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jlacosta   0
May 1, 2025
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0 replies
1 viewing
jlacosta
May 1, 2025
0 replies
Popular children at camp with algebra and geometry
Assassino9931   2
N 11 minutes ago by cj13609517288
Source: RMM Shortlist 2024 C3
Fix an odd integer $n\geq 3$. At a maths camp, there are $n^2$ children, each of whom selects
either algebra or geometry as their favourite topic. At lunch, they sit at $n$ tables, with $n$ children
on each table, and start talking about mathematics. A child is said to be popular if their favourite
topic has a majority at their table. For dinner, the students again sit at $n$ tables, with $n$ children
on each table, such that no two children share a table at both lunch and dinner. Determine the
minimal number of young mathematicians who are popular at both mealtimes. (The minimum is across all sets of topic preferences and seating arrangements.)
2 replies
Assassino9931
Friday at 11:07 PM
cj13609517288
11 minutes ago
Interesting inequalities
sqing   1
N 19 minutes ago by sqing
Source: Own
Let $ a,b> 0 $ and $  a^2+ab+b^2=a+b   $. Prove that
$$   \frac{a }{2b^2+1}+ \frac{b }{2a^2+1}+ \frac{1}{2ab+1} \geq \frac{21}{17}$$Let $ a,b> 0 $ and $ a^2+ab+b^2=a+b+1   $. Prove that
$$   \frac{a }{2b^2+1}+ \frac{b }{2a^2+1}+ \frac{1}{2ab+1} \geq1$$
1 reply
sqing
39 minutes ago
sqing
19 minutes ago
Interesting inequalities
sqing   0
29 minutes ago
Source: Own
Let $ a,b> 0 $ and $  a^2+ab+b^2=k(a+b)   $. Prove that
$$   \frac{a }{2b^2+1}+ \frac{b }{2a^2+1}+ \frac{1}{2ab+1} \geq \frac{12k+9}{8k^2+9}$$Where $ k\in N^+.$
Let $ a,b> 0 $ and $  a^2+ab+b^2=3(a+b)   $. Prove that
$$   \frac{a }{2b^2+1}+ \frac{b }{2a^2+1}+ \frac{1}{2ab+1} \geq \frac{5}{9}$$
0 replies
sqing
29 minutes ago
0 replies
Queue geo
vincentwant   6
N 36 minutes ago by ihategeo_1969
Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $Y, Z$ be the feet of the altitudes from $B, C$ to $AC, AB$ respectively. Let $D$ be the midpoint of $BC$. Let $\omega_1$ be the circle with diameter $AD$. Let $Q\neq A$ be the intersection of $(ABC)$ and $\omega$. Let $H$ be the orthocenter of $ABC$. Let $K$ be the intersection of $AQ$ and $BC$. Let $l_1,l_2$ be the lines through $Q$ tangent to $\omega,(AYZ)$ respectively. Let $I$ be the intersection of $l_1$ and $KH$. Let $P$ be the intersection of $l_2$ and $YZ$. Let $l$ be the line through $I$ parallel to $HD$ and let $O'$ be the reflection of $O$ across $l$. Prove that $O'P$ is tangent to $(KPQ)$.
6 replies
vincentwant
Apr 30, 2025
ihategeo_1969
36 minutes ago
No more topics!
cos k theta and cos(k + 1) theta are both rational
N.T.TUAN   12
N Apr 12, 2025 by Ilikeminecraft
Source: USA Team Selection Test 2007
Let $ \theta$ be an angle in the interval $ (0,\pi/2)$. Given that $ \cos \theta$ is irrational, and that $ \cos k \theta$ and $ \cos[(k + 1)\theta ]$ are both rational for some positive integer $ k$, show that $ \theta = \pi/6$.
12 replies
N.T.TUAN
Dec 8, 2007
Ilikeminecraft
Apr 12, 2025
cos k theta and cos(k + 1) theta are both rational
G H J
Source: USA Team Selection Test 2007
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N.T.TUAN
3595 posts
#1 • 4 Y
Y by Davi-8191, Adventure10, Mango247, cubres
Let $ \theta$ be an angle in the interval $ (0,\pi/2)$. Given that $ \cos \theta$ is irrational, and that $ \cos k \theta$ and $ \cos[(k + 1)\theta ]$ are both rational for some positive integer $ k$, show that $ \theta = \pi/6$.
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epitomy01
240 posts
#2 • 3 Y
Y by Adventure10, Mango247, cubres
Here's an idea:
We can prove easily by induction, that for each integer $ k$ there exists a polynomial $ R_{k}$ so that $ R_{k} (cos x) = cos (kx)$.
