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Bosnia and Herzegovina TST 1997 Day 2 Problem 1

Source: Bosnia and Herzegovina Team Selection Test 1997

September 20, 2018
geometryincirclealgebrainequalities

Problem Statement

a)a) In triangle ABCABC let A1A_1, B1B_1 and C1C_1 be touching points of incircle ABCABC with BABA, CACA and ABAB, respectively. Let l1l_1, l2l_2 and l3l_3 be lenghts of arcs B1C1 B_1C_1, A1C1A_1C_1, B1A1B_1A_1 of incircle ABCABC, respectively, which does not contain points A1A_1, B1B_1 and C1C_1, respectively. Does the following inequality hold: al1+bl2+cl3≥93π \frac{a}{l_1}+\frac{b}{l_2}+\frac{c}{l_3} \geq \frac{9\sqrt{3}}{\pi}
b)b) Tetrahedron ABCDABCD has three pairs of equal opposing sides. Find length of height of tetrahedron in function od lengths of sides
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