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China Team Selection Test
2023 China Team Selection Test
P6
P6
Part of
2023 China Team Selection Test
Problems
(1)
2023 China TST Problem 6
Source: 2023 China TST Problem 6
3/14/2023
Prove that: (1) In the complex plane, each line (except for the real axis) that crosses the origin has at most one point
z
{z}
z
, satisfy
1
+
z
23
z
64
∈
R
.
\frac {1+z^{23}}{z^{64}}\in\mathbb R.
z
64
1
+
z
23
∈
R
.
(2) For any non-zero complex number
a
{a}
a
and any real number
θ
\theta
θ
, the equation
1
+
z
23
+
a
z
64
=
0
1+z^{23}+az^{64}=0
1
+
z
23
+
a
z
64
=
0
has roots in
S
θ
=
{
z
∈
C
∣
Re
(
z
e
−
i
θ
)
⩾
∣
z
∣
cos
π
20
}
.
S_{\theta}=\left\{ z\in\mathbb C\mid\operatorname{Re}(ze^{-i\theta })\geqslant |z|\cos\frac{\pi}{20}\right\}.
S
θ
=
{
z
∈
C
∣
Re
(
z
e
−
i
θ
)
⩾
∣
z
∣
cos
20
π
}
.
Proposed by Yijun Yao
algebra
China TST