Difference between revisions of "2007 Alabama ARML TST Problems/Problem 2"
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− | ===== | + | == Problem == |
+ | The points <math>A, B, C ,</math> and <math>D</math> lie in that order on a line. Point <math>E</math> lies in a plane with <math>A, B, C ,</math> and <math>D</math> such that <math>\angle BEC = 78^{\circ}</math> . Given that <math>\angle EBC > \angle ECB</math>, <math>\angle ABE = 4x + y</math>, and <math>\angle ECB = x + y</math>, compute the number of positive integer values that <math>y</math> can take on. | ||
+ | == Solution == | ||
<math>\angle ABE</math> is external to <math>\triangle BEC</math> at <math>\angle B</math>. Therefore it is equal to the sum: <math>\angle E + \angle C</math> | <math>\angle ABE</math> is external to <math>\triangle BEC</math> at <math>\angle B</math>. Therefore it is equal to the sum: <math>\angle E + \angle C</math> | ||
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<math>\angle ABE = 4x + y = 78 + x + y</math> | <math>\angle ABE = 4x + y = 78 + x + y</math> | ||
+ | |||
<math>x = 26</math> | <math>x = 26</math> | ||
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However we note that by the problem statement <math>\angle ECB</math> cannot be greater than <math>\angle EBC</math>. | However we note that by the problem statement <math>\angle ECB</math> cannot be greater than <math>\angle EBC</math>. | ||
− | The two angles sum to <math>102^\circ</math>, thus <math>m\angle ECB < | + | The two angles sum to <math>102^\circ</math>, thus <math>m\angle ECB < 51^\circ</math> |
− | Noting that <math>m\angle ECB = 26 + y</math>, it becomes clear that <math>1 \le m\angle ECB \le | + | Noting that <math>m\angle ECB = 26 + y</math>, it becomes clear that <math>1 \le m\angle ECB \le 24</math> <math>\longrightarrow \boxed {24}</math> |
Latest revision as of 01:54, 21 December 2021
Problem
The points and
lie in that order on a line. Point
lies in a plane with
and
such that
. Given that
,
, and
, compute the number of positive integer values that
can take on.
Solution
is external to
at
. Therefore it is equal to the sum:
Then, according to the problem statement:
As y cancels, its value is not bounded by this algebraic relation.
However we note that by the problem statement cannot be greater than
.
The two angles sum to , thus
Noting that , it becomes clear that