Difference between revisions of "1973 IMO Problems/Problem 1"
Durianaops (talk | contribs) (Created page with "==Problem== Point <math>O</math> lies on line <math>g;</math> <math>\overrightarrow{OP_1}, \overrightarrow{OP_2},\cdots, \overrightarrow{OP_n}</math> are unit vectors such tha...") |
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==Solution== | ==Solution== | ||
| − | {{ | + | We prove it by induction on the number <math>2n+1</math> of vectors. The base step (when we have one vector) is clear, and for the induction step we use the hypothesis for the <math>2n-1</math> vectors obtained by disregarding the outermost two vectors. We thus get a vector with norm <math>\ge 1</math> betwen two with norm <math>1</math>. The sum of the two vectors of norm <math>1</math> makes an angle of <math>\le\frac\pi 2</math> with the vector of norm <math>\ge 1</math>, so their sum has norm <math>\ge 1</math>, and we're done. |
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| + | The above solution was posted and copyrighted by grobber. The original thread for this problem can be found here: [https://aops.com/community/p357939] | ||
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| + | {{alternate solutions}} | ||
==See Also== | ==See Also== | ||
Latest revision as of 15:45, 29 January 2021
Problem
Point 
lies on line 
are unit vectors such that points 
all lie in a plane containing 
and on one side of 
Prove that if 
is odd, ![\[\left|\overrightarrow{OP_1}+\overrightarrow{OP_2}+\cdots+ \overrightarrow{OP_n}\right|\ge1.\]](http://latex.artofproblemsolving.com/7/c/2/7c261196979f62bc5b704df4bace1553f2760e4b.png)
Here 
denotes the length of vector 
Solution
We prove it by induction on the number 
of vectors. The base step (when we have one vector) is clear, and for the induction step we use the hypothesis for the 
vectors obtained by disregarding the outermost two vectors. We thus get a vector with norm 
betwen two with norm 
. The sum of the two vectors of norm makes an angle of 
with the vector of norm , so their sum has norm , and we're done.
The above solution was posted and copyrighted by grobber. The original thread for this problem can be found here: [1]
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
| 1973 IMO (Problems) • Resources | ||
| Preceded by First Question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
| All IMO Problems and Solutions | ||
