2009_Solutions_Grades 11-12-2009_Solutions_Levels

2009_Solutions_Levels_11&12

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The document presents the solutions to the Math Kangaroo 2009 competition for levels 11-12. Here’s a concise summary of the solutions provided for each problem: <br /><br />1. The solution involves setting up and solving an equation to find the value of \( k \), leading to an increment sequence result.<br /><br />2. The property of the square root function is discussed as increasing yet concave down, selecting a specific option as the answer.<br /><br />3. Examines properties of numbers, concluding that there's only one even prime number.<br /><br />4. Involves divisibility rules, particularly by 3, to arrive at the correct calculation.<br /><br />5. A percentage calculation determines how many fish need to be removed to achieve a specific percentage. <br /><br />6. The strategy for choosing socks ensures a pair without a hole through a series of steps in the worst-case scenario.<br /><br />7. Solves for integers \( n \) and \( m \) by working through an equation to find a relationship in financial terms.<br /><br />8. Discusses circles and geometric shapes, employing the Pythagorean Theorem and comparisons between diagonals.<br /><br />9. Uses logical steps to deduce the minimum number of participants who solved all problems given different non-solving groups.<br /><br />10. Deals with geometry, specifically angles within polygons, concluding that a right triangle cannot exist within these constraints.<br /><br />11. Uses properties of triangles and quadrilaterals to calculate the area.<br /><br />12. Mathematical reasoning is used to find the number of pale kangaroos.<br /><br />13-30: Each further problem involves complexities such as geometry, algebra, combinatorics, and logical deductions, all employing advanced problem-solving skills indicative of the Math Kangaroo aims.<br /><br />This set of solutions reflects high-level mathematical reasoning and a variety of strategies to solve diverse problems typically featured in Math Kangaroo contests. Each solution takes into account the properties and axioms of mathematics tailored to the problem's requirements.

Keywords

Math Kangaroo 2009

competition solutions

levels 11-12

equation solving

square root function

even prime number

divisibility rules

percentage calculation

geometry and angles

problem-solving strategies