MathDB

1

Part of 1990 Iran MO (2nd round)

Problems(2)

Maximum value of the sum [Iran Second Round 1990]

Source:

12/1/2010
(a) Consider the set of all triangles ABCABC which are inscribed in a circle with radius R.R. When is AB2+BC2+CA2AB^2+BC^2+CA^2 maximum? Find this maximum.
(b) Consider the set of all tetragonals ABCDABCD which are inscribed in a sphere with radius R.R. When is the sum of squares of the six edges of ABCDABCD maximum? Find this maximum, and in this case prove that all of the edges are equal.
geometry3D geometryspheregeometry proposed
Prove that RS || Delta [Iran Second Round 1990]

Source:

12/1/2010
Let ABCDABCD be a parallelogram. The line Δ\Delta meets the lines AB,BC,CDAB, BC, CD and DADA at M,N,PM, N, P and Q,Q, respectively. Let RR be the intersection point of the lines AB,DNAB,DN and let SS be intersection point of the lines AD,BP.AD, BP. Prove that RS∥Δ.RS \parallel \Delta.
[asy] import graph; size(400); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); pen qqzzcc = rgb(0,0.6,0.8); pen wwwwff = rgb(0.4,0.4,1); draw((2,2)--(6,2),qqzzcc+linewidth(1.6pt)); draw((6,2)--(4,0),qqzzcc+linewidth(1.6pt)); draw((-1.95,(+12-2*-1.95)/2)--(12.24,(+12-2*12.24)/2),qqzzcc+linewidth(1.6pt)); draw((-1.95,(-0+3*-1.95)/3)--(12.24,(-0+3*12.24)/3),qqzzcc+linewidth(1.6pt)); draw((-1.95,(-0-0*-1.95)/6)--(12.24,(-0-0*12.24)/6),qqzzcc+linewidth(1.6pt)); draw((4,0)--(4,4),wwwwff+linewidth(1.2pt)+linetype("3pt 3pt")); draw((2,2)--(8.14,0),wwwwff+linewidth(1.2pt)+linetype("3pt 3pt")); draw((-1.95,(+32.56-4*-1.95)/4.14)--(12.24,(+32.56-4*12.24)/4.14),qqzzcc+linewidth(1.6pt)); dot((0,0),ds); label("AA", (0,-0.3),NE*lsf); dot((4,0),ds); label("BB", (4.02,-0.33),NE*lsf); dot((2,2),ds); label("DD", (1.81,2.07),NE*lsf); dot((6,2),ds); label("CC", (6.16,2.08),NE*lsf); dot((3,3),ds); label("QQ", (2.97,3.22),NE*lsf); dot((5,1),ds); label("NN", (4.99,1.19),NE*lsf); label("Δ\Delta", (1.7,3.76),NE*lsf); dot((6,0),ds); label("MM", (5.9,-0.33),NE*lsf); dot((4,2),ds); label("PP", (4.02,2.08),NE*lsf); dot((4,4),ds); label("SS", (3.94,4.12),NE*lsf); dot((8.14,0),ds); label("EE", (8.2,0.09),NE*lsf); clip((-1.95,-6.96)--(-1.95,4.99)--(12.24,4.99)--(12.24,-6.96)--cycle); [/asy]
geometry proposedgeometry