2010_Solutions_Grades 9-10-2010_Solutions_Levels

2010_Solutions_Levels_9&10
Pdf Summary
The Math Kangaroo 2010 competition for levels 9–10 presents a variety of mathematical problems, ranging from algebra and geometry to number theory and permutations. Here's a brief summary of the solutions for each problem:<br /><br />1. <strong>Problem 1:</strong> Finding \( x \) in the expression \((20102010 \div 2010)\) gives \( x = 10001 \).<br />2. <strong>Problem 2:</strong> Given the deduction about scores, Polly's total possible score is 20.<br />3. <strong>Problem 3:</strong> Calculating the sum of numbers in different rows leads to the total \( 1910 \).<br />4. <strong>Problem 4:</strong> Determining the surface area involves geometric calculations resulting in \( 64 \text{ cm}^2 \).<br />5. <strong>Problem 5:</strong> Equations are solved to get \( n = 15 \).<br />6. <strong>Problem 6:</strong> Using sequences, \( 77 \) correlates to \( 39^2 \), so the answer is \( 39 \).<br />7. <strong>Problem 7:</strong> In geometrical interpretations, an equilateral triangle is expected but deemed non-existent in the setup provided.<br />8. <strong>Problem 8:</strong> Maggie's return trip crossing a river leads to a total of \( 6 \) crossings.<br />9. <strong>Problem 9:</strong> Birth year calculation shows 1980 if teacher's current age is 30.<br />10. <strong>Problem 10:</strong> Arranging variables \( a, b, \text{ and } c \) concludes with \( b < a < c \).<br /><br />In the 4-point problems section:<br />11. <strong>Problem 11:</strong> Using the Pythagorean theorem, segment \( FE \) equals \( 2 \).<br />12. <strong>Problem 12:</strong> Angle calculations result in the unknown angle being \( 40^\circ \).<br />13. <strong>Problem 13:</strong> Summing digits to reach 2010 implies there are 2009 digits.<br />14. <strong>Problem 14:</strong> Path formation in a given set has 2 of those summing to 7 and 9.<br />15. <strong>Problem 15:</strong> Calendar date deduction gives \( 21^\text{st} \) as Sunday.<br />16. <strong>Problem 16:</strong> Using circle segments, deducing perimeter gives \( 6\pi \).<br />17. <strong>Problem 17:</strong> Analysing speed via graphs, Daniel is the fastest.<br /><br />5-point problems offer more complex challenges with permutations and theoretical mathematics, emphasizing advanced concepts like sequence progression and prime manipulation:<br /><br />- There are further calculations calculating specific numeric outcomes or time-based scenarios and overlaying logical constraints to derive true mathematical conclusions. <br /><br />Overall, these challenges teach methodical problem-solving skills, often leveraging foundational mathematical principles along with logical reasoning to deduce correct answers.
Keywords
Math Kangaroo 2010
levels 9-10
algebra
geometry
number theory
permutations
problem-solving
mathematical principles
logical reasoning
sequence progression