97 CHINESE MATH OLYMPIAD

First Day 8:00--12:30 A.M 13 January 1997
 

Problem 1.
Real numbers X1, X2, ... , X1997 satisfies:

  1. -1/Ö3 £ Xi £ Ö3 ( i=1,2,..,1997 );
  2. X1+X2+...+X1997= -318 Ö3.
Find the maximum value of X112+X212+...+X199712. Where Ö3 means the positive root of 3.

Problem 2.
Let A1B1C1D1 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 Ak, Bk, Ck, Dk be the symmetry points of P by the lines A(k-1)B(k-1), B(k-1)C(k-1), C(k-1)D(k-1), D(k-1)A(k-1) respectively ( k=2,3,...). Think about the sequece AjBjCjDj (j=1,2,...) :

  1. Among the first 12 quadrilateral, which is similar to the 1997th quadrilateral and which not?
  2. Suppose the 1997th 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:
  1. A1+B1+C1=A2+B2+C2=...=An+Bn+Cn is divided by 6;
  2. A1+A2+...An=B1+B2+...+Bn=C1+C2+...Cn is also divided by 6.

    3.  
Second Day 8:00--12:30 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 one-to-one 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 one-to-one mapping F satisfies that there exists a natural number M, which:

  1. When m < M, 1 £ i £ 16,

    2. Fm(i+1)-Fm(i)<>1 or -1 (mod 17),
    Fm(1)-Fm(17)<>1 or -1 (mod 17);
  2. When 1 £ i £ 16,

    3. 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.
Non-positive sequence A1, A2, ... satisfies An+m £ An+Am, where m, n are natural numbers.
Prove: For any n ³ m, An £ mA1+((n/m)-1)Am.

Problems File

© 1997 Enrique Valeriano Cuba