MathDB

5

Part of 2011 NZMOC Camp Selection Problems

Problems(2)

computational with a square, collinear given (2011 NZOMC Camp Sel. J5)

Source:

1/3/2021
Let a square ABCDABCD with sides of length 11 be given. A point XX on BCBC is at distance dd from CC, and a point YY on CDCD is at distance dd from CC. The extensions of: ABAB and DXDX meet at PP, ADAD and BYBY meet at Q,AXQ, AX and DCDC meet at RR, and AYAY and BCBC meet at SS. If points P,Q,RP, Q, R and SS are collinear, determine dd.
squaregeometrydiastancecollinear
[(a^2-b^2)^3 + (b^2- c^2)^3 + (c^2 - a2)^3][(a-b)^3 + (b-c)^3 + (c-a)^3]> 8sbc

Source: New Zealand NZMOC Camp Selection Problems 2011 Seniors 5

9/18/2021
Prove that for any three distinct positive real numbers a,ba, b and cc: (a2−b2)3+(b2−c2)3+(c2−a2)3(a−b)3+(b−c)3+(c−a)3>8abc.\frac{(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3}{(a - b)^3 + (b - c)^3 + (c - a)^3}> 8abc.
algebrainequalities