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Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2019 China National Olympiad
2019 China National Olympiad
Part of
China National Olympiad
Subcontests
(6)
5
1
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Operation on n by n grid with integers
Given is an
n
×
n
n\times n
n
×
n
board, with an integer written in each grid. For each move, I can choose any grid, and add
1
1
1
to all
2
n
−
1
2n-1
2
n
−
1
numbers in its row and column. Find the largest
N
(
n
)
N(n)
N
(
n
)
, such that for any initial choice of integers, I can make a finite number of moves so that there are at least
N
(
n
)
N(n)
N
(
n
)
even numbers on the board.
4
1
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Unique rhombus of minimum area tangent to ellipse
Given an ellipse that is not a circle. (1) Prove that the rhombus tangent to the ellipse at all four of its sides with minimum area is unique. (2) Construct this rhombus using a compass and a straight edge.
6
1
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China Mathematical Olympiad 2019 Q6
The point
P
1
,
P
2
,
⋯
,
P
2018
P_1, P_2,\cdots ,P_{2018}
P
1
,
P
2
,
⋯
,
P
2018
is placed inside or on the boundary of a given regular pentagon. Find all placement methods are made so that
S
=
∑
1
≤
i
<
j
≤
2018
∣
P
i
P
j
∣
2
S=\sum_{1\leq i<j\leq 2018}|P_iP_j| ^2
S
=
1
≤
i
<
j
≤
2018
∑
∣
P
i
P
j
∣
2
takes the maximum value.
2
1
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Chain of pythagorean triplets
Call a set of 3 positive integers
{
a
,
b
,
c
}
\{a,b,c\}
{
a
,
b
,
c
}
a Pythagorean set if
a
,
b
,
c
a,b,c
a
,
b
,
c
are the lengths of the 3 sides of a right-angled triangle. Prove that for any 2 Pythagorean sets
P
,
Q
P,Q
P
,
Q
, there exists a positive integer
m
≥
2
m\ge 2
m
≥
2
and Pythagorean sets
P
1
,
P
2
,
…
,
P
m
P_1,P_2,\ldots ,P_m
P
1
,
P
2
,
…
,
P
m
such that
P
=
P
1
,
Q
=
P
m
P=P_1, Q=P_m
P
=
P
1
,
Q
=
P
m
, and
∀
1
≤
i
≤
m
−
1
\forall 1\le i\le m-1
∀1
≤
i
≤
m
−
1
,
P
i
∩
P
i
+
1
≠
∅
P_i\cap P_{i+1}\neq \emptyset
P
i
∩
P
i
+
1
=
∅
.
3
1
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Prove line tangent to circle
Let
O
O
O
be the circumcenter of
△
A
B
C
\triangle ABC
△
A
BC
(
A
B
<
A
C
AB<AC
A
B
<
A
C
), and
D
D
D
be a point on the internal angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
. Point
E
E
E
lies on
B
C
BC
BC
, satisfying
O
E
∥
A
D
OE\parallel AD
OE
∥
A
D
,
D
E
⊥
B
C
DE\perp BC
D
E
⊥
BC
. Point
K
K
K
lies on
E
B
EB
EB
extended such that
E
K
=
E
A
EK=EA
E
K
=
E
A
. The circumcircle of
△
A
D
K
\triangle ADK
△
A
DK
meets
B
C
BC
BC
at
P
≠
K
P\neq K
P
=
K
, and meets the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at
Q
≠
A
Q\neq A
Q
=
A
. Prove that
P
Q
PQ
PQ
is tangent to the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
.
1
1
Hide problems
China Mathematical Olympiad 2019 Q1
Let
a
,
b
,
c
,
d
,
e
≥
−
1
a,b,c,d,e\geq -1
a
,
b
,
c
,
d
,
e
≥
−
1
and
a
+
b
+
c
+
d
+
e
=
5.
a+b+c+d+e=5.
a
+
b
+
c
+
d
+
e
=
5.
Find the maximum and minimum value of
S
=
(
a
+
b
)
(
b
+
c
)
(
c
+
d
)
(
d
+
e
)
(
e
+
a
)
.
S=(a+b)(b+c)(c+d)(d+e)(e+a).
S
=
(
a
+
b
)
(
b
+
c
)
(
c
+
d
)
(
d
+
e
)
(
e
+
a
)
.