so many truly able mathematicians among my readers, I've decided to give them something to keep busy with! This set will start at the hardest and work backwards.
Difficulty: hard.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Answer is here.
Difficulty: hard.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Answer is here.
Saturday, January 20, 2007
2007: 20, 21, and 22.
Problem 20
Difficulty: Easy, but fun. A new way to look at averages and the effect of an addition on the average, depending on n.
Problem 21
Difficulty: Linear programming.Any bets this will somehow work out to an integer?
Problem 22
Difficulty: Easy. Once you've got the details right.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 20, answer 21 and answer 21.
Difficulty: Easy, but fun. A new way to look at averages and the effect of an addition on the average, depending on n.
Problem 21
Difficulty: Linear programming.Any bets this will somehow work out to an integer?
Problem 22
Difficulty: Easy. Once you've got the details right.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 20, answer 21 and answer 21.
Thursday, January 18, 2007
2007: 18 and 24
Problem 2*3²
Difficulty: easy-ish. A little trigonometric - algebraic manipulation and "Surprise!"
Problem 2³*3
Difficulty: Easy. At least once in every test, in some form or another, the creators work the year into a problem. This test was for 2007.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 18 and answer 24.
Difficulty: easy-ish. A little trigonometric - algebraic manipulation and "Surprise!"
Problem 2³*3
Difficulty: Easy. At least once in every test, in some form or another, the creators work the year into a problem. This test was for 2007.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 18 and answer 24.
Wednesday, January 17, 2007
2007: 17 and 25.
Day Seventeen
Difficulty: easy-schmeasy, if you subtract. More complicated if you try to add.
Problem 25 = 5².
(I wonder how often you can switch the digits and still write an equivalent expression.)
Difficulty: Medium. More damned logarithms, but there's a neat little prime factorization going on, too.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 17 and answer 25.
Difficulty: easy-schmeasy, if you subtract. More complicated if you try to add.
Problem 25 = 5².
(I wonder how often you can switch the digits and still write an equivalent expression.)
Difficulty: Medium. More damned logarithms, but there's a neat little prime factorization going on, too.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 17 and answer 25.
Tuesday, January 16, 2007
2007: 16 and 26
Sweet Sixteen
Difficulty: easy-ish. Another SAT-type reasoning question. Fun to watch the students work out the ramifications.
Problem 26
Difficulty: Medium. Is it wrong of me to get really annoyed at this question? It's not terribly difficult, but it is so ... artificial.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 16 and answer 26.
Difficulty: easy-ish. Another SAT-type reasoning question. Fun to watch the students work out the ramifications.
Problem 26
Difficulty: Medium. Is it wrong of me to get really annoyed at this question? It's not terribly difficult, but it is so ... artificial.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 16 and answer 26.
Saturday, January 13, 2007
2007: 13 and 29
Day Thirteen
Difficulty: easy-ish. Mostly, it's an interpretation question.
I love this next problem. So very cool.
Problem 29
i.e., find the shaded area.
Difficulty: Medium.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 13 and answer 29.
Difficulty: easy-ish. Mostly, it's an interpretation question.
I love this next problem. So very cool.
Problem 29
i.e., find the shaded area.
Difficulty: Medium.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 13 and answer 29.
Thursday, January 11, 2007
2007: 11 and 31
First, an easy one.
And then something more complicated, though personally, I'm not a fan of probability questions.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 11 and answer 31.
And then something more complicated, though personally, I'm not a fan of probability questions.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 11 and answer 31.
Wednesday, January 10, 2007
2007. 10 and 32
Day Ten
Difficulty: easy-ish.
Problem 32:
Difficulty: Tricky. It seems easy, but then tries to kneecap you.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 10 and answer 32.
Difficulty: easy-ish.
Problem 32:
Difficulty: Tricky. It seems easy, but then tries to kneecap you.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 10 and answer 32.
Tuesday, January 9, 2007
2007: 9 and 33
Day Nine.
Difficulty: easy-ish.
Number 33:
Difficulty: Hard.
(or as the kids said, OMFG)
It's actually more complicated than hard.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 9 and answer 33.
Difficulty: easy-ish.
Number 33:
Difficulty: Hard.
(or as the kids said, OMFG)
It's actually more complicated than hard.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 9 and answer 33.
Sunday, January 7, 2007
2007. 7 and 35
Problem 7
Difficulty: so easy the kids kept second-guessing themselves. ;-)
Problem 35
Difficulty: Hard (last page)
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 7 and answer 35.
Difficulty: so easy the kids kept second-guessing themselves. ;-)
Problem 35
Difficulty: Hard (last page)
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 7 and answer 35.
Saturday, January 6, 2007
2007: 6 and 36
Day Six will make you think a bit.
Difficulty: easy - ish.
Difficulty: Hard (last page)
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 6 and answer 36.
Difficulty: easy - ish.
Difficulty: Hard (last page)
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
answer 6 and answer 36.
Thursday, January 4, 2007
2007. 4 and 39
Problem (4)
Difficulty: first page.
Problem 39
Note: Remainder.
Difficulty: Hard. (last page)
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Yeah, I know, it should read "larger" and "smaller" but I didn't have the energy to fix it.
answer 4 and answer 39.
Difficulty: first page.
Problem 39
Note: Remainder.
Difficulty: Hard. (last page)
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Yeah, I know, it should read "larger" and "smaller" but I didn't have the energy to fix it.
answer 4 and answer 39.
Tuesday, January 2, 2007
2007. 2
Day two.
Difficulty: easy.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Answer is here.
Difficulty: easy.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Answer is here.
Sunday, January 18, 2004
2004 #18 and 24
Surprisingly easy for this late in the test.
Difficulty: easy.
Difficulty: Changing everything to a common base goes a long way here.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 2004
answer 18, answer 24.
Difficulty: easy.
Difficulty: Changing everything to a common base goes a long way here.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 2004
answer 18, answer 24.
Saturday, January 17, 2004
2004 #17 and 25
Fraction fun. Maybe simple enough for the SAT but maybe not.
Difficulty: easy.
Difficulty: Another one of the problems that needs you to manipulate one expression until it looks like another. Try cubing.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 2004
answer 17, answer 25.
Difficulty: easy.
Difficulty: Another one of the problems that needs you to manipulate one expression until it looks like another. Try cubing.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest 2004
answer 17, answer 25.
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