Suppose $ x = cos \theta$ is irrational. Then $ R_{k} (x) = - p$ and $ R_{k + 1} (x) = - q$ are both rational. Thus $ R_{k} (x) + p$ and $ R_{k + 1} (x) + q$ are polynomials with rational coefficients, which have zeros at $ x$. Consider the monic polynomial with rational coefficients $ Q$ of minimal degree such that $ S(x) = 0$. Using the well-known Lemma:[ if for an algebraic number $ x$ the rational polynomial of minimal degree which has $ x$ as a 0 is $ Q(x)$, then if $ R(x) = 0$ for some rational polynomial $ R(x)$, we have $ Q(x)$ is a factor of $ R(x)$.], we have $ Q(x)$ is a factor of both $ R_{k} (x) + p$ and $ R_{k + 1} (x) + q$, so $ R(x)$ is a factor of $ (R_{k} (x) + p, R_{k + 1} (x) + q)$.
Since $ x$ is irrational, the degree of $ Q(x)$ is at least $ 2$. Our goal is to prove that the degree of $ (R_{k} (x) + p, R_{k + 1} (x) + q)$, and hence $ Q(x)$ is $ 2$. To do that, it may be possible to apply Euclidean algorithms to those 2 polynomials. Hopefully, there will be a special relationship between the coefficients of the polynomials that we obtain, which will make this easier to do. I think we can prove, using this method that the degree of $ (R_{k} (x) + p, R_{k + 1} (x) + q)$ is at most $ 2$, and if it is $ 2$ then $ p$ and $ q$ must take specific values (depending on the value of $ k (mod 6)$). If $ k \equiv 1 (mod 3)$, using this method I think we should get a contradictory relationships between $ p,q$, if the degree of the LCD polynomial is 2, that means $ k \equiv 1 (mod 3)$ is not possible. Once, for a certain value of $ k$, we find out what $ p$ and $ q$ are (one should be $ 0.5$ or $ - 0.5$, the other should be $ 0$), the problem will be finished.
----
Here's how we can kill $ k = 3$ using this method:
Since $ P_{3} (x) = 4x^3 - 3x$ and $ P_{4} (x) = 8x^4 - 8x^2 + 1$, let $ 4x^3 - 3x + a = 0, 8x^4 - 8x^2 + b = 0$ for rational $ a,b$. Then if $ Q(x)$ is minimal integer polynomial of $ x$, $ Q(x)$ divides $ 4x^3 - 3x + a$ and $ 8x^4 - 8x^2 + b$, $ Q(x)$ has degree at least $ 2$. We have:
$ (4x^3 - 3x + a, 8x^4 - 8x^2 + b) = (4x^3 - 3x + a, 2x^2 + 2ax - b) = (2x^2 + 2ax - b, 4ax^2 - (2b - 3) x - a)$.
Here we have used typical Euclidean algorithm: using fact that $ 8x^4 - 8x^2 + b = (4x^3 - 3x + a)2x - (2x^2 + 2ax - b)$, etc. Since the GCD of those 2 polynomials, degree $ 2$, is a polynomial of degree $ 2$, either the $ 2$-nd one is $ 0$ - giving us $ b = 3/2, a = 0$ from where we get that $ cos (3 \theta) = 0$, $ cos (4 \theta) = - 0.5$ and $ \theta = \frac {\pi}{6}$ as required: OR the second quadratic is a rational multiple of the first. In this case, a routine calculation gives a contradic tion.
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chess
43 posts
#3 • 3 Y
Y by Adventure10, Mango247, cubres
N.T.TUAN, excuse me, where did you find USA TST 2007? :?:
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epitomy01
240 posts
#4 • 3 Y
Y by Adventure10, Mango247, cubres
cheese: they've been posted here: http://www.mathlinks.ro/viewtopic.php?t=177771
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Little Gauss
200 posts
#5 • 2 Y
Y by Adventure10, cubres
Hm.. in last year, I saw this problem and tried to solve this problem by same method as yours.... I checked it for $ k=3,4,5$ but I failed to prove for all cases.
in my case, It isn't enough to prove it, but it needs more facts such as $ |x|\le 1$...., please post details.. I want to see a solution of this problem.
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Little Gauss
200 posts
#6 • 3 Y
Y by Adventure10, Mango247, cubres
I don't know why I couldn't think about this......
Let $ T_{n}$ be a integer coefficient polynomial such that $ T_{n}(\cos \theta) = \cos n\theta$.
It is easy to prove that $ T_{n + 1}(x) + T_{n - 1}(x) = 2xT_{n}(x)$,
Therfore $ deg( T_{n}) = n$ , $ [x^{n}]T_{n} = 2^{n - 1}$ , $ [x^{n - 2}]T_{n} = - 2^{n - 3}n$ .........

