Math Contest Problems

Kevin Hopkins, Jul 26, 2014

What follows is my attempt at creating worksheets that produce problems like math contest problems. This is not easy to do as most math contests pride themselves on offering original problems to challenge students, so it is difficult to find "families" of problems that can be algorithmically generated. In most problems you see Answer followed by a ? . Replace the ? with the answer you come up with. On hitting enter, the page should grade your submitted answer. Most ideas for these pages originated with MATHCOUNTS minis videos at [url]http://www.artofproblemsolving.com/Videos/index.php?type=minis[/url], MAA Curriculum Inspiration video based off AMC competitions at [url]http://www.maa.org/math-competitions/teachers/curriculum-inspirations[/url], Art of Problem Solving videos at [url]http://www.artofproblemsolving.com/Videos/index.php[/url], Art of Problem Solving Books at [url]http://www.artofproblemsolving.com/Store/curriculum.php[/url] or Dr. Hopkins' work with hosting competitions written by [url]http://mathleague.org[/url]. Art of Problem Solving also has a free resource known as Alcumus, [url]http://www.artofproblemsolving.com/liz/Alcumus/index.php[/url], which teaches some of this material and presents problems about it. Many math contests exist. You can find many of them at [url]https://sites.google.com/site/sbumathcon/home/other-math-contests[/url]. Dr. Hopkins has been interested in contests for years, even running a camp called Math Contest Camp. 2018 will be the 13th year of this camp. You can find information about 2018 at [url]https://sites.google.com/site/sbumathcon/math-contest-camp-2018[/url]. Check the camp website for more information. If you have other suggestions for families of problems to include, please email those to Dr. Hopkins, khopkins@sbuniv.edu . If there is a source of the problem, or a source of an explanation for the problem, please send me that as well so I can include it with the worksheet. Also, please report any errors. If you think you have entered a correct answer and the worksheet says it is incorrect, please take a screenshot of this and email it to me. There may be a special case in the worksheet that I failed to account for and so the worksheet doesn't operate in that special case. It is also possible that I just did the problem wrong and only thought I was correct. This was certainly unintentional. In math, no one chooses to do a problem wrong. We may do it wrong because we misread the problem, made a careless error, are trying to mislead someone (which I am not) or some other reason. In life that is not always the case. We unfortunately make poor decisions to do something wrong knowing full well we are doing something wrong (and ignoring the possible consequences of that action). God knew this was possible and I believe He has provided a way for us to deal with that problem. For my explanation of that solution, you can view a Facebook video of a presentation of that: [url]https://www.facebook.com/photo.php?v=569730479736877&set=vb.417886931587900&type=3&theater[/url] (you do have to be logged into Facebook to view this video).

Table of Contents

  1. Elementary Type Contests
    1. Estimating Sums
    2. Estimating Products
    3. Estimating Products Part 2
    4. Estimating Quotients
    5. Estimating Square Roots
    6. Estimating Squares of Numbers
    7. Estimation Golf with Timer
    8. Solve Simple Equations
    9. Finding middle Number
    10. Counting Elements in a List
    11. Counting Multiples
    12. Probability of Selecting a Multiple from a List
    13. How Many Outcomes?
    14. Counting with Venn Diagrams
    15. Spinner Probability
    16. More Spinner Probability
    17. Signed Subtraction
    18. Extended Subtraction
    19. Visual Patterns
  2. Middle School Type Contests
    1. Unit Digit of Powers
    2. Using Ratios
    3. Pythagorean Theorem
    4. Curve Stitching
    5. Converting to Another Base
    6. Other Number Bases
    7. Recognizing squares
    8. Function Rule-Section 1.3
    9. Function Table-Section 1.3
    10. Function Graphs-Section 1.3
    11. New Operations
    12. Probability
    13. Probability and Multiplication
    14. Probability with Lengths
    15. Summing a List
    16. How Many Divisors
    17. How many EVEN (or ODD) Divisors does a number have?
    18. Sum of all divisors of a number
    19. Greatest Common Divisor
    20. Least Common Multiple
    21. Simplify Factorials
    22. Find the Largest Prime Factor of expression with factorials
    23. Handshake Problem
    24. From 2015 National Countdown Round
    25. Counting Paths
  3. High School Type Contests
    1. Checkerboard Relay Problem
    2. Hexagonal Numbers as Relay Question
    3. Recursive Function as Relay Question
    4. Counting Handshakes as a Relay Question
    5. Sum of List Relay
    6. Base Relay
    7. Angle Sum Relay
    8. Triangular Numbers Relay
    9. Counting Committees as Relay Question
    10. Counting More Complicated Committees
    11. Harmonic Mean Relay
    12. Factors Relay
    13. More on Counting Committees
    14. Counting Marbles
    15. Probability with Area
    16. Area Between Concentric Circles
  4. Useful Geometry Theorems
    1. Pick's Theorem
    2. Intersecting Chords Theorem
    3. Heron's Formula for Area, then used to find inradius
    4. Angle Bisector Theorem
    5. Vivani's Theorem
    6. External Tangents Theorem
    7. Pythagorean Theorem in 3D
  5. Useful Theorems, not Geometry
    1. Chicken Nugget Theorem

1.

