Chinese IMO Team Selection Tests 1986
First Test
1. Let ABCD be a cyclic convex quadrilateral and let IA , IB , IC , ID be the incenters
of the triangles BCD, ACD, ABD, ABC, respectively. Show that IA IB IC ID is a rectangle.
2. Let ai , bi (i = 1, 2, . . . , n) be real numbers. Prove that the following two statements are equivalent:
(i) ∑nk=1 ak xk ≤ ∑nk=1 bk xk whenever x1 ≤ x2 ≤ · · · ≤ xn ;
(ii) ∑Kk=1 ak ≤ ∑Kk=1 bk for K = 1, 2, . . . , n − 1 and ∑nk=1 ak = ∑nk=1 bk .
3. For a natural number A = an an−1 . . . a0 in decimal expansion, denote f (A) =
2n a0 + 2n−1a1 + · · · + an . Define A1 = f (A) and Ai+1 = f (Ai ) for i ∈ N. Show
that:
(a) There exists a positive integer k for which Ak+1 = Ak .
(b) Determine the value of this Ak if A = 1986 .
4. A triangle ABC has a right angle at C. Show that, given any n points inside the
triangle, we can denote them by P1 , P2 , . . . , Pn such that
P1 P22 + P2P32 + · · · + Pn−1Pn2 ≤ AB2 .
Second Test
1. Points P and Q on sides AB, AD respectively of a unit square are taken so that
the perimeter of triangle APQ is 2. Find the angle PCQ.
2. Points E, F, G are chosen on the respective edges AB, AC and AD of a tetrahedron
ABCD. Let SXY Z denote the area and PXY Z the perimeter of triangle XY Z. Prove
that:
(a) SEFG ≤ max SABC , SABD , SACD , SBCD ;
(b) SEFG ≤ max PABC , PABD , PACD , PBCD .
3. For n ≥ 3 real numbers x1 , x2 , . . . , xn , denote p = ∑ni=1 xi and q = ∑1≤i< j≤n xi x j .
Prove that:
n−1 2
p − 2q ≥ 0;
n
r
p n − 1
2n
q for each i = 1, . . . , n.
(b) xi − ≤
p2 −
n
n
n−1
(a)
1
The IMO Compendium Group,
D. Djukić, V. Janković, I. Matić, N. Petrović
Typed in LATEX by Ercole Suppa
www.imomath.com
4. On a circle are marked 4k distinct points, arbitrarily labelled from 1 to 4k.
(a) Show that one can draw 2k pairwise disjoint chords with endpoints at the
marked points so that the labels of the endpoints of each chord differ by at
most 3k − 1.
(b) Show that the bound 3k − 1 cannot be improved.
2
The IMO Compendium Group,
D. Djukić, V. Janković, I. Matić, N. Petrović
Typed in LATEX by Ercole Suppa
www.imomath.com