1963 AHSME Problems/Problem 40
Problem
If is a number satisfying the equation , then is between:
Solution 1
Let and . Cubing these equations, we get and , so . The left-hand side factors as
However, from the given equation , we get . Then , so .
Squaring the equation , we get . Subtracting this equation from the equation , we get , so . But and , so , so . Cubing both sides, we get , so . The answer is .
Solution 2
i.e,
if the sum of three numbers is zero, then their sum of cubes is thrice the product of each number. then, . by solving this, we get . this gives the step what we had done in solution 1.The answer is .
Solution 3
Cubing both sides, we get so Dividing both sides by 3 and cubing, we find , which is between .
~pfalcon
Solution 4
We consider the formula . Factoring out , and the commutative property gives . Cubing both sides gives us Using our formula, we have Solving this gives , therefore, the answer is . ~zixuan12
See Also
1963 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 39 |
Followed by Last Problem | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.