2007_Solutions_Grades 11-12-2007_Solutions_Levels

2007_Solutions_Levels_11&12
Pdf Summary
The suggested solutions for the Math Kangaroo 2007 Levels 11-12 focus on problems that involve angles, marbles, geometry, trigonometry, arithmetic sequences, probability, and more. Here's a brief summary of the problems and their solutions:<br /><br />1. The ball reflects off at an angle equal to the incident angle and continues at 45-degree angles.<br />2. The solution involves logarithmic calculations, simplifying grouped numbers.<br />3. By writing equations for the marbles Adam, Barry, and Carl have, their amounts are determined.<br />4. Equal-area triangles are identified using base and height.<br />5. A trigonometry identity simplifies the expression.<br />6. A percentage problem resolves the number of unanswered questions.<br />7. Ratio problems involving semicircle circumferences are solved.<br />8. A sequence problem calculates a maximal integer based on conditions.<br />9. A choosing problem with letters.<br />10. Using Pythagorean triples to solve a geometric arrangement.<br />11. Total squares are found by adding edge-sharing and diagonal-sharing squares.<br />12. Angle relationships give a clear comparison between C and A.<br />13. Simplifying an algebraic equation to find integer solutions.<br />14. Trigonometric identities cancel out terms in a sequence, resulting in a leftover value.<br />15. Use of geometry and Pythagorean theorem to relate semicircle radii.<br />16. Inequalities in triangles determine feasible integer side values.<br />17. Divisibility factors determine milestones in a birthday-related problem.<br />18. Logic problem involving statements about knights and liars.<br />19. Integer values of a sum of squares equation are calculated.<br />20. Factorizing equations finds specific constant values.<br />21. Relationships in diagonals and angles of a rhombus are analyzed.<br />22. Rational functions are examined for feasible x values.<br />23. Finding integer x values that satisfy a square equation.<br />24. Solving for a variable in an equation by substitution.<br />25. Factorization problem with specific given factors.<br />26. Resolves a sum of a series using algebraic manipulations.<br />27. A computation of total ways to assign gifts per specific rules.<br />28. Arithmetic sequences explain the layout and numbering of blocks.<br />29. Binary sequences correspond to positions in a powers-of-3 series.<br />30. A probability scenario is expressed through calculating a series sum.<br /><br />Overall, the problems touch on a range of mathematical concepts that require a mix of logic, arithmetic, geometry, and algebra to solve.
Keywords
Math Kangaroo 2007
angles
marbles
geometry
trigonometry
arithmetic sequences
probability
logarithmic calculations
Pythagorean theorem
rational functions