97 CHINESE MATH OLYMPIAD
First Day 8:0012:30 A.M 13 January 1997
Problem 1.
Real numbers X1, X2, ... , X1997 satisfies:

1/Ö3 £
Xi £ Ö3
( i=1,2,..,1997 );

X_{1}+X_{2}+...+X_{1997}= 318 Ö3.
Find the maximum value of X_{1}^{12}+X_{2}^{12}+...+X_{1997}^{12}.
Where Ö3 means the positive root of 3.
Problem 2.
Let A_{1}B_{1}C_{1}D_{1} be any convex
quadrilateral, P is an interior point, and for any vertex of the quadrilateral,
the line joining P and the vertex and the two sides from the vertex formed
two angels, they are both acute. Let A_{k}, B_{k}, C_{k},
D_{k} be the symmetry points of P by the lines A_{(k1)}B_{(k1)},
B_{(k1)}C_{(k1)}, C_{(k1)}D_{(k1)},
D_{(k1)}A_{(k1)} respectively ( k=2,3,...). Think about
the sequece A_{j}B_{j}C_{j}D_{j} (j=1,2,...)
:

Among the first 12 quadrilateral, which is similar to the 1997^{th}
quadrilateral and which not?

Suppose the 1997^{th} is an inscribed quadrilateral, which ones
among the first 12 quadrilaterals is also inscribed and which not?
Problem 3.
Prove that there exists infinitely natural numbers n, so that we can
arrange 1,2,3,...,3n as a table:
A1 A2 ... An 
B1 B2 ... Bn 
C1 C2 ... Cn 
satisfies the following:

A1+B1+C1=A2+B2+C2=...=An+Bn+Cn is divided by 6;

A1+A2+...An=B1+B2+...+Bn=C1+C2+...Cn is also divided by 6.
Second Day 8:0012:30 A.M 14 January 1997
Problem 4.
Quadrilateral ABCD is inscribed in a circle, the extension lines of
AB and DC intersects at P, the extensions lines of AD and BC intersects
at Q, draw the two tangent lines of the circle QE and QF which E, F is
the points of tangency. Prove: P, E, F are collinear.
Problem 5.
A={1,2,3,...,17}. For a onetoone mapping F from A onto A. Let F1(x)=F(x),
F(k+1)(x)=F(Fk(x)) (k is a natural number) Then there are some onetoone
mapping F satisfies that there exists a natural number M, which:

When m < M, 1 £ i £
16,
Fm(i+1)Fm(i)<>1 or 1 (mod 17),
Fm(1)Fm(17)<>1 or 1 (mod 17);

When 1 £ i £
16,
FM(i+1)FM(i)=1 or 1 (mod 17),
FM(1)FM(17)=1 or 1 (mod 17).
For all such mappping F, find the maximum value of M.
Problem 6.
Nonpositive sequence A1, A2, ... satisfies A_{n+m} £
A_{n}+A_{m}, where m, n are natural numbers.
Prove: For any n ³ m, An £
mA1+((n/m)1)Am.
© 1997 Enrique Valeriano
Cuba