2004 #20, 21 and 22

Probably my favorite problem.  Looks so fierce and imposing, yet so tame.

Difficulty: easy.

Difficulty: easy.

Difficulty: Pretty complex.

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 20, answer 21, answer 22.

2004 #19 and 23

This year was 2004.  They had to work that in somehow.

Difficulty: easy.

Difficulty: Pretty complex. I was chasing log rules around for a while trying to get all this worked out.

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 19, answer 23.

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.

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.

2004 #16 and 26

Simple algebra -- as soon as you've stated the problem with algebra.

Difficulty: easy.

Difficulty: Functions. Meh.

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 16, answer 26.

2004 #15 and 27

Slope. Parallel and perpendicular lines and a big honking radical. What joy!

Difficulty: easy.

Difficulty: Pretty complex.

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 15, answer 27.

2004 #14 and 28

Thinking is fun.

Difficulty: easy.

Difficulty: Pretty complex.

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 14, answer 28.

2004 #13 and 29

Multiply or divide? You decide..

 Difficulty: Algebra rules.

Difficulty: Well, do we count her different personalities as different people?

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 13, answer 29.

2004 #12 and 30

I find it probable that even sum of your odd students will get this one.

Difficulty: Easy.

Difficulty: This one took a minute to even get started.

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 12, answer 30.

2004 #11 and 31

What percent will get this correct answer? Correct!

Difficulty: Quick.  Only the reckless will get this wrong.

Difficulty: Oddly difficult.

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 11, answer 31.

2004 #10 and 32

Algebra and number theory.  Easy day..

Difficulty: Algebra rules.

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 2004

answer 10, answer 32.

2004: 9 and 33

Round and Round we go!

Difficulty: Can they complete the square?

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 2004

answer 09, answer 33.

2004 #08 and 34

We're mixing it up today.

Difficulty: Algebra rules.

Difficulty: Boring ... but that's solely my personal feelings.

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 08, answer 34.

2004 #07 and 35

Wordiness.

Difficulty: Too simple? Maybe.

Difficulty: Interesting. Don't count ties.

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 07, answer 35.

2004 #06 and 36

How well do yours remember their geometry rules?

Difficulty: Similar triangles.  Simple.

Difficulty: It's the obligatory infinite series, but what's the ratio? That's what will trip them up.

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 06, answer 36.

2004 #05 and 37

Time for circles and triangles.

Difficulty: "I never thought it would be THAT."

Difficulty: You can hear it now ... "Why did they just ask for the sum of the areas of the triangles?"

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 05, answer 37.

2004 #04 and 38

How well do yours remember exponents?

Difficulty: Simple, maybe.

Difficulty: Number theory. Tough little bastard until the AHA! moment.

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 04, answer 38.

2004 #03 and 39

Every year, they manage to work the date into question.  Here's this year's:

Difficulty: Pretty Simple.

Difficulty: Relies on a bunch of work and a couple AHA! moments.  Tough.

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 03, answer 39.

2004 #02 and 40

A nice fraction and a seemingly simply area question. Nice!

Difficulty: Simple.

Difficulty: Easy after the AHA! moments. 

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 02, answer 40.

2004 #01 and 41

Start of a new series. A nice fraction and a holy craptastic function thing. Wow!

Difficulty: Simple.

Difficulty: Relies on a bunch of work and a couple AHA! moments.  Tough.

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 01, answer 41.

Answers to 2004

Problem 01:

Just an extended fraction: $\dfrac{53}{22}$

Problem 02: 3
Problem 03: 2
Problem 04: 1/25
Problem 05: 4pi
Problem 06: 8/3
Problem 07: 30°
Problem 08: 125
Problem 09: 5

Problem 10: 10
Problem 11: 25% decrease
Problem 12: 1/2
Problem 13: 630
Problem 14: 6
Problem 15: 25/4
Problem 16: 6
Problem 17: 35/36
Problem 18: 225
Problem 19: 4008

Problem 20: 5
Problem 21: 6√3-3π
Problem 22: -4/5
Problem 23: 33/4
Problem 24: 0 or 4/3
Problem 25: 46
Problem 26: $\sqrt[3]{2}$
Problem 27: 139
Problem 28: 240
Problem 29: 35

Problem 30: (x,y) = (66, 55)
Problem 31: 3/4
Problem 32: 704
Problem 33: 1003
Problem 34: 1/6
Problem 35: 41
Problem 36: 9/8
Problem 37: 8√3
Problem 38: 42
Problem 39: 76

Problem 40: 1/20
Problem 41:

$(x,y)=\dfrac{3}{2},\dfrac{1}{2}$