Difference between revisions of "1993 AHSME Problems/Problem 30"
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<math>2x_0 = a_1(a_2 a_3 \cdots)_2</math> | <math>2x_0 = a_1(a_2 a_3 \cdots)_2</math> | ||
− | If <math>2x_0 < 1</math>, then <math>x_0 < \frac{1}{2}</math> which means that <math> | + | If <math>2x_0 < 1</math>, then <math>x_0 < \frac{1}{2}</math> which means that <math>a_1 = 0</math> and so <math>x_1 = (.a_2 a_3 a_4 \cdots)_2</math> |
If <math>2x_0 \geq 1</math> then <math>x \geq \frac{1}{2}</math> which means that <math>x_1 = 2x_0 - 1 = (.a_2 a_3 a_4 \cdots)_2</math>. | If <math>2x_0 \geq 1</math> then <math>x \geq \frac{1}{2}</math> which means that <math>x_1 = 2x_0 - 1 = (.a_2 a_3 a_4 \cdots)_2</math>. |
Revision as of 01:09, 3 January 2016
Problem
Given , let
for all integers
. For how many
is it true that
?
Solution
We are going to look at this problem in binary.
If , then
which means that
and so
If then
which means that
.
Using the same logic, we notice that this sequence cycles and that since we notice that
.
We have possibilities for each of
to
but we can't have
so we have
-mathman523
See also
1993 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 29 |
Followed by Problem 30 | |
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 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.