Let $ T_{k}(\cos \theta) = p$, $ T_{k + 1}(\cos \theta) = q$. It is easy to prove that
if $ R(x) = x^{2} - 2pqx + p^{2} + q^{2} - 1$, $ R(\cos \theta) = R(\cos (2k + 1)\theta) = 0$....
It is obvious that $ R$ is a minimal degree polynomial which has root $ \cos \theta$.

So $ R(x)|T_{k}(x) - p, T_{k + 1}(x) - q$,
$ \cos k\theta = \cos (2k^{2} + k)\theta = p$,
$ \cos (k + 1)\theta = \cos (2k + 1)(k + 1)\theta = q$,

Let's Suppose that $ \frac {k(k + 1)\theta}{\pi} \not\in \mathbb{Z}$.
then $ \frac {k^{2}\theta}{\pi}, \frac {(k + 1)^{2}\theta}{\pi} \in \mathbb{Z}$
$ \Rightarrow \frac {(2k + 1)\theta}{\pi}\in\mathbb{Z} \rightarrow \cos (2k + 1)\theta = \pm 1$, but $ \cos \theta$ is irrational, contradiction!

Now $ \frac {k(k + 1)\theta}{\pi} \in \mathbb{Z}$.
$ \Rightarrow T_{k}(q) = T_{k + 1}(p) = \pm 1$.
$ q = \frac {a}{b}$ ($ (a,b) = 1$). if $ k\ge3$, then we can easily show that $ q = 0$ or $ a = \pm 1$ and $ b = \pm 2^{i}$.
Therfore $ p, q = \pm \frac {1}{2^{i}}$ or $ 0$.

If one can prove that $ T_{n}(\frac {1}{2^{i}})\not = \pm1$ if $ i\ge 2$, this is the end..
I think it isn't very hard.... but I don't have times to check it..
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hxy09
602 posts
#7 • 3 Y
Y by Adventure10, Mango247, cubres
Little Gauss wrote:
I don't know why I couldn't think about this......
Let $ T_{n}$ be a integer coefficient polynomial such that $ T_{n}(\cos \theta) = \cos n\theta$.
It is easy to prove that $ T_{n + 1}(x) + T_{n - 1}(x) = 2xT_{n}(x)$,
Therfore $ deg( T_{n}) = n$ , $ [x^{n}]T_{n} = 2^{n - 1}$ , $ [x^{n - 2}]T_{n} = - 2^{n - 3}n$ .........

Let $ T_{k}(\cos \theta) = p$, $ T_{k + 1}(\cos \theta) = q$. It is easy to prove that
if $ R(x) = x^{2} - 2pqx + p^{2} + q^{2} - 1$, $ R(\cos \theta) = R(\cos (2k + 1)\theta) = 0$....
It is obvious that $ R$ is a minimal degree polynomial which has root $ \cos \theta$.

So $ R(x)|T_{k}(x) - p, T_{k + 1}(x) - q$,
$ \cos k\theta = \cos (2k^{2} + k)\theta = p$,
$ \cos (k + 1)\theta = \cos (2k + 1)(k + 1)\theta = q$,

Let's Suppose that $ \frac {k(k + 1)\theta}{\pi} \not\in \mathbb{Z}$.
then $ \frac {k^{2}\theta}{\pi}, \frac {(k + 1)^{2}\theta}{\pi} \in \mathbb{Z}$
$ \Rightarrow \frac {(2k + 1)\theta}{\pi}\in\mathbb{Z} \rightarrow \cos (2k + 1)\theta = \pm 1$, but $ \cos \theta$ is irrational, contradiction!

Now $ \frac {k(k + 1)\theta}{\pi} \in \mathbb{Z}$.
$ \Rightarrow T_{k}(q) = T_{k + 1}(p) = \pm 1$.
$ q = \frac {a}{b}$ ($ (a,b) = 1$). if $ k\ge3$, then we can easily show that $ q = 0$ or $ a = \pm 1$ and $ b = \pm 2^{i}$.
Therfore $ p, q = \pm \frac {1}{2^{i}}$ or $ 0$.

If one can prove that $ T_{n}(\frac {1}{2^{i}})\not = \pm1$ if $ i\ge 2$, this is the end..
I think it isn't very hard.... but I don't have times to check it..
A wondeful approach :lol:
We needn't to prove $ T_{n}(\frac {1}{2^{i}})\not = \pm1$ if $ i\ge 2$
We just take your conclusion $ \frac {k(k + 1)\theta}{\pi} \in \mathbb{Z}$
We get:$ \theta$ is a rational multiple of $ \pi$
From $ R$ is a minimal degree polynomial which has root $ \cos \theta$ yields $ \cos \theta$ is a number whose algebraic degree is 2
From an old assertion of Hsiao (easy to prove)
$ \theta = \frac {p}{q}\pi$ that $ q = 4,5,6$
Easy to check $ q\neq 4,5$ yields $ q = 6,p = 1$ that finished our proof :)
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silversheep
77 posts
#8 • 8 Y
Y by v_Enhance, djmathman, Aryan-23, rayfish, Adventure10, Mango247, cubres, and 1 other user
The problem statement is still true if $k+1$ is replaced by an integer $l$ relatively prime to $k$.

Lemma 1
The only rational $p\in [0,1]$ such that $\cos p\pi$ is rational are $p=0, \frac{1}{3},\frac{1}{2},\frac{2}{3},1$.
Proof
Let $\omega=\cos (p\pi)+i\sin(p\pi)$. Then $\omega$ is a root of unity and hence an algebraic integer; similarly $\overline{\omega}$ is as well. Hence $2\cos(p\pi) =\omega + \overline{\omega}$ is an algebraic integer; since it is rational it is a rational (i.e. regular) integer. Hence we must have $2\cos(p\pi)=0,\pm 1,\pm 2$, giving $p=0, \frac{1}{3},\frac{1}{2},\frac{2}{3},1$.

Lemma 2
If $\cos k\theta$ and $\cos l\theta $ are rational for relatively prime positive integers $k,l$, then either $\cos \theta $ is rational or $\theta$ is a rational multiple of $\pi$.
Proof
Let $\cos k\theta=p$ and $\cos l\theta =q$. Noting that $e^{i\phi}+\frac{1}{e^{i\phi}}=2\cos \phi$, we have that $e^{i\theta}$ is a root of
\begin{align}
x^k+\frac{1}{x^k}=2p&\implies x^{2k}-2px^k+1=0\\
x^l+\frac{1}{x^l}=2q&\implies x^{2l}-2qx^l+1=0
\end{align}
Suppose $\theta$ is not a rational multiple of $\pi$. Then the numbers $e^{\pm i\theta+\frac{2\pi i j}{k}}$ for $0\leq j<k$ are all distinct. They are the $2k$ roots of (1). Similarly, $e^{\pm i\theta+\frac{2\pi i j}{l}}$ are the $2l$ roots of (2). Since $\theta$ is not a rational multiple of $\pi$ and $k,l$ are relatively prime, we have that the only solution to $e^{\pm i\theta+\frac{2\pi i j_1}{k}}= e^{\pm i\theta+\frac{2\pi i j_2}{l}}$ for $0\leq j_1<k$ and $0\leq j_2<l$ for some choices of signs is when $j_1=j_2=0$. Thus the only roots the two polynomials share in common are $\omega=e^{i\theta}$ and $\bar{\omega}=-e^{i\theta}$, and the polynomial
$g(x):=(x-e^{i\theta})(x-e^{-i\theta})$
is the greatest common divisor of polynomials $x^{2k}-2px^k+1$ and $x^{2l}-2qx^l+1$, which have rational coefficients. Hence $g(x)$ has rational coefficients, and $e^{i\theta}+e^{-i\theta}=2\cos \theta$ is rational.

By Lemma 2, $\theta$ is a rational multiple of $\pi$. Since by Lemma 1 the only rational multiples of $\pi$ that give a rational values for cosine are multiples of $\frac{\pi}{6}$, $k\theta$ and $l\theta$ are both multiples of $\frac{\pi}{6}$. By Bezout's Theorem $\theta=m(k\theta)+n(l\theta)$ for some integers $m,n$, so $\theta$ is a multiple of $\frac{\pi}{6}$. Since $\cos\theta$ is irrational, $\theta=\frac{\pi}{6}$.

Note: This solution is based on the idea of the solution to the problem here: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=589&t=290391
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62861
3564 posts
#9 • 3 Y
Y by pieater314159, Adventure10, cubres
Let $x = \cos \theta + i \sin \theta$, and let $z_m = x^m + \tfrac{1}{x^m} = 2 \cos m\theta$ for all integer $m$. We are told that $z_k$ and $z_{k + 1}$ are both rational but $z_1$ is not.

Let $T_n(z)$ be the monic polynomial with integer coefficients satisfying $T_n(2 \cos x) = 2 \cos nx$ for all $x$.

Let $q(z) = \gcd(T_n(z) - z_n, T_{n + 1}(z) - z_{n + 1})$. Then $q$ has rational coefficients and $z_1$ is a root of $q$. Thus, if $\deg q = 1$, then $z_1$ is rational, contradiction.

Thus, $\deg q \ge 2$. If $q$ has root $z_1$ with multiplicity $\deg q$, then it's easy to see $z_1$ is rational, contradiction. Thus, $q$ has a root $w \neq z_1$.

Let $y$ be a number with $y + \tfrac{1}{y} = w$. Then since $x + \tfrac{1}{x} = z_1 \neq w = y + \tfrac{1}{y}$, $x \neq y$ and $xy \neq 1$.

Now, $x^k + \tfrac{1}{x^k} = y^k + \tfrac{1}{y^k} = z_k$ and $x^{k + 1} + \tfrac{1}{x^{k + 1}} = y^{k + 1} + \tfrac{1}{y^{k + 1}} = z_{k + 1}$. Thus, we obtain
  • $x^k = y^k$ or $(xy)^k = 1$, and
  • $x^{k + 1} = y^{k + 1}$ or $(xy)^{k + 1} = 1$.
By setting $y \to \tfrac{1}{y}$ if necessary, assume $x^k = y^k$. Then $x^{k + 1} = y^{k + 1}$ implies $x = y$, bad. Thus $(xy)^{k + 1} = 1$.

Since $(x/y)^k = (xy)^{k + 1} = 1$, both $x/y$ and $xy$ are roots of unity, so $x$ and $y$ are too. It follows that angles $\alpha_1 = k \theta$ and $\alpha_2 = (k + 1) \theta$ are a rational number of degrees and have rational cosines, so
\[\cos k\theta, \cos (k + 1)\theta \in \{-1, -\tfrac{1}{2}, 0, \tfrac{1}{2}, 1\}.\]It follows that $k \theta$ and $(k + 1) \theta$ are both multiples of $30^{\circ}$, so $\theta$ is too.

Since $\theta \in (0^{\circ}, 90^{\circ})$, we obtain $\theta = 30^{\circ}$ or $\theta = 60^{\circ}$. Since $\cos 60^{\circ}$ is rational, it follows that $\theta = 30^{\circ}$.
The solution also works for the following problem:
Quote:
Let $ \theta$ be an angle in the interval $ (0,\pi/2)$. Given that $ \cos \theta$ is irrational, and that $ \cos p \theta$ and $ \cos q \theta$ are both rational for some relatively prime positive integers $ p, q$, show that $ \theta = \pi/6$.
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IAmTheHazard
5001 posts
#10 • 1 Y
Y by cubres
Nice problem

The difficulty of the problem lies in proving that $\theta$ is a rational multiple of $\pi$. Suppose otherwise, and let $z=\operatorname{cis}(\theta)$. Then since $\cos k\theta:=q_1$ is rational, we have
$$\operatorname{Re}(z^k)=q_1 \implies z^k+z^{-k}=2q_1 \implies z^{2k}-2q_1z^k+1=0,$$so $z$ is a root of $P_1(x):=x^{2k}-2q_1x^k+1$. Likewise, $z$ is a root of $P_2(x):=x^{2k+2}-2q_2x^{k+1}+1$, where $\cos (k+1)\theta=q_2$, so it's a root of $\gcd(P_1(x),P_2(x)) \in \mathbb{Q}[x]$.
It turns out that we can find all roots of $P_1(x)$ without great difficulty: observe that all numbers of the form $\operatorname{cis}(\pm\theta+\tfrac{2n\pi}{k})$ are roots, where $n=0,\ldots,2k-1$. Since there are $2n$ of these (all such roots are distinct because $\theta$ isn't a rational multiple of $\pi$), they are actually precisely the roots of $P_1$. Likewise, the roots of $P_2$ are precisely $\operatorname{cis}(\pm \theta+\frac{2n\pi}{k+1})$. Because $\theta$ isn't a rational multiple of $\pi$ and $k \perp k+1$, the only shared roots between $P_1$ and $P_2$ are $\operatorname{cis}(\pm\theta)=z,\overline{z}$. Thus
$$\gcd(P_1(x),P_2(x))=(x-z)(x-\overline{z}) \in \mathbb{Q}[x],$$which implies that $z+\overline{z} \in \mathbb{Q} \implies \operatorname{Re}(z)=\cos \theta \in \mathbb{Q}$: illegal. Thus $\theta$ must be a rational multiple of $\pi$.

By Niven's Theorem, if $\phi$ is a rational multiple of $\pi$ and $\cos(\phi) \in \mathbb{Q}$, then $\phi \in \{0,\pi/3,\pi/2,2\pi/3\} \pmod{\pi}$, so modulo $\pi$ the only possible differences between two rational multiples of $\pi$ whose cosines are rational are $0,\pi/6,\pi/3,\pi/2,2\pi/3$. Since the cosines of $k\theta$ and $(k+1)\theta$ are both rational, it follows that
$$(k+1)\theta-k\theta=\theta \in \{0,\pi/6,\pi/3,\pi/2,2\pi/3\} \pmod{\pi},$$so if $\theta \in (0,\pi/2)$ then we must have $\theta \in \{\pi/6,\pi/3\}$. But $\cos \pi/3$ is rational, hence $\theta=\pi/6$ as desired. $\blacksquare$
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Saucepan_man02
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#11 • 1 Y
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Here we go:

If $\theta$ is a rational multiple of $\pi$, by Niven's theorem, $k \theta , (k+1) \theta \in \{0,\pi/3,\pi/2,2\pi/3\} \pmod{\pi} $ Thus, their difference $(k+1) \theta - k \theta = \theta \in \{0,\pi/6,\pi/3,\pi/2,2\pi/3\} \pmod \pi$. Since $\theta \in (0, \pi/2)$, and $\cos \theta$ is irrational, $\theta = \pi/6$.

If $\theta$ is not a rational multiple of $\pi $: Consider the following two polynomials: $$P(x)= x^{2k}-2 \cos (k \theta) x^k+1, G(x) = x^{2(k+1)}-2 \cos ((k+1) \theta) x^{k+1}+1.$$Their roots are respectively: $$z = \text{cis}\left( \pm \theta + \frac{2\pi s}{k} \right)$$$$ z =  \text{cis} \left( \pm \theta + \frac{2\pi t}{k+1} \right).$$Since $\theta$ is not a rational multiple of $\pi$: the only common roots are $\text{cis} ( \pm \theta )$. Thus: $H(x) = \gcd(P, G) = (x-\alpha)(x-\bar{\alpha})$ where $\alpha = \text{cis}(\theta)$. Since $P, G \in \mathbb Q [x ]$, we have: $H \in \mathbb Q [ x ]$. Thus: $\alpha + \bar{\alpha} = 2 \cos \theta \in \mathbb Q$ which contradicts that $\cos \theta$ is irrational.
This post has been edited 1 time. Last edited by Saucepan_man02, Jan 15, 2025, 1:35 AM
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Mathandski
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#12 • 1 Y
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Please contact westskigamer@gmail.com if there is an error with any of my solution for cash bounties by 3/18/2025.
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Ilikeminecraft
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#13 • 1 Y
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Assume $\theta$ wasn’t a rational multiple of $\pi.$ Denote $z = \cos\theta + i\sin\theta$
Consider the polynomials $P(X) = X^{2k} - 2\cos(k\theta) X^k + 1$ and $Q(X) = X^{2k + 2} - 2\cos[(k + 1)\theta]X^{k + 1} + 1.$
Note that $P$ has zeroes when $X^k = e^{\pm \pi i k\theta}$ and $Q$ has zeroes when $X^{k + 1} = e^{\pm \pi i(k+1)\theta}.$
It is obvious that they only share the roots $z, z^{-1}.$
Hence, $\gcd(P, Q) = (x - z)(x - z^{-1}) = x^2 - 2\cos\theta x + 1.$ However, note that $P, Q \in \mathbb Q[x]$ from the given constraints. Therefore, the gcd, $x^2 - 2\cos\theta x + 1$ must also be in $\mathbb Q[x].$ This implies that $\cos\theta\in\mathbb Q,$ which gives us a contradiction.

Thus, $\theta$ is a rational multiple of $\pi.$ By Nixen, it follows $\cos(k\theta) = \{-1,-1/2,0,1/2,1\}.$ Obviously, this implies $\cos\theta = \frac{\sqrt3}{2},$ or $\theta = \frac\pi6.$
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