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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2010 Korea Junior Math Olympiad
2010 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(7)
2
1
Hide problems
0,1 in nxn board, ways that sum