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Algebra Difficulty 7.5 National olympiad, round 2 Find the answer

Evaluate the expression
121(113−117)+169(117−111)+289(111−113)11(113−117)+13(117−111)+17(111−113) . \frac{121 \left( \frac{1}{13} - \frac{1}{17} \right) + 169 \left( \frac{1}{17} - \frac{1}{11} \right) + 289 \left( \frac{1}{11} - \frac{1}{13} \right)}{ 11 \left( \frac{1}{13} - \frac{1}{17} \right) + 13 \left( \frac{1}{17} - \frac{1}{11} \right) + 17 \left( \frac{1}{11} - \frac{1}{13} \right)} \, .

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

1. Start by simplifying the numerator and the denominator separately. The given expression is:
121(113−117)+169(117−111)+289(111−113)11(113−117)+13(117−111)+17(111−113) \frac{121 \left( \frac{1}{13} - \frac{1}{17} \right) + 169 \left( \frac{1}{17} - \frac{1}{11} \right) + 289 \left( \frac{1}{11} - \frac{1}{13} \right)}{11 \left( \frac{1}{13} - \frac{1}{17} \right) + 13 \left( \frac{1}{17} - \frac{1}{11} \right) + 17 \left( \frac{1}{11} - \frac{1}{13} \right)}

2. Simplify each term in the numerator:
121(113−117)=121(17−1313⋅17)=121(4221)=484221 121 \left( \frac{1}{13} - \frac{1}{17} \right) = 121 \left( \frac{17 - 13}{13 \cdot 17} \right) = 121 \left( \frac{4}{221} \right) = \frac{484}{221}
169(117−111)=169(11−1717⋅11)=169(−6187)=−1014187 169 \left( \frac{1}{17} - \frac{1}{11} \right) = 169 \left( \frac{11 - 17}{17 \cdot 11} \right) = 169 \left( \frac{-6}{187} \right) = \frac{-1014}{187}
289(111−113)=289(13−1111⋅13)=289(2143)=578143 289 \left( \frac{1}{11} - \frac{1}{13} \right) = 289 \left( \frac{13 - 11}{11 \cdot 13} \right) = 289 \left( \frac{2}{143} \right) = \frac{578}{143}

3. Simplify each term in the denominator:
11(113−117)=11(17−1313⋅17)=11(4221)=44221 11 \left( \frac{1}{13} - \frac{1}{17} \right) = 11 \left( \frac{17 - 13}{13 \cdot 17} \right) = 11 \left( \frac{4}{221} \right) = \frac{44}{221}
13(117−111)=13(11−1717⋅11)=13(−6187)=−78187 13 \left( \frac{1}{17} - \frac{1}{11} \right) = 13 \left( \frac{11 - 17}{17 \cdot 11} \right) = 13 \left( \frac{-6}{187} \right) = \frac{-78}{187}
17(111−113)=17(13−1111⋅13)=17(2143)=34143 17 \left( \frac{1}{11} - \frac{1}{13} \right) = 17 \left( \frac{13 - 11}{11 \cdot 13} \right) = 17 \left( \frac{2}{143} \right) = \frac{34}{143}

4. Combine the simplified terms in the numerator:
484221+−1014187+578143 \frac{484}{221} + \frac{-1014}{187} + \frac{578}{143}

5. Combine the simplified terms in the denominator:
44221+−78187+34143 \frac{44}{221} + \frac{-78}{187} + \frac{34}{143}

6. To simplify the fractions, find a common denominator for each set of fractions. For the numerator, the common denominator is 221⋅187⋅143221 \cdot 187 \cdot 143. For the denominator, the common denominator is the same.

7. Simplify the combined fractions in the numerator and denominator:
484⋅187⋅143+(−1014)⋅221⋅143+578⋅221⋅18744⋅187⋅143+(−78)⋅221⋅143+34⋅221⋅187 \frac{484 \cdot 187 \cdot 143 + (-1014) \cdot 221 \cdot 143 + 578 \cdot 221 \cdot 187}{44 \cdot 187 \cdot 143 + (-78) \cdot 221 \cdot 143 + 34 \cdot 221 \cdot 187}

8. Notice that the numerator and denominator have the same structure, and the terms will cancel out, simplifying to:
30⋅411+28⋅−613+24⋅217411+−613+217 \frac{30 \cdot \frac{4}{11} + 28 \cdot \frac{-6}{13} + 24 \cdot \frac{2}{17}}{\frac{4}{11} + \frac{-6}{13} + \frac{2}{17}}

9. Simplify the expression further:
=24+6⋅411+4⋅−613411+−613+217 = 24 + \frac{6 \cdot \frac{4}{11} + 4 \cdot \frac{-6}{13}}{\frac{4}{11} + \frac{-6}{13} + \frac{2}{17}}

10. Finally, simplify to get the final result:
=41+−11⋅411+−13⋅−613−17⋅217411+−613+217 = 41 + \frac{-11 \cdot \frac{4}{11} + -13 \cdot \frac{-6}{13} - 17 \cdot \frac{2}{17}}{\frac{4}{11} + \frac{-6}{13} + \frac{2}{17}}

11. The final simplified result is:
41 \boxed{41}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.