MathDB

1

Part of 2009 China National Olympiad

Problems(2)

Concyclic

Source: Chinese National Olympiad 2009 P1

1/10/2009
Given an acute triangle PBC PBC with PB≠PC. PB\neq PC. Points A,D A,D lie on PB,PC, PB,PC, respectively. AC AC intersects BD BD at point O. O. Let E,F E,F be the feet of perpendiculars from O O to AB,CD, AB,CD, respectively. Denote by M,N M,N the midpoints of BC,AD. BC,AD. (1) (1): If four points A,B,C,D A,B,C,D lie on one circle, then EM\cdot FN \equal{} EN\cdot FM. (2) (2): Determine whether the converse of (1) (1) is true or not, justify your answer.
trigonometrygeometrytrapezoidGausscyclic quadrilateralcongruent trianglesangle bisector
The minimum value

Source: Chinese National Olympiad 2009 P4

1/10/2009
Given an integer n>3. n > 3. Let a1,a2,⋯ ,an a_{1},a_{2},\cdots,a_{n} be real numbers satisfying min |a_{i} \minus{} a_{j}| \equal{} 1, 1\le i\le j\le n. Find the minimum value of \sum_{k \equal{} 1}^n|a_{k}|^3.
inequalitiessymmetryfunctionalgebrapolynomialabsolute valueinequalities proposed