Difference between revisions of "2001 AIME I Problems/Problem 11"
Claudeaops (talk | contribs) (Sidenote) |
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\end{align*}</cmath> | \end{align*}</cmath> | ||
As these are all positive and less than <math>N</math>, <math>\boxed{149}</math> is the solution. | As these are all positive and less than <math>N</math>, <math>\boxed{149}</math> is the solution. | ||
+ | |||
+ | == Sidenote == | ||
+ | If we express all the <math>c_i</math> in terms of <math>N</math>, we have | ||
+ | <cmath>24c_1=5N+23</cmath> | ||
+ | <cmath>24c_2=N+19</cmath> | ||
+ | <cmath>124c_3=117N+51</cmath> | ||
+ | <cmath>124c_4=73N+35</cmath> | ||
+ | <cmath>124c_5=89N+7</cmath> | ||
+ | |||
+ | It turns out that there exists such an array satisfying the problem conditions if and only if | ||
+ | <cmath>N\equiv 149 \pmod{744}</cmath> | ||
== See also == | == See also == |
Revision as of 21:32, 4 December 2015
Contents
Problem
In a rectangular array of points, with 5 rows and columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through the second row is numbered through and so forth. Five points, and are selected so that each is in row Let be the number associated with Now renumber the array consecutively from top to bottom, beginning with the first column. Let be the number associated with after the renumbering. It is found that and Find the smallest possible value of
Solution
Let each point be in column . The numberings for can now be defined as follows.
We can now convert the five given equalities. Equations and combine to form Similarly equations , , and combine to form Take this equation modulo 31 And substitute for N
Thus the smallest might be is and by substitution
The column values can also easily be found by substitution As these are all positive and less than , is the solution.
Sidenote
If we express all the in terms of , we have
It turns out that there exists such an array satisfying the problem conditions if and only if
See also
2001 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.