2015_Solutions_Grades 11-12-2015_Solutions_Levels

2015_Solutions_Levels_11&12
Pdf Summary
The Math Kangaroo 2015 solutions document for Levels 11 and 12 provides a comprehensive set of suggested answers and detailed solution approaches for each question presented in the competition held on March 19, 2015. These solutions cover a wide array of mathematical problems involving concepts like algebra, geometry, number theory, combinatorics, and logic. Here is a summary of the solutions provided for a selection of problems:<br /><br />1. **Age Problem**: A difference in birth years yields an age difference of not less than 3 years.<br />   <br />2. **Equation**: Simplifying an algebraic expression results in zero.<br />   <br />3. **Contradiction**: Determining an impossible inequality confirms that the statement yields no valid solution.<br />   <br />5. **Graph Theory**: By employing Eulerian paths principles, figure configurations are considered.<br />   <br />6. **Summation**: Calculations involving sequential numerical additions yield a result using formulaic approaches to large-number handling.<br /><br />8. **Geometrical Shape**: A flattened cone's lateral surface results in a circle sector with a part removed, emphasizing spatial visualization skills.<br />   <br />10. **Quadrilateral Acute Angles**: Analyzing different constructions of quadrilaterals helps determine feasible numbers of acute angles.<br />   <br />18. **Series Summation**: Clearly demonstrating n-row addition provides the total for an arithmetic series.<br /><br />22. **Logic Puzzle**: Analyzing statements about truth values deduces which assertion is true under given conditions.<br /><br />23. **Coloring Problem**: Describing planes and faces on a cube illustrates combinatorial counting and symmetry.<br /><br />25. **Digit Work**: Focuses on number representation in terms of digit sums and finding those that support divisibility criteria, specifically by 11.<br /><br />29. **Counting Problem**: In a counting-out game scenario, backward-working calculations determine the speaking order to find the last remaining participant.<br /><br />These problems exemplify problem-solving methodologies including algebraic manipulation, geometric reasoning, logical deduction, and symmetry in combinatorics. The document thoroughly breaks down each problem, primarily to improve understanding and teaching or tutoring in preparatory mathematics for competitions or complex problem solving.
Keywords
Math Kangaroo 2015
Levels 11 and 12
algebra
geometry
number theory
combinatorics
logic
problem-solving
mathematical competition
solution approaches