2001_Solutions_Grades 9-10-2001_Solutions_Levels

2001_Solutions_Levels_9&10
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The Math Kangaroo 2001 Solution Suggestions for Levels 9-10 offers answers and methods for solving a range of mathematical problems typical of the competition for high school students. Here is a summary of the problem solutions provided in the document:<br /><br />1. <strong>Dice Sum</strong>: The sum of dots on three dice can range from 3 to 18, yielding 16 possible values.<br />   <br />2. <strong>Student Arrangement</strong>: Student 'A' is positioned third from an end in both possible seating arrangements.<br /><br />3. <strong>Polygon Diagonal</strong>: Using given conditions, the length of the diagonal in the polygon is calculated to be 10 cm.<br /><br />4. <strong>Cube Completion</strong>: To complete the figure into a 5x5x5 cube, 110 more cubes are needed.<br /><br />5. <strong>Trapezoid Area</strong>: After removing the triangles, the calculated remaining area is 10 cm².<br /><br />6. <strong>Expression Evaluation</strong>: With x = 0, possible outcomes for the given expression are -2, 0, or 2.<br /><br />7. <strong>Prime Numbers</strong>: Only 2, 11, and 101 are prime with digit sums of 2 within the range less than 2001.<br /><br />8. <strong>Fence Length</strong>: Using the Pythagorean Theorem, the missing fence side is determined to be 13 m, totaling a length of 46 m.<br /><br />9. <strong>Divisibility</strong>: If the GCF of m and 35 is 35, then m must be a multiple of 35.<br /><br />10. <strong>Ring Cutting</strong>: Ring C, which holds all the other rings, must be cut to separate them.<br /><br />11. <strong>Natural Number Sets</strong>: For different conditions of d, five possible natural numbers are determined.<br /><br />12. <strong>Angle Calculation</strong>: From given angle conditions in geometry, an angle is calculated to be 40°.<br /><br />13. <strong>Time Calculation</strong>: Calculates watch delay over one week using seconds per hour.<br /><br />14. <strong>Remaining Money</strong>: After purchasing candies, Adam remains with $56.<br /><br />15. <strong>Digit Product</strong>: Based on divisibility by 9 and 25 constraints, a viable 11-digit number is created.<br /><br />16. <strong>Camel's Mass</strong>: Determines the thirsty camel’s mass to be 750 kg.<br /><br />17. <strong>Family Age</strong>: By analyzing factorization of a number, three children’s ages are concluded.<br /><br />18. <strong>Price Reduction</strong>: The reduction of the calculator's price is calculated to be 40%.<br /><br />19. <strong>Candy Distribution</strong>: Peter and Mary’s candies are determined from provided daily consumption.<br /><br />20. <strong>Division Remainder</strong>: Concludes the remainder when 2001 is divided is 9.<br /><br />21. <strong>Divisibility by 4</strong>: The expression results in numbers divisible by 4; identifies an exception.<br /><br />22. <strong>Similar Triangles</strong>: Uses triangle similarity to equate areas as S4 = 9S1.<br /><br />23. <strong>Polynomial Zeros</strong>: Proves impossibility for one zero in given conditions.<br /><br />24. <strong>Equation Solve</strong>: Solves for \(q\) using Pythagorean equation.<br /><br />25. <strong>Power Digit</strong>: Identifies the units digit of a large power expression as 5.<br /><br />26. <strong>Area Calculation</strong>: Determines the area of triangle PQR using side length claims.<br /><br />27. <strong>Line Segments</strong>: Counts 36 line segments in a grid, including horizontals, verticals, and diagonals.<br /><br />28. <strong>Digit Expression</strong>: Finds two possible numbers satisfying the digit equation 14a = 10a + b.<br /><br />29. <strong>Shaded Area</strong>: Computes shaded area in a configuration involving tangent circles.<br /><br />30. <strong>Box Problem</strong>: Utilizes constraint equations to find the total number of boxes as 115.<br /><br />These solutions demonstrate various mathematical skills, including algebra, geometry, number theory, and logical reasoning, geared toward helping participants successfully solve competition-style math problems.
Keywords
Math Kangaroo
mathematical problems
high school students
competition solutions
algebra
geometry
number theory
logical reasoning
problem solving
math competition