Problem 4

Inside triangle

P_1P_2P_3

a point

P

is given. Let

Q_1,Q_2,Q_3

be the intersections of

PP_1,PP_2,PP_3

with the opposite sides. Prove that among

\frac{PP_1}{PQ_1},\frac{PP_2}{PQ_2},\frac{PP_3}{PQ_3}

there is one not larger than

2

and one not smaller than

2

Step 3 of 5: Use the average of the three ratios

\min_i r_i\le\frac13\le\max_i r_i

Detailed analysis

Three positive numbers with sum $1$ cannot all exceed $1/3$ , nor can they all be below $1/3$ . Hence one $r_i\le1/3$ and one $r_j\ge1/3$ .