2009_Solutions_Grades 9-10-2009_Solutions_Levels

2009_Solutions_Levels_9&10
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The Math Kangaroo 2009 Solutions document for levels 9-10 outlines solutions to 30 challenging math problems presented during the competition. Below is a summary of the problems and their corresponding solutions:<br /><br />1. **Divisibility**: Identify numbers divisible by 3 by summing their digits. Solution: 29 is not divisible.<br />2. **Removing Dots**: Calculation of dots removed to maintain symmetry. Solution: 3.<br />3. **Finishing Position**: Solve for Peter's race position using a formula. Solution: 503.<br />4. **Numerical Calculation**: Step-by-step calculation using given equations. Solution: 60.<br />5. **Counting Appearances**: Adjust a count based on divisibility error. Solution: 18072.<br />6. **Face Value**: Determine values using geometric shapes. Solution: 18.<br />7. **Integer Check**: Compare square and cube numbers. Solution: 3.<br />8. **Angles**: Calculate necessary angle measures. Solution: 80 - 2π.<br />9. **Sequence Prediction**: Determine next number in a sequence. Solution: 24.<br />10. **Angle Sum**: Derive angle from triangle properties. Solution: 124°.<br />11. **Score Analysis**: Find possible scores given constraints. Solution: Betty scored 3 points on exactly 3 tests.<br />12. **Connectivity of Rings**: Remove to retain connectivity. Solution: Specific connectivity based on ring color.<br />13. **Truth-Telling**: Solve deduction-based sequence. Solution: 13 liars.<br />14. **Equation Solving**: Solve for x in a linear equation. Solution: 7.<br />15. **Circle and Square Relationship**: Calculate k from geometric relationships. Solution: \(1/\sqrt{2}\).<br />16. **Number Range**: Find integer range between squares. Solution: 39.<br />17. **Sequence Construction**: Work with number combinations. Solution: 9.<br />18. **Bee Movement**: Return to starting point after steps. Solution: 6 turns.<br />19. **Decimal Calculation**: Number of zeros required. Solution: 3.<br />20. **Numerical Comparison**: Determine greatest number. Solution: c < b < a.<br />21. **Number Possibilities**: Count number possibilities based on conditions. Solution: 64.<br />22. **Sticker Usage**: Calculate unused stickers. Solution: 763 stickers.<br />23. **Piece Arrangement**: Find minimum configuration. Solution: 14 pieces.<br />24. **Word Construction**: Example given. Solution: OABOP BAOBP APO.<br />25. **Prime Factorization**: Make squares from factorization. Solution: 8.<br />26. **Divisibility Check**: Analyze even and odd divisors. Solution: 2.<br />27. **Kangaroo Jumps**: Count possible jump positions. Solution: 121.<br />28. **Geometric Angles**: Calculate angles using properties. Solution: 30°.<br />29. **Sequence of Scores**: Determine possible score sequence. Solution: 8.<br />30. **Special Prime Numbers**: Identify special prime numbers. Solution: 9 special primes.<br /><br />The solutions explore mathematical reasoning, number theory, geometry, logic puzzles, and sequential analysis used for solving complex problems often seen in competitive math environments.
Keywords
Math Kangaroo
2009 Solutions
levels 9-10
divisibility
geometry
number theory
logic puzzles
sequential analysis
competitive math
problem-solving