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1973 IMO Problems/Problem 1

Revision as of 13:02, 15 May 2025 by Pf02 (talk | contribs)

Problem

Point $O$ lies on line $g;$ $\overrightarrow{OP_1}, \overrightarrow{OP_2},\cdots, \overrightarrow{OP_n}$ are unit vectors such that points $P_1, P_2, \cdots, P_n$ all lie in a plane containing $g$ and on one side of $g.$ Prove that if $n$ is odd, \[\left|\overrightarrow{OP_1}+\overrightarrow{OP_2}+\cdots+ \overrightarrow{OP_n}\right|\ge1.\] Here $\left|\overrightarrow{OM}\right|$ denotes the length of vector $\overrightarrow{OM}.$


Solution

We prove it by induction on the number $2n+1$ of vectors. The base step (when we have one vector) is clear, and for the induction step we use the hypothesis for the $2n-1$ vectors obtained by disregarding the outermost two vectors. We thus get a vector with norm $\ge 1$ betwen two with norm $1$. The sum of the two vectors of norm $1$ makes an angle of $\le\frac\pi 2$ with the vector of norm $\ge 1$, so their sum has norm $\ge 1$, and we're done.

The above solution was posted and copyrighted by grobber. The original thread for this problem can be found here: [1]


Remarks (added by pf02, May 2025)

1. The "solution" given above is so incomplete that it can not be called a solution. It simply shoves the difficulty of the problem into the phrase "... so their sum has norm $\ge 1$, and we're done." Indeed, we don't know that their sum has norm $\ge 1$. This is exactly what we have to prove. To make this difficulty clear, imagine $n = 3$ (I am referring to $n$ from the statement of the problem, not the one from the "solution", which should have been "k", so that $n = 2k - 1$; so the parameter from the proof is $k = 3$). Take $P_1 = (\cos \epsilon_1, \sin \epsilon_1)$, $P_2 = (\cos (\pi/3), \sin (\pi/3))$, $P_3 = (\cos (\pi - \epsilon_2), \sin (\pi - \epsilon_2))$, where $\epsilon_1, \epsilon_2$ are very small positive numbers (and, assume $O = (0, 0)$, and $g$ given by $y = 0$.) Then $\overrightarrow{OP_1} + \overrightarrow{OP_3}$ has a very small norm, and its direction depends on $\epsilon_1, \epsilon_2$. It is not clear at all that adding $\overrightarrow{OP_1} + \overrightarrow{OP_3}$ to $\overrightarrow{OP_2}$ has norm $\ge 1$. (It is true, but it needs a proof!)

2. Below, I will give a proof, which follows the idea from the proof given above in the sense that it uses induction, but it fills in the details of the essential step.

3. Then, I will give a second proof, which uses the same argument, but it proves the problem directly, without using induction.


Solution 2

[TO BE CONTINUED]


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
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