MathDB

2

Part of 2001 239 Open Mathematical Olympiad

Problems(2)

Inequality with geometric mean and 1+\frac{1}{x(1+x)}

Source: 239 2001 J7 S2

5/19/2020
For any positive numbers a1,a2,…,an a_1 , a_2 , \dots, a_n prove the inequality  ⁣( ⁣1 ⁣+ ⁣1a1(1+a1) ⁣) ⁣( ⁣1 ⁣+ ⁣1a2(1+a2) ⁣) ⁣… ⁣( ⁣1 ⁣+ ⁣1an(1+an) ⁣)≥( ⁣1 ⁣+ ⁣1p(1+p) ⁣) ⁣n ⁣,\! \left(\!1\!+\!\frac{1}{a_1(1+a_1)} \!\right)\! \left(\!1\!+\!\frac{1}{a_2(1+a_2)} \! \right) \! \dots \! \left(\!1\!+\!\frac{1}{a_n(1+a_n)} \! \right) \geq \left(\!1\!+\!\frac{1}{p(1+p)} \! \right)^{\! n} \! , where p=a1a2…annp=\sqrt[n]{a_1 a_2 \dots a_n}.
inequalities
intersection of circumcircles lies on diagonal of ABCD, equal angles given

Source: 239 MO 2001 VIII-IX p2

5/7/2020
In a convex quadrangle ABCD ABCD , the rays DA DA and CB CB intersect at point Q Q , and the rays BA BA and CD CD at the point P P . It turned out that ∠AQB=∠APD \angle AQB = \angle APD . The bisectors of the angles ∠AQB \angle AQB and ∠APD \angle APD intersect the sides quadrangle at points X X , Y Y and Z Z , T T respectively. Circumscribed circles of triangles ZQT ZQT and XPY XPY intersect at K K inside quadrangle. Prove that K K lies on the diagonal AC AC .
geometrycircumcirclediagonalequal angles