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MMO 209 Moscow MO 1952 (ax+by+cz)^2+(bx+cy+az)^2+(cx+ay+bz)^2=..

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August 8, 2019
algebraidentity

Problem Statement

Prove the identity:
a) (ax+by+cz)2+(bx+cy+az)2+(cx+ay+bz)2=(cx+by+az)2+(bx+ay+cz)2+(ax+cy+bz)2(ax + by + cz)^2 + (bx + cy + az)^2 + (cx + ay + bz)^2 =(cx + by + az)^2 + (bx + ay + cz)^2 + (ax + cy + bz)^2
b) (ax+by+cz+du)2+(bx+cy+dz+au)2+(cx+dy+az+bu)2+(dx+ay+bz+cu)2=(ax + by + cz + du)^2+(bx + cy + dz + au)^2 +(cx + dy + az + bu)^2 + (dx + ay + bz + cu)^2 = (dx+cy+bz+au)2+(cx+by+az+du)2+(bx+ay+dz+cu)2+(ax+dy+cz+bu)2(dx + cy + bz + au)^2+(cx + by + az + du)^2 +(bx + ay + dz + cu)^2 + (ax + dy + cz + bu)^2.
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