Difference between revisions of "1963 AHSME Problems/Problem 31"
Rockmanex3 (talk | contribs) (Solution to Problem 31) |
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\textbf{(E)}\ 0 </math> | \textbf{(E)}\ 0 </math> | ||
− | ==Solution== | + | ==Solution 1== |
− | Solving for <math>x</math> in the equation yields <math>x = \frac{ | + | Solving for <math>x</math> in the equation yields <math>x =rfthe meaning of theta 0</math>. Solving the inequality results in <math>y \le 254 \frac{1}{3}</math>. From the two conditions, <math>y</math> can be an odd number from <math>1</math> to <math>253</math>, so there are <math>127</math> solutions where <math>x</math> and <math>y</math> are integers. The answer is <math>\boxed{\textbf{(D)}}</math>. |
+ | ==Solution 2== | ||
+ | We will prove that <math>y</math> is an odd number by contradiction. If <math>y</math> is even, then we know that <math>y = 2m</math> where <math>m</math> is some integer. However, this immediately assumes that <math>\text{ even } + \text{ even } = \text{ odd }</math> which is impossible. therefore <math>y</math> must ben odd. | ||
− | + | then we can easily prove that <math>x</math> ..... | |
==See Also== | ==See Also== |
Latest revision as of 18:42, 18 January 2021
Contents
Problem
The number of solutions in positive integers of is:
Solution 1
Solving for in the equation yields . Solving the inequality results in . From the two conditions, can be an odd number from to , so there are solutions where and are integers. The answer is .
Solution 2
We will prove that is an odd number by contradiction. If is even, then we know that where is some integer. However, this immediately assumes that which is impossible. therefore must ben odd.
then we can easily prove that .....
See Also
1963 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 30 |
Followed by Problem 32 | |
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 |
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