Selection and Training Session

Question

Point P inside an acute-angled triangle A_1A_2A_3 is chosen so that its projections P_1, P_2, P_3 onto the sides A_1A_2, A_2A_3, A_3A_1 respectively lie on the sides of the triangle. Prove that for points X_1, X_2, X_3 on the sides A_1A_2, A_2A_3, A_3A_1 respectively, X_1X_2P_1P_2, X_2X_3P_2P_3, X_3X_1P_3P_1 1 if a) X_1, X_2, X_3 are the midpoints of the corresponding sides; b) X_1, X_2, X_3 are the feet of the corresponding altitudes; c) X_1, X_2, X_3 are arbitrary points on the corresponding sides. (IMO-2010 Shortlist, Problem G3, modified)

Accepted Answer

a), b) (Solution of M. Mankevich, A. Nekrashevich, A. Tanana.) The statement from a) and b) simply follow from the general

Lemma. If there is a point

X

such that

X_{1}

,

X_{2}

,

X_{3}

are the orthogonal projections of

X

onto the sides

A_{2} A_{3}

,

A_{3} A_{1}

,

A_{1} A_{2}

respectively, then the statement of the problem is valid.

Indeed,

P

belongs to one of the quadrilaterals

A_{1} X_{2} X X_{3}

,

A_{2} X_{3} X X_{1}

,

A_{3} X_{1} X X_{2}

; say,

P

belongs to

A_{2} X_{1} X X_{3}

. Then

P_{1} P_{3} = A_{2} P sin ∠ A_{2} \leq A_{2} X sin ∠ A_{2},

i.e.

X_{1} X_{3} / P_{1} P_{3} \geq 1

. Thus the lemma is proved.