Elementary Type Contests

  • 1. Estimating Sums
  • 2. Estimating Products
  • 3. Estimating Products Part 2
  • 4. Estimating Quotients
  • 5. Estimating Square Roots
  • 6. Estimating Squares of Numbers
  • 7. Estimation Golf with Timer
  • 8. Solve Simple Equations
  • 9. Finding middle Number
  • 10. Counting Elements in a List
  • 11. Counting Multiples
  • 12. Probability of Selecting a Multiple from a List
  • 13. How Many Outcomes?
  • 14. Counting with Venn Diagrams
  • 15. Spinner Probability
  • 16. More Spinner Probability
  • 17. Signed Subtraction
  • 18. Extended Subtraction
  • 19. Visual Patterns

1.1.

Estimating Sums

Estimation problem like what might be seen on [url]mathleague.org[/url] Number Sense Exam (among their Elementary Contest tests). Need to get the answer to within 5% of the exact answer.[br][br]The goal is to quickly come to an estimate within 5%, not to spend the extra time to get the exact answer.
Estimating Sums
– Kevin Hopkins

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2.

Middle School Type Contests

  • 1. Unit Digit of Powers
  • 2. Using Ratios
  • 3. Pythagorean Theorem
  • 4. Curve Stitching
  • 5. Converting to Another Base
  • 6. Other Number Bases
  • 7. Recognizing squares
  • 8. Function Rule-Section 1.3
  • 9. Function Table-Section 1.3
  • 10. Function Graphs-Section 1.3
  • 11. New Operations
  • 12. Probability
  • 13. Probability and Multiplication
  • 14. Probability with Lengths
  • 15. Summing a List
  • 16. How Many Divisors
  • 17. How many EVEN (or ODD) Divisors does a number have?
  • 18. Sum of all divisors of a number
  • 19. Greatest Common Divisor
  • 20. Least Common Multiple
  • 21. Simplify Factorials
  • 22. Find the Largest Prime Factor of expression with factorials
  • 23. Handshake Problem
  • 24. From 2015 National Countdown Round
  • 25. Counting Paths

2.1.

Unit Digit of Powers

Units Digit of Powers
Find the units digits of a large number raised to a large power. Somewhat similar to [url]https://www.youtube.com/watch?v=gjfJSNAoqjU&feature=youtu.be[/url]
– Kevin Hopkins

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3.

High School Type Contests

  • 1. Checkerboard Relay Problem
  • 2. Hexagonal Numbers as Relay Question
  • 3. Recursive Function as Relay Question
  • 4. Counting Handshakes as a Relay Question
  • 5. Sum of List Relay
  • 6. Base Relay
  • 7. Angle Sum Relay
  • 8. Triangular Numbers Relay
  • 9. Counting Committees as Relay Question
  • 10. Counting More Complicated Committees
  • 11. Harmonic Mean Relay
  • 12. Factors Relay
  • 13. More on Counting Committees
  • 14. Counting Marbles
  • 15. Probability with Area
  • 16. Area Between Concentric Circles

3.1.

Checkerboard Relay Problem

Relay questions can come in many forms. They typically start with let T = TNYWR (The Number You Will Receive). The idea is you can do a lot of work on solving the problem before you know what T is. Then when you receive T from a teammate, you can finish up your problem and get your answer.[br][br]Relays appear on [url]mathleague.org[/url] High School competitions and on ARML, [url]www.arml.com[/url], competitions.[br][br]This particular question is explained at [url]http://puzzles.nigelcoldwell.co.uk/twentyseven.htm[/url], along with lots of other places. A video explanation is at [url]https://www.youtube.com/watch?v=HUkN-AuYs2I[/url][br][br]An extension of the problem is explained on a video at [url]http://mathcounts.org/resources/video-library/mathcounts-minis/mini-21-counting-rectangles-gridsquares-lattice-grid[/url]
Checkerboard Relay Problem
– Kevin Hopkins

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4.

Useful Geometry Theorems

Some contest problems rely heavily on an application of a specific theorem. Once you realize the theorem you need and apply it successfully the problem is more straightforward.

  • 1. Pick's Theorem
  • 2. Intersecting Chords Theorem
  • 3. Heron's Formula for Area, then used to find inradius
  • 4. Angle Bisector Theorem
  • 5. Vivani's Theorem
  • 6. External Tangents Theorem
  • 7. Pythagorean Theorem in 3D

4.1.

Pick's Theorem

[url=https://www.math.hmc.edu/funfacts/ffiles/10002.2.shtml]https://www.math.hmc.edu/funfacts/ffiles/10002.2.shtml[/url] gives an explanation of Pick's Theorem. When all vertices of a polygon have integer coordinates, the area of the polygon is given by the formula[br][br][math]Area\left(P\right)=i+\frac{b}{2}-1[/math][br]where [math]i[/math] is the number of interior lattice points (points with integer coefficients) and [math]b[/math] is the number of boundary lattice points.[br][br]Compute the area of the given polygon. You can check the box to see the correct answer for the area enclosed. It is possible that a non-polygon might be produced here. If so, produce a new problem and try that one.
– Kevin Hopkins

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5.

Useful Theorems, not Geometry

Some contest problems rely heavily on an application of a specific theorem. Once you realize the theorem you need and apply it successfully the problem is more straightforward.

  • 1. Chicken Nugget Theorem

5.1.

Chicken Nugget Theorem

The Chicken Nugget Theorem has some other names (Postage Stamp Problem or Frobenius Coin Problem, see [url]https://artofproblemsolving.com/wiki/index.php/Chicken_McNugget_Theorem[/url] for a detailed explanation). [url]https://www.youtube.com/watch?v=vNTSugyS038[/url] has a video explanation where
– Kevin Hopkins

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