I have no idea to solve this problem about circles and the intermediate value theorem

Question

I have looked at this problem and literally cannot solve it. Please help.

It’s number 26

Answer 1

It's entirely in the hint.

1) Let $\theta$ be be an angle in the circle. Every point on the circle corresponds to an angle in the circle.

2) If $\theta$ corresponds to one point then $\theta + \pi$ corresponds to the point on the opposite side of the circle.

3) Let $f(\theta)$ be the difference in temperature of the point and the temperature of the opposite point. (The the temperature at the point corresponding at $\theta$ and subtract the temperature at the point corresponding at $\theta + \pi$, the opposite side.

Now, simple observations.

a) $f$ is continuous.

Temperature is a continuum based on location and so the differences in two temperatures based on location will be a continuum too.

b) $f(\theta+\pi) = -f(\theta)$

Let $T_1$ be the temperature at the point corresponding to $\theta$ and let $T_2$ be the temperature at the point corresponding to $\theta +\pi$.

The $f(\theta) = T_1 - T_2$.

And $f(\theta + \pi) = T_2 - T_1$.

c) The temperature and point $\theta$ and at the opposite point $\theta + \pi$ are equal if and only if the differences in temperature is zero if and only if $f(\theta) = 0$.

.....

So now it is up to you to prove there is some angle $\theta$ where $f(\theta) = 0$.

And to do that you have

$A-\Omega$ THE INTERMEDIATE VALUE THEOREM

If $f$ is a continuous function on a closed interval $[a,b]$ and $t: \min(f(a),f(b))\le t \le \max(f(a),f(b))$ then there is $c: a \le c \le b$ so that $f(c) = t$

TRUMPETS AND FANFARE

Well, our difference of temperature function is continuous; And it is continuous on $[0,\pi]$; and if $f(0) = r$ then $f(\pi) = -r$ and $\min(f(0),f(\pi)) = -|r| \le 0 \le |r|$..... so........

....There is a $\theta: 0 \le \theta\le \pi$ so that $f(\theta) = 0$ so the difference between the temperature at $\theta$ and the temperature at the point opposite of $\theta$ is $0$, so the temperatures at $\theta$ and the point opposite of $\theta$ are the same.

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Analogies:

Supppose you walk from the bottom of a mountain to to the top of a mountain. Suppose at the same time that you start another person walks from the top of the mountain to the bottom on the same path. At the end of the day you are at the top and she is at the bottom. Do you two ever meet?

Well obviously. You are on the same path and in the beginning you were below and she was above. In the end you were above and she was below. Ther had to be a point where you cross.

Suppose you are a point $A$ in a circle and it is hot. There is a person at point $B$ on the opposite side of the circle and it is cold. You are both on cell phones to each other.

You both decide to walk around the circle clock-wise. Each step of the way you calculate which one of you is hotter. At the beginning it is you who is hotter and $A$ and she is colder at $B$. Half way around the circle you have reached $B$ and she has reached $A$. It is now she who is hotter at $A$ than you who is colder at $B$.

Was there some point where you were both the same? Is it possible you went from hotter to colder in an instant without passing through a point where you were both the same?