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Difference between revisions of "1969 IMO Problems/Problem 2"

m
Line 11: Line 11:
 
This implies that if <math>f(x_1)=0</math>,
 
This implies that if <math>f(x_1)=0</math>,
 
<cmath>\tan{x_1}=\frac{\cos{a_1}+\frac{1}{2}\cos{a_2}+\frac{1}{4}\cos{a_3}+\cdots+\frac{1}{2^{n-1}}\cos{a_n}}{\sin{a_1}+\frac{1}{2}\sin{a_2}+\frac{1}{4}\sin{a_3}+\cdots+\frac{1}{2^{n-1}}\sin{a_n}}</cmath>
 
<cmath>\tan{x_1}=\frac{\cos{a_1}+\frac{1}{2}\cos{a_2}+\frac{1}{4}\cos{a_3}+\cdots+\frac{1}{2^{n-1}}\cos{a_n}}{\sin{a_1}+\frac{1}{2}\sin{a_2}+\frac{1}{4}\sin{a_3}+\cdots+\frac{1}{2^{n-1}}\sin{a_n}}</cmath>
−
Since the period of <math>\tan{x}</math> is <math>\pi</math>, this means that <math>\tan{x_1}=\tan{x_1+\pi}=\tan{x_1+m\pi}</math> for any natural number <math>m</math>. That implies that every value <math>x_1+m\pi</math> is a zero of <math>f(x)</math>.
+
Since the period of <math>\tan{x}</math> is <math>\pi</math>, this means that <math>\tan{x_1}=\tan{(x_1+\pi)}=\tan{(x_1+m\pi)}</math> for any natural number <math>m</math>. That implies that every value <math>x_1+m\pi</math> is a zero of <math>f(x)</math>.
 +
 
 +
==Remarks (added by pf02, August 2024)==
 +
 
 +
Both solutions given above are incorrect.
 +
 
 +
The first solution is hopelessly incorrect.  It states that (and relies on it)
 +
if <math>f(x)</math> has period <math>2\pi</math> and <math>f(x_1) = f(x_2)</math> then <math>x_2 - x_1 = m\pi</math> for
 +
some integer <math>m</math>.  This is plainly wrong (think of <math>\sin{\pi/3} = \sin{2\pi/3}</math>).
 +
There is an obvious "red flag" as far as solutions go, namely the solution did
 +
not use that <math>f(x_1) = 0</math> and <math>f(x_2) = 0</math>.
 +
 
 +
The second solution starts promising, but then it goes on to prove the converse
 +
of the given problem, namely that if <math>f(x_1) = 0</math> then <math>f(x_1 + m\pi) = 0</math> for
 +
any <math>m</math>.
 +
 
 +
Below, I will give a solution to the problem.  I feel a little uneasy about it,
 +
it has an "orange flag", namely I make no use of the fact that the coefficients
 +
if <math>\cos{(a_k + x)}</math> are the given powers of <math>1/2</math>.
 +
 
 +
==Solution==
 +
 
 +
Do the proof for the slightly more general function
 +
<math>f(x) = b_1\cos{(a_1 + x)} + b_1\cos{(a_1 + x)} + \cdots + b_n\cos{(a_n + x)}</math>
 +
 
 +
Just like the previous attempt to solve the problem, start by using the formula
 +
<math>\cos{(\alpha + \beta)} = \cos{\alpha}\cos{\beta} - \sin{\alpha}\sin{\beta}</math>.
 +
 
 +
We get
 +
<math>f(x)=(b_1\cos{a_1} + b_2\cos{a_2} + \cdots + b_n\cos{a_n})\cos{x} -
 +
b_1\sin{a_1} + b_2\sin{a_2} + \cdots + b_n\sin{a_n})\sin{x}</math>.
 +
 
 +
 
 +
 
 +
 
 +
 
 +
 
 +
TO BE CONTINUED. I AM JUST SAVING NOW SO THAT I DON'T LOSE WORK DONE SO FAR.
  
