Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.5 AIME Find the answer

Example 2. (7th American Mathematical Competition) In the scalene $\triangle ABC$ , two altitudes are 4 and 12. If the third altitude is also an integer, then it is most likely to be ( ).
(A) 4.
(B) 5.
(C) 6.
(D) 7.
(E) An answer different from (A)-(D).

Solve the system of inequalities

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Solution

$\left\{\begin{array}{l} \frac{1}{4}<\frac{1}{12}+\frac{1}{h} \\ \frac{1}{h}<\frac{1}{4}+\frac{1}{12} \end{array}\right.$

We get $3<h<6$ . It is clear that $h_{\max }=5$ (since $h$ is a positive integer).

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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.