MathDB
Problems
Contests
National and Regional Contests
China Contests
China Girls Math Olympiad
2024 China Girls Math Olympiad
2024 China Girls Math Olympiad
Part of
China Girls Math Olympiad
Subcontests
(7)
8
1
Hide problems
Splitting people into pairs
It is known that there are
2024
2024
2024
pairs of friends among
100
100
100
people. Show that is possible to split them into
50
50
50
pairs so that: (a) There are at most
20
20
20
pairs that are friends with each other; (b) There are at least
23
23
23
pairs that are friends with each other; (c) There are exactly
22
22
22
pairs that are friends with each other.
7
1
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n-variable inequality in CGMO
Let
n
n
n
be a positive integer. If
x
1
,
x
2
,
…
,
x
n
≥
0
x_1, x_2, \ldots, x_n \geq 0
x
1
,
x
2
,
…
,
x
n
≥
0
,
x
1
+
x
2
+
…
+
x
n
=
1
x_1+x_2+\ldots+x_n=1
x
1
+
x
2
+
…
+
x
n
=
1
and, assuming
x
n
+
1
=
x
1
x_{n+1}=x_1
x
n
+
1
=
x
1
, find the maximal value of
∑
k
=
1
n
1
+
x
k
2
+
x
k
4
1
+
x
k
+
1
+
x
k
+
1
2
+
x
k
+
1
3
+
x
k
+
1
4
.
\sum_{k=1}^n \frac{1+x_k^2+x_k^4}{1+x_{k+1}+x_{k+1}^2+x_{k+1}^3+x_{k+1}^4}.
k
=
1
∑
n
1
+
x
k
+
1
+
x
k
+
1
2
+
x
k
+
1
3
+
x
k
+
1
4
1
+
x
k
2
+
x
k
4
.
6
1
Hide problems
n^2+r and m^2+r are powers of 2
Let
n
,
m
,
r
n,m,r
n
,
m
,
r
be positive integers such that
n
>
m
n>m
n
>
m
and both
n
2
+
r
,
m
2
+
r
n^2+r, m^2+r
n
2
+
r
,
m
2
+
r
are powers of
2
2
2
. Show that
n
>
2
m
2
r
n>\frac{2m^2}{r}
n
>
r
2
m
2
.
5
1
Hide problems
Right triangle covered by two unit circles
If a right triangle can be covered by two unit circles, find the maximal area of the right triangle.
1
1
Hide problems
Maximum term of a weird sequence
Let
{
a
n
}
\{a_n\}
{
a
n
}
be a sequence defined by
a
1
=
0
a_1=0
a
1
=
0
and
a
n
=
1
n
+
1
⌈
n
2
⌉
∑
k
=
1
⌈
n
2
⌉
a
k
a_n=\frac{1}{n}+\frac{1}{\lceil \frac{n}{2} \rceil}\sum_{k=1}^{\lceil \frac{n}{2} \rceil}a_k
a
n
=
n
1
+
⌈
2
n
⌉
1
k
=
1
∑
⌈
2
n
⌉
a
k
for any positive integer
n
n
n
. Find the maximal term of this sequence.
3
1
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NT with fractional parts
Find the smallest real
λ
\lambda
λ
, such that for any positive integers
n
,
a
,
b
n, a, b
n
,
a
,
b
, such that
n
∤
a
+
b
n \nmid a+b
n
∤
a
+
b
, there exists a positive integer
1
≤
k
≤
n
−
1
1 \leq k \leq n-1
1
≤
k
≤
n
−
1
, satisfying
{
a
k
n
}
+
{
b
k
n
}
≤
λ
.
\{\frac{ak} {n}\}+\{\frac{bk} {n}\} \leq \lambda.
{
n
ak
}
+
{
n
bk
}
≤
λ
.
4
1
Hide problems
Geo with an angle condition and fixed angle
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
<
B
C
<
C
A
AB<BC<CA
A
B
<
BC
<
C
A
and let
D
D
D
be a variable point on
B
C
BC
BC
. The point
E
E
E
on the circumcircle of
A
B
C
ABC
A
BC
is such that
∠
B
A
D
=
∠
B
E
D
\angle BAD=\angle BED
∠
B
A
D
=
∠
BE
D
. The line through
D
D
D
perpendicular to
A
B
AB
A
B
meets
A
C
AC
A
C
at
F
F
F
. Show that the measure of
∠
B
E
F
\angle BEF
∠
BEF
is constant as
D
D
D
varies.