1976 USAMO Problems
Problems from the 1976 USAMO.
Problem 1
- (a) Suppose that each square of a chessboard, as shown above, is colored either black or white. Prove that with any such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board such as the one outlined in the figure) whose four distinct unit corner squares are all of the same color.
- (b) Exhibit a black-white coloring of a board in which the four corner squares of every rectangle, as described above, are not all of the same color.
Problem 2
If and are fixed points on a given circle and is a variable diameter of the same circle, determine the locus of the point of intersection of lines and . You may assume that is not a diameter.
Problem 3
Determine all integral solutions of .
Problem 4
If the sum of the lengths of the six edges of a trirectangular tetrahedron (i.e., ) is , determine its maximum volume.
Problem 5
If , , , and are all polynomials such that prove that is a factor of .
See Also
1976 USAMO (Problems • Resources) | ||
Preceded by 1975 USAMO |
Followed by 1977 USAMO | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.