2017_Solutions_Grades 9-10-2017_Solutions_Levels

2017_Solutions_Levels_9&10
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The document provides solution suggestions for the Math Kangaroo 2017 competition for Levels 9-10. Here’s a brief summary of the solutions and strategies provided:<br /><br />1. **Question 1** deals with solving for a missing number in a mathematical pyramid using the equation \( u - w = 16 \).<br /><br />2. **Question 2** involves a visual rotation analogy, where the image is manipulated as if viewed through transparent glass, resulting in the answer being the mirrored image rotated by 180 degrees.<br /><br />3. **Question 3** calculates the total gray area of asteroid shapes to be 10 cm².<br /><br />4. **Question 4** determines that each of Maria's siblings receives 3 euros from her after splitting 312 euros evenly.<br /><br />5. **Question 5** describes the path of a wheel moving along alternating straight and circular paths, resulting in a total path that matches option (E).<br /><br />6. **Question 6** calculates there are 13 girls by counting how many are on either side of Bianca and Antonia.<br /><br />7. **Question 7** shows that after an incremental move, a circle changes orientation, resulting in an option (E) appearance.<br /><br />8. **Question 8** highlights that winning 5 more games results in a 70% win ratio.<br /><br />9. **Question 9** uses fractions to conclude that half the guests at a party are women.<br /><br />10. **Question 10** finds that among 7 buttons, you will always have at least three of the same color.<br /><br />11. **Question 11** uses trapezoid area calculations to find that AE equals 35.<br /><br />12. **Question 12** concludes that valid values of A form either a 3-digit or 4-digit number, resulting in 40 options.<br /><br />13. **Question 13** proves through geometric symmetry that certain medians split an equilateral triangle into regions with equal areas, resulting in a hexagon being half the area of the triangle.<br /><br />14. **Question 14** solves an algebraic expression to find the middle of three consecutive numbers equating to 770 as 17.<br /><br />15. **Question 15** involves rotational math to find the circumference of a circle, calculated to be 28 cm.<br /><br />16. **Question 16** schools the interpretation of rotation schedules, arriving at 7 valid schedules providing consecutive jogging days.<br /><br />17. **Question 17** calculates Oscar's height as 160 cm based on the differential series provided in the problem setup.<br /><br />18. **Question 18** determines a minimum of 9 days is required to distribute half-days of sunny and rainy conditions.<br /><br />19. **Questions 19 through 30** involve more geometric and algebraic reasoning solving questions involving symmetry, parity, probabilities, pyramid configurations, age factorization with primes, and angle theorems to derive logical and mathematical conclusions.<br /><br />Each problem is dissected with a step-by-step approach to derive the intended answers, showcasing reasoning, algebra, geometry, and combinatorial logic prevalent in such competitive exams.