Difference between revisions of "1969 IMO Problems/Problem 2"
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==Solution== | ==Solution== | ||
− | Because the period of <math>cos(x)</math> is <math>2\pi</math>, the period of <math>f(x)</math> is also <math>2\pi</math>. | + | Because the period of <math>\cos(x)</math> is <math>2\pi</math>, the period of <math>f(x)</math> is also <math>2\pi</math>. |
<cmath>f(x_1)=f(x_2)=f(x_1+x_2-x_1)</cmath> | <cmath>f(x_1)=f(x_2)=f(x_1+x_2-x_1)</cmath> | ||
We can get <math>x_2-x_1 = 2k\pi</math> for <math>k\in N^*</math>. Thus, <math>x_2-x_1=m\pi</math> for some integer <math>m.</math> | We can get <math>x_2-x_1 = 2k\pi</math> for <math>k\in N^*</math>. Thus, <math>x_2-x_1=m\pi</math> for some integer <math>m.</math> | ||
+ | ==Solution 2 (longer)== | ||
+ | By the cosine addition formula, | ||
+ | <cmath>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}</cmath> | ||
+ | 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> | ||
+ | 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>. | ||
==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 11:29, 4 April 2024
Problem
Let be real constants,
a real variable, and
Given that
prove that
for some integer
Solution
Because the period of is
, the period of
is also
.
We can get
for
. Thus,
for some integer
Solution 2 (longer)
By the cosine addition formula,
This implies that if
,
Since the period of
is
, this means that
for any natural number
. That implies that every value
is a zero of
.
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 |