2021 USAMO Problems/Problem 1

Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that $\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.$ Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.

Solution

Let $D$ be the second point of intersection of the circles $AB_1B$ and $AA_1C.$ Then $\angle ADB = 180^\circ – \angle AB_1B,\angle ADC = 180^\circ – \angle AA_1C \implies$ $\angle BDC = 360^\circ – \angle ADB – \angle ADC =$ $= 360^\circ – (180^\circ – \angle AB_1B) – (180^\circ – \angle AA_1C) =$ $=\angle AB_1B + \angle AA_1C \implies \angle BDC + \angle BC_1C = 180^\circ \implies$ $BDCC_1B_2$ is cyclic with diameters $BC_1$ and $CB_2 \implies \angle CDB_2 = 90^\circ.$ Similarly, $\angle CDA_1 = 90^\circ \implies$ points $A_1, D,$ and $B_2$ are collinear.