MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2017 China Second Round Olympiad
2017 China Second Round Olympiad
Part of
(China) National High School Mathematics League
Subcontests
(5)
3
1
Hide problems
Minimal number of separating edges in a tri-colored grid
Each square of a
33
×
33
33\times 33
33
×
33
square grid is colored in one of the three colors: red, yellow or blue, such that the numbers of squares in each color are the same. If two squares sharing a common edge are in different colors, call that common edge a separating edge. Find the minimal number of separating edges in the grid.
4
1
Hide problems
2017 China Second Round Olympiad Test 2 Problem 4
Let
m
,
n
m,n
m
,
n
be integers greater than 1,
m
≥
n
m \geq n
m
≥
n
,
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots,a_n
a
1
,
a
2
,
…
,
a
n
are
n
n
n
distinct numbers not exceed
m
m
m
,which are relatively primitive.Show that for any real
x
x
x
,there exists
i
i
i
for which
∣
∣
a
i
x
∣
∣
≥
2
m
(
m
+
1
)
∣
∣
x
∣
∣
||a_ix|| \geq \frac{2}{m(m+1)} ||x||
∣∣
a
i
x
∣∣
≥
m
(
m
+
1
)
2
∣∣
x
∣∣
,where
∣
∣
x
∣
∣
||x||
∣∣
x
∣∣
denotes the distance between
x
x
x
and the nearest integer to
x
x
x
.
2
2
Hide problems
China Second Round Olympiad 2017 Test 1 Q2
Let
x
,
y
x,y
x
,
y
are real numbers such that
x
2
+
2
c
o
s
y
=
1
x^2+2cosy=1
x
2
+
2
cosy
=
1
. Find the ranges of
x
−
c
o
s
y
x-cosy
x
−
cosy
.
Sequence
Given a sequence
{
a
n
}
\{a_n\}
{
a
n
}
: a_1=1, a_{n+1}=\left\{ \begin{array}{lcr} a_n+n, a_n\le n, \\ a_n-n, a_n>n, \end{array} \right. n=1,2,\cdots. Find the number of positive integers
r
r
r
satisfying
a
r
<
r
≤
3
2017
a_r<r\le 3^{2017}
a
r
<
r
≤
3
2017
.
10
1
Hide problems
China Second Round Olympiad 2017 Test 1 Q10
Let
x
1
,
x
2
,
x
3
≥
0
x_1,x_2,x_3\geq 0
x
1
,
x
2
,
x
3
≥
0
and
x
1
+
x
2
+
x
3
=
1
x_1+x_2+x_3=1
x
1
+
x
2
+
x
3
=
1
. Find the minimum value and the maximum value of
(
x
1
+
3
x
2
+
5
x
3
)
(
x
1
+
x
2
3
+
x
3
5
)
.
(x_1+3x_2+5x_3)\left(x_1+\frac{x_2}{3}+\frac{x_3}{5}\right).
(
x
1
+
3
x
2
+
5
x
3
)
(
x
1
+
3
x
2
+
5
x
3
)
.
1
1
Hide problems
2017 China Second Round Test 2 Olympiad Problem 1
Given an isocleos triangle
A
B
C
ABC
A
BC
with equal sides
A
B
=
A
C
AB=AC
A
B
=
A
C
and incenter
I
I
I
.Let
Γ
1
\Gamma_1
Γ
1
be the circle centered at
A
A
A
with radius
A
B
AB
A
B
,
Γ
2
\Gamma_2
Γ
2
be the circle centered at
I
I
I
with radius
B
I
BI
B
I
.A circle
Γ
3
\Gamma_3
Γ
3
passing through
B
,
I
B,I
B
,
I
intersects
Γ
1
\Gamma_1
Γ
1
,
Γ
2
\Gamma_2
Γ
2
again at
P
,
Q
P,Q
P
,
Q
(different from
B
B
B
) respectively.Let
R
R
R
be the intersection of
P
I
PI
P
I
and
B
Q
BQ
BQ
.Show that
B
R
⊥
C
R
BR \perp CR
BR
⊥
CR
.