A tribute to the creators of the triangle !!!

Geometry Level 3

The lengths of the sides of a triangle with positive area are log ⁡ 10 12 , log ⁡ 10 75 \log_{10} 12, \log_{10} 75 , and log ⁡ 10 n \log_{10} n , where n n is a positive integer. Find the number of possible values for n n


The answer is 893.

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Parth Lohomi by Parth Lohomi , 20, India

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2 solutions

Pranjal Jain
Feb 2, 2015

By triangle inequality, log ⁡ 10 75 + log ⁡ 10 12 > log ⁡ 10 n > ∣ log ⁡ 10 75 − log ⁡ 10 12 ∣ \log_{10} 75+\log_{10} 12>\log_{10} n>|\log_{10} 75-\log_{10} 12| log ⁡ 10 900 > log ⁡ 10 n > log ⁡ 10 6.25 \log_{10} 900>\log_{10} n>\log_{10} 6.25 900 > n > 6.25 900>n>6.25

Number of values of n = 899 − 6 = 893 n=899-6=\boxed{893}

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Lu Chee Ket
Feb 4, 2015

Sum in logarithm is product in domain; difference in logarithm is division.

75/ 12 < n < 75 x 12

6.25 < n < 900

N = 899 - 7 + 1 = 893

Note: Sum of two side lengths must be greater than the one side length remained for sides of any triangle.

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