2004_Solutions_Grades 11-12-2004_Solutions_Levels

2004_Solutions_Levels_11&12
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The Math Kangaroo 2004 solution suggestions for levels 11 and 12 offer a detailed walkthrough of complex mathematical problems, with solutions provided for each of the 30 questions. Here's a summary of the solutions:<br /><br />1. For calculating the total edges of a polyhedron formed by a 16-gon base and its lateral faces: 32 edges.<br />2. Solving the inequality \(x^2 \leq 2004\): the solution is between \(-\sqrt{2004}\) and \(\sqrt{2004}\).<br />3. Determining that the sum of angles provided indicates a square.<br />4. Calculation of unshaded squares: \(s^2 - 2s + 1\).<br />5. In selecting cards, ensure the product of 52 cards includes even numbers.<br />6. Finding the average cost in a scenario involving costs and quantities of pens.<br />7. Area of a square formed by arranging smaller squares: 81 square units.<br />8. Determining the percentage of Martians with more antennas than the average.<br />9. Identifying that a triangle with a 30-degree angle due to the properties of circles.<br />10. Solving for the total number of shepherds and sheep based on legs counted.<br />11. Calculating numbers with specific properties in dealing with last digits.<br />12. Counting right triangles formed by a 14-gon inscribed in a circle: 84 triangles.<br />13. Identifying symmetry in vertex placement of a square: 5 possible configurations.<br />14. Graphing parts of a circle based on inequalities.<br />15. Determining positive numbers possible from a set condition: 121 numbers.<br />16. Area calculation in geometry involving triangles and squares.<br />17. Proving expressions as the fourth power of integers.<br />18. Finding the radius of a circle based on geometric properties.<br />19. Geometric problem involving a tangent circle and calculations for its radius.<br />20. Proving a pattern with powers using mathematical induction.<br />21. Manipulating sequences and solving based on given recursive step properties.<br />22. Calculating number sums and possibilities based on number rearrangement.<br />23. Mapping transformations of functions involving translations and reflections.<br />24. Selecting numbers with specific subset equality conditions in arrangements.<br />25. Simplifying trigonometric expressions and solving for specific values.<br />26. Calculating the number of triangles formable by selecting points from a set.<br />27. Estimating voting shares based on given fractional relationships.<br />28. Summing digits of a large number calculated through powers and differences.<br />29. Determining the number of divisors of 120, leading to calculations of steps.<br />30. Solving for areas in a cyclic quadrilateral using rectilinear properties. <br /><br />These solutions address different concepts from geometry, algebra, number theory, and trigonometry, showcasing how these areas of math intersect in problem-solving.
Keywords
Math Kangaroo 2004
polyhedron edges
inequality solution
angles square
unshaded squares
card selection
average cost
circle properties
right triangles
trigonometric expressions