MathDB

2

Part of 2021 Czech-Polish-Slovak Junior Match

Problems(2)

AF _|_ DE wanted, projections on external angle bisectors

Source: Czech-Polish-Slovak Match Junior 2021, individual p2 CPSJ

9/12/2021
An acute triangle ABCABC is given. Let us denote by DD and EE the orthogonal projections, respectively of points B B and CC on the bisector of the external angle BACBAC. Let FF be the point of intersection of the lines BEBE and CDCD. Show that the lines AFAF and DEDE are perpendicular.
perpendicularangle bisectorgeometry
4 divides n if x_1x_2 +x_2x_3+... +x_{n-1}x_n +x_nx_1=0 with x_i \in {-1,1}

Source: 2021 Czech-Polish-Slovak Match Junior, team p2 CPSJ

9/12/2021
Let the numbers xi∈{−1,1}x_i \in \{-1, 1\} be given for i=1,2,...,ni = 1, 2,..., n, satisfying x1x2+x2x3+...+xn−1xn+xnx1=0.x_1x_2 + x_2x_3 +... + x_{n-1}x_n + x_nx_1 = 0. Prove that nn is divisible by 44.
number theorydivisibledivides