Difference between revisions of "2001 AIME I Problems/Problem 11"
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We solve the system (the first two equations, and then the latter three) to get | We solve the system (the first two equations, and then the latter three) to get | ||
<cmath> | <cmath> | ||
− | \left(a_{1},a_{2},a_{3},a_{4},a_{5}\right) = \left(\frac {23 + 5N}{24},\frac {19 + N}{24},\frac {51 + 117N}{124},\frac { | + | \left(a_{1},a_{2},a_{3},a_{4},a_{5}\right) = \left(\frac {23 + 5N}{24},\frac {19 + N}{24},\frac {51 + 117N}{124},\frac {35 + 73N}{124},\frac {7 + 89N}{124}\right). |
</cmath> | </cmath> | ||
<math>a_{1},a_{2}</math> will be integers iff <math>N\equiv 5\pmod{24}</math> and <math>a_{3},a_{4},a_{5}</math> will be integers iff <math>N\equiv 25\pmod{124}</math>. Solving these [[Congruent (modular arithmetic)|congruence]]s simultaneously by standard methods gives <math>N\equiv \boxed{149}\pmod{744}</math>. | <math>a_{1},a_{2}</math> will be integers iff <math>N\equiv 5\pmod{24}</math> and <math>a_{3},a_{4},a_{5}</math> will be integers iff <math>N\equiv 25\pmod{124}</math>. Solving these [[Congruent (modular arithmetic)|congruence]]s simultaneously by standard methods gives <math>N\equiv \boxed{149}\pmod{744}</math>. |
Revision as of 17:13, 18 February 2012
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 , where the first coordinate represents the row number and
represents the column number. Then
,
,
, etc. and
,
, etc. Now we get the system of equations:
We solve the system (the first two equations, and then the latter three) to get
will be integers iff
and
will be integers iff
. Solving these congruences simultaneously by standard methods gives
.
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 |