1986 AHSME Problems/Problem 30
Problem
The number of real solutions of the simultaneous equations is
Solution
Consider the cases and , and also note that by AM-GM, for any positive number , we have , with equality only if . Thus, if , considering each equation in turn, we get that , and finally .
Now suppose . Then , so that . Similarly, we can get , , and , and combining these gives , an obvious contradiction.
Thus we must have , but , so if , the only possibility is , and analogously from the other equations we get ; indeed, by substituting, we verify that this works.
As for the other case, , notice that is a solution if and only if is a solution, since this just negates both sides of each equation and so they are equivalent. Thus the only other solution is , so that we have solutions in total, and therefore the answer is .
See also
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