MathDB

3

Part of 2000 239 Open Mathematical Olympiad

Problems(2)

$\frac{a_1 + a_2}{2}$ $\frac{a_1+ a_2 +a_3}{2 \sqrt{2}}$

Source: 239 2000 S3

5/18/2020
For all positive real numbers a1,a2,…,ana_1, a_2, \dots, a_n, prove that a1 ⁣+ ⁣a22⋅a2 ⁣+ ⁣a32⋅⋯⋅an ⁣+ ⁣a12≤a1 ⁣+ ⁣a2 ⁣+ ⁣a322⋅a2 ⁣+ ⁣a3 ⁣+ ⁣a422⋅⋯⋅an ⁣+ ⁣a1 ⁣+ ⁣a222. \frac{a_1\! +\! a_2}{2} \cdot \frac{a_2\! +\! a_3}{2} \cdot \dots \cdot \frac{a_n\! +\! a_1}{2} \leq \frac{a_1\!+\!a_2\!+\!a_3}{2 \sqrt{2}} \cdot \frac{a_2\!+\!a_3\!+\!a_4}{2 \sqrt{2}} \cdot \dots \cdot \frac{a_n\!+\!a_1\!+\!a_2}{2 \sqrt{2}}.
Inequalityalgebra
equal segments wanted, line connecting incenters related, altitudes

Source: 239 MO VIII-IX 2000 p3

5/12/2020
Let AA1 AA_1 and CC1 CC_1 be the altitudes of the acute-angled triangle ABC ABC . A line passing through the centers of the inscribed circles the triangles AA1C AA_1C and CC1A CC_1A intersect the sides of AB AB and BC BC triangle ABC ABC at points X X and Y Y . Prove that BX=BY BX = BY .
geometryincenteraltitudeequal segments