Problem:
Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to
Answer Choices:
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Solution:
Suppose that a quadrilateral with sides and with perimeter 32 exists. By the triangle inequality , so . Reciprocally, if is a quadruple of positive integers whose sum equals 32 , and whose maximum entry is , then , so the triangle inequality is satisfied. This is the only condition required to guarantee the existence of a convex quadrilateral with given side lengths. Moreover, if the cyclic order of the sides is specified, then there is exactly one such cyclic quadrilateral.
The problem reduces to counting all the quadruples of positive integers with , and where two quadruples are considered the same if they generate the same quadrilateral, that is if one is a cyclic permutation of the other one. For example and generate the same quadrilateral.
The number of quadruples with can be counted as follows: consider 31 spots on a line to be filled with 28 ones and 3 plus signs. There are ways to choose the locations of the plus signs, and every such assignment is in one-to-one correspondence to the quadruple ( ), where each entry indicates the number of ones between consecutive plus signs. Setting gives precisely all quadruples where and . To count those where the maximum entry is 16 or more, consider 13 ones and 3 plus signs. There are quadruples ( ) where and , there are 4 ways to choose one of the coordinates, say , to be the maximum. Then the quadruple satisfies our requirements. Thus there are exactly quadruples where , and\
; consequently, there are
quadruples where , and 15.
If consists of distinct entries, then it has exactly 4 cyclic permutations. The same occurs if only two entries are equal to each other, or three entries are equal to each other and the remaining entry is not. If has two pairs of entries equal to each other ordered , then it has 4 cyclic permutations, but if they are ordered then it has only 2 cyclic permutations. Finally, if all entries are equal then there is only one cyclic permutation.
There are exactly quadruples of the form with and and there is only one quadruple with four equal entries. Adding to (1) the number of quadruples of the form and 3 times the number of quadruples of the form ( , guarantees that all classes of equivalence under cyclic permutations are counted exactly 4 times. Therefore the required number of cyclic quadrilaterals is
The problems on this page are the property of the MAA's American Mathematics Competitions