2004_Solutions_Grades 9-10-2004_Solutions_Levels

2004_Solutions_Levels_9&10
Pdf Summary
The provided document contains solutions and explanations for the Math Kangaroo 2004 competition questions designed for Levels 9 and 10. Here is a concise summary of each problem and solution:<br /><br />1. The expression simplifies to 1 repeated 49 times for a total difference of 48.<br />2. The number of cars of other colors is calculated to be 167.<br />3. Calculating the sides of a pyramid with a hexagonal base results in 12 edges.<br />4. The scale ratio of a plan with a 200 m perimeter swimming pool is 1:200.<br />5. Andrzej originally had 11 coins after setting up and solving equations based on given conditions.<br />6. Using triangle properties, the measure of angle ADC is 65 degrees.<br />7. Given conditions about mushroom picking indicate 11 white mushrooms in a basket.<br />8. For shaded and unshaded areas in a square grid, the unshaded area is calculated as 20022.<br />9. Comparing areas, the ratio of the area of the grey ring to the black circle is 5.<br />10. The greatest expression derived from given variables a and b with different signs is \((a + b)^2\).<br />11. Using proportions, Ania received 198 nuts in an allocation problem.<br />12. Identifying valid multiplication results, only the number 256 fits the conditions.<br />13. Given equilateral and right isosceles triangles, the radius ratio of circles is found to be 2:1.<br />14. Correcting for ring overlap, a chain's total length calculates as 42 rings.<br />15. The ratio of traveled distances of clock hands from time-based movement is 1:24.<br />16. Determining water levels in tanks after overflow, the resulting water depth is 3 cm.<br />17. Using geometric properties and the Pythagorean theorem, a segment length is calculated to be 8.<br />18. There are 4 possible ways to fill a specific numerical grid while meeting conditions.<br />19. Locating a vertex position in a polygon involves computations resulting in vertex 27.<br />20. Numbers expressible as \(2^n \times 3^k\) yielding five appropriate numbers between 100 and 200.<br />21. Analyzing rectangle area properties yields a mere combination resulting in rectangle sides 4, 5, 8, 9.<br />22. Calculating correct answers given a test score results in 5 unanswered questions.<br />23. Using symmetry and proportionality, concludes a segment length of 6.<br />24. The perimeter-defined square within a dodecagon computes an area of 72.<br />25. Calculating divisibility results in 42 numbers fitting specific modulo conditions.<br />26. Analysis of step-based divisor calculations concludes with 16 divisors.<br />27. Solving for digit combinations in an equation provides 35 valid numbers.<br />28. After analyzing a geometric setup in circles, a segment concludes a length of 4.<br />29. The motion of a point on a circle yields a pattern proving linearity.<br />30. Sequence analysis in multiples yields the 2004th term: 7348.<br /><br />These solutions demonstrate geometric, algebraic, and combinatorial reasoning across a variety of mathematical challenges.
Keywords
Math Kangaroo 2004
Levels 9 and 10
mathematical solutions
geometric reasoning
algebraic reasoning
combinatorial reasoning
problem solving
competition questions
math challenges
educational content