Difference between revisions of "Mock AIME 1 Pre 2005 Problems/Problem 3"
(solution by krsattack) |
Jabberwock2 (talk | contribs) (Corrected absolute values.) |
||
Line 3: | Line 3: | ||
== Solution == | == Solution == | ||
− | Let the altitude from <math>P</math> onto <math>AE</math> at <math>Q</math> have lengths <math>PQ = h</math> and <math>AQ = r</math>. It is clear that, for a given <math>r</math> value, <math>AP</math>, <math>BP</math>, <math>CP</math>, <math>DP</math>, and <math>EP</math> are all minimized when <math>h = 0</math>. So <math>P</math> is on <math>AE</math>, and therefore, <math>P = Q</math>. Thus, <math>AP</math>=r, <math>BP = | + | Let the altitude from <math>P</math> onto <math>AE</math> at <math>Q</math> have lengths <math>PQ = h</math> and <math>AQ = r</math>. It is clear that, for a given <math>r</math> value, <math>AP</math>, <math>BP</math>, <math>CP</math>, <math>DP</math>, and <math>EP</math> are all minimized when <math>h = 0</math>. So <math>P</math> is on <math>AE</math>, and therefore, <math>P = Q</math>. Thus, <math>AP</math>=r, <math>BP = |r - 1|</math>, <math>CP = |r - 2|</math>, <math>DP = |r - 4|</math>, and <math>EP = |r - 13|.</math> Squaring each of these gives: |
<math>AP^2 + BP^2 + CP^2 + DP^2 + EP^2 = r^2 + (r - 1)^2 + (r - 2)^2 + (r - 4)^2 + (r - 13)^2 = 5r^2 - 40r + 190</math> | <math>AP^2 + BP^2 + CP^2 + DP^2 + EP^2 = r^2 + (r - 1)^2 + (r - 2)^2 + (r - 4)^2 + (r - 13)^2 = 5r^2 - 40r + 190</math> |
Latest revision as of 06:18, 2 July 2015
Problem
and are collinear in that order such that and . If can be any point in space, what is the smallest possible value of ?
Solution
Let the altitude from onto at have lengths and . It is clear that, for a given value, , , , , and are all minimized when . So is on , and therefore, . Thus, =r, , , , and Squaring each of these gives:
This reaches its minimum at , at which point the sum of the squares of the distances is .
See also
Mock AIME 1 Pre 2005 (Problems, Source) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |