MathDB

3

Part of 2000 Hungary-Israel Binational

Problems(2)

locus of the orthocenter of the touchpoints of the incircle

Source: Hungary-Israel Binational 2000 day P3

9/27/2017
Let ABC{ABC} be a non-equilateral triangle. The incircle is tangent to the sides BC,CA,AB{BC,CA,AB} at A1,B1,C1{A_1,B_1,C_1}, respectively, and M is the orthocenter of triangle A1B1C1{A_1B_1C_1}. Prove that M{M} lies on the line through the incenter and circumcenter of △ABC{\vartriangle ABC}.
geometryorthocenterincircle
kl positive integers

Source: 11-th Hungary-Israel Binational Mathematical Competition 2000

4/22/2007
Let kk and ll be two given positive integers and aij(1≤i≤k,1≤j≤l)a_{ij}(1 \leq i \leq k, 1 \leq j \leq l) be klkl positive integers. Show that if q≥p>0q \geq p > 0, then (∑j=1l(∑i=1kaijp)q/p)1/q≤(∑i=1k(∑j=1laijq)p/q)1/p.(\sum_{j=1}^{l}(\sum_{i=1}^{k}a_{ij}^{p})^{q/p})^{1/q}\leq (\sum_{i=1}^{k}(\sum_{j=1}^{l}a_{ij}^{q})^{p/q})^{1/p}.
inequalitiesinductioninequalities unsolved