 
==See Also==
 
==See Also==
  
 
{{IMO box|year=1969|num-b=1|num-a=3}}
 
{{IMO box|year=1969|num-b=1|num-a=3}}

Revision as of 14:49, 8 August 2024

Problem

Let $a_1, a_2,\cdots, a_n$ be real constants, $x$ a real variable, and \[f(x)=\cos(a_1+x)+\frac{1}{2}\cos(a_2+x)+\frac{1}{4}\cos(a_3+x)+\cdots+\frac{1}{2^{n-1}}\cos(a_n+x).\] Given that $f(x_1)=f(x_2)=0,$ prove that $x_2-x_1=m\pi$ for some integer $m.$

Solution

Because the period of $\cos(x)$ is $2\pi$, the period of $f(x)$ is also $2\pi$. \[f(x_1)=f(x_2)=f(x_1+x_2-x_1)\] We can get $x_2-x_1 = 2k\pi$ for $k\in N^*$. Thus, $x_2-x_1=m\pi$ for some integer $m.$

Solution 2 (longer)

By the cosine addition formula, \[f(x)=(\cos{a_1}+\frac{1}{2}\cos{a_2}+\frac{1}{4}\cos{a_3}+\cdots+\frac{1}{2^{n-1}}\cos{a_n})\cos{x}-(\sin{a_1}+\frac{1}{2}\sin{a_2}+\frac{1}{4}\sin{a_3}+\cdots+\frac{1}{2^{n-1}}\sin{a_n})\sin{x}\] This implies that if $f(x_1)=0$, \[\tan{x_1}=\frac{\cos{a_1}+\frac{1}{2}\cos{a_2}+\frac{1}{4}\cos{a_3}+\cdots+\frac{1}{2^{n-1}}\cos{a_n}}{\sin{a_1}+\frac{1}{2}\sin{a_2}+\frac{1}{4}\sin{a_3}+\cdots+\frac{1}{2^{n-1}}\sin{a_n}}\] Since the period of $\tan{x}$ is $\pi$, this means that $\tan{x_1}=\tan{(x_1+\pi)}=\tan{(x_1+m\pi)}$ for any natural number $m$. That implies that every value $x_1+m\pi$ is a zero of $f(x)$.

Remarks (added by pf02, August 2024)

Both solutions given above are incorrect.

The first solution is hopelessly incorrect. It states that (and relies on it) if $f(x)$ has period $2\pi$ and $f(x_1) = f(x_2)$ then $x_2 - x_1 = m\pi$ for some integer $m$. This is plainly wrong (think of $\sin{\pi/3} = \sin{2\pi/3}$). There is an obvious "red flag" as far as solutions go, namely the solution did not use that $f(x_1) = 0$ and $f(x_2) = 0$.

The second solution starts promising, but then it goes on to prove the converse of the given problem, namely that if $f(x_1) = 0$ then $f(x_1 + m\pi) = 0$ for any $m$.

Below, I will give a solution to the problem. I feel a little uneasy about it, it has an "orange flag", namely I make no use of the fact that the coefficients if $\cos{(a_k + x)}$ are the given powers of $1/2$.

Solution

Do the proof for the slightly more general function $f(x) = b_1\cos{(a_1 + x)} + b_1\cos{(a_1 + x)} + \cdots + b_n\cos{(a_n + x)}$

Just like the previous attempt to solve the problem, start by using the formula $\cos{(\alpha + \beta)} = \cos{\alpha}\cos{\beta} - \sin{\alpha}\sin{\beta}$.

We get $f(x)=(b_1\cos{a_1} + b_2\cos{a_2} + \cdots + b_n\cos{a_n})\cos{x} - b_1\sin{a_1} + b_2\sin{a_2} + \cdots + b_n\sin{a_n})\sin{x}$.




TO BE CONTINUED. I AM JUST SAVING NOW SO THAT I DON'T LOSE WORK DONE SO FAR.

See Also

1969 IMO (Problems) • Resources
Preceded by
Problem 1 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 3
All IMO Problems and Solutions
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