MathDB

1

Part of 2009 Poland - Second Round

Problems(2)

Inequality with products of a_1,a_2, ... a_n

Source: Polish Second Round 2009

8/4/2011
Let a1≥a2≥…≥an>0a_1\ge a_2\ge \ldots \ge a_n>0 be nn reals. Prove the inequality a1a2…an−1+(2a2−a1)(2a3−a2)…(2an−an−1)≥2a2a3…ana_1a_2\ldots a_{n-1}+(2a_2-a_1)(2a_3-a_2)\ldots (2a_n-a_{n-1})\ge 2a_2a_3\ldots a_n
inequalitiesinductionfunctioninequalities proposed
If tangents at C,D meet at P then PC=PE

Source: Polish Second Round 2009

8/4/2011
ABCDABCD is a cyclic quadrilateral inscribed in the circle Γ\Gamma with ABAB as diameter. Let EE be the intersection of the diagonals ACAC and BDBD. The tangents to Γ\Gamma at the points C,DC,D meet at PP. Prove that PC=PEPC=PE.
geometrycyclic quadrilateralgeometry unsolved