2010_Solutions_Grades 11-12-2010_Solutions_Levels

2010_Solutions_Levels_11&12
Pdf Summary
The 2010 Math Kangaroo competition solutions for Levels 11-12 offer step-by-step answers and reasoning for a variety of mathematical problems. Here is a summary of the key points:<br /><br /><strong>3-Point Problems:</strong><br />1. The equation represents a point (3, 2).<br />2. Valid integer side lengths for triangles adding up to seven, given the triangle inequality, are 1,3,3 and 2,3,2.<br />3. The volume ratio between two containers is 8:1.<br />4. There are 125 possible four-digit numbers composed of odd digits ending in 5.<br />5. To contradict the CEO's claim, at least one employee must be younger than 25.<br />6. By geometry, the angle is 120.<br />7. 2010 is the only number divisible by 3 among the edges of a prism.<br />8. Analysis of digit sums leads to specific values.<br />9. The set of four-digit numbers {2010, 4020, 6030, 8040} are divisible by 30.<br />10. The fourth sequence term is 1.<br /><br /><strong>4-Point Problems:</strong><br />11. The scenarios given lead to 8 possible dice configurations.<br />12. Calculations of circle areas yield \(4\pi(2-1)^2\).<br />13. By system calculations, the sum of 2010 is reached using provided equations.<br />14. The area of a shaded circle region solved through Pythagorean and circle formulas is \(64\pi\).<br />15. A configuration of digits, specifically 257, satisfies the given prime constraint.<br />16. Only one blue and one green marble fit the condition to always draw a desired red marble combination.<br />17. Using geometric properties of a 14-gon, 84 configurations satisfy the condition.<br />18. Solving for triangle sides results in a perimeter of 35.<br />19. By coordinate analysis, the graph forms particular intercepts.<br />20. Calculation changes lead to 3628801, thus not divisible by typical numerals like 2 or 3.<br /><br /><strong>5-Point Problems:</strong><br />21. The complex arrangement yields a final angle measure of 120.<br />22. For minimal incorrect answers, 12 incorrect responses fit the given score.<br />23. By using geometric formulas, the area calculation results in 1/15.<br />24. Use of Lagrange multipliers demonstrates bounds and critical values in certain equations amount to a max of 6.<br />25. Ratio calculations lead to a solution of \((2 + \sqrt{3}):1\).<br />26. Examining the sum adjustments over several steps results in a final number of 46.<br />27. Expressions yield complex number calculations simplifying to the set criteria.<br />28. Square pattern evaluations show 6 as the optimal choice.<br />29. Solving equations determines \(f(6)\) to be 993.<br />30. By determining structure and alignment, calculations resolve geometric configurations. <br /><br />These solutions cover various mathematical techniques, including algebra, geometry, and numerical reasoning, aimed at high-level problem-solving skills.
Keywords
Math Kangaroo
Levels 11-12
step-by-step solutions
mathematical problems
geometry
algebra
numerical reasoning
triangle inequality
volume ratio
digit sums