2008_Solutions_Grades 11-12-2008_Solutions_Levels

2008_Solutions_Levels_11&12

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The Math Kangaroo 2008 Suggested Solutions document provides answers and detailed explanations for problems aimed at levels 11-12. Here's a summary of some of the solutions:<br /><br />1. Problem 1 focuses on inequalities with exponents, resulting in a comparison involving \(x^{2008}\) and \(y^{2008}\) to conclude \(x\) must be greater than or equal to \(y\).<br /><br />2. Problem 2 involves geometry, utilizing angle sums in triangles to deduce specific angle measures, leading to the solution that angle \(\angle BDC\) measures 108 degrees.<br /><br />3. In Problem 3, to ensure a product involves even numbers, the only even prime number must be chosen, resulting in \(p = 2\).<br /><br />4. Problem 4 uses properties of circles and coordinates to apply the Pythagorean theorem, determining a side length in a geometrical configuration as 4.<br /><br />5. Problem 5 calculates the fraction of remaining juice using algebraic simplifications, finding that half of the juice remains.<br /><br />6. Problem 6 focuses on functions with absolute values, identifying a minimum value manipulation without crossing into negative values.<br /><br />7. Problem 7 involves constructing a sum with specific values in a table, where constraints are placed on the maximum—yielding 6 as the largest value.<br /><br />8. The solution for Problem 8 counts combinatorial arrangements in a table setup with specific labeled rows and columns, resulting in 121 cells.<br /><br />9. Problem 9 discusses age intervals based on birthday party patterns, pinpointing that Bart's 7th birthday is 28 years post-birth.<br /><br />10. Problem 10 explores even and odd score outcomes, computing Mark's sum exclusively via even card selection.<br /><br />Additional problems deal with various mathematical concepts: number divisibility, geometric configurations, symmetry, algebraic equations, and logical deductions—culminating in comprehensive reasoning aligning with mathematical principles to offer respective answers and validations.

Keywords

Math Kangaroo

2008 solutions

inequalities

geometry

even numbers

Pythagorean theorem

algebraic simplifications

absolute values

combinatorial arrangements

mathematical concepts