MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2021 China Second Round A1
2021 China Second Round A1
Part of
(China) National High School Mathematics League
Subcontests
(4)
4
1
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Coloring Points on a circle
There are 100 points on a circle that are about to be colored in two colors: red or blue. Find the largest number
k
k
k
such that no matter how I select and color
k
k
k
points, you can always color the remaining
100
−
k
100-k
100
−
k
points such that you can connect 50 pairs of points of the same color with lines in a way such that no two lines intersect.
3
1
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Sequence Problem
Let
{
a
n
}
\{a_n\}
{
a
n
}
,
{
b
n
}
\{b_n\}
{
b
n
}
be sequences of positive real numbers satisfying
a
n
=
1
100
∑
j
=
1
100
b
n
−
j
2
a_n=\sqrt{\frac{1}{100} \sum\limits_{j=1}^{100} b_{n-j}^2}
a
n
=
100
1
j
=
1
∑
100
b
n
−
j
2
and
b
n
=
1
100
∑
j
=
1
100
a
n
−
j
2
b_n=\sqrt{\frac{1}{100} \sum\limits_{j=1}^{100} a_{n-j}^2}
b
n
=
100
1
j
=
1
∑
100
a
n
−
j
2
For all
n
≥
101
n\ge 101
n
≥
101
. Prove that there exists
m
∈
N
m\in \mathbb{N}
m
∈
N
such that
∣
a
m
−
b
m
∣
<
0.001
|a_m-b_m|<0.001
∣
a
m
−
b
m
∣
<
0.001
[url=https://zhuanlan.zhihu.com/p/417529866] Link
1
1
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Quite easy geometry
In triangle ABC,X,Y are on the angle bisector of ∠BAC and ∠ABX=∠ACY.BX intersects CY at P and circles (BYP) and (CXP) intersect at Q different from P. Prove that A,P,Q are on a line.
2
1
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A nice number theory problem.
Find a necessary and sufficient condition of
a
,
b
,
n
∈
N
∗
a,b,n\in\mathbb{N^*}
a
,
b
,
n
∈
N
∗
such that for
S
=
{
a
+
b
t
∣
t
=
0
,
1
,
2
,
⋯
,
n
−
1
}
S=\{a+bt\mid t=0,1,2,\cdots,n-1\}
S
=
{
a
+
b
t
∣
t
=
0
,
1
,
2
,
⋯
,
n
−
1
}
, there exists a one-to-one mapping
f
:
S
→
S
f: S\to S
f
:
S
→
S
such that for all
x
∈
S
x\in S
x
∈
S
,
gcd
(
x
,
f
(
x
)
)
=
1
\gcd(x,f(x))=1
g
cd
(
x
,
f
(
x
))
=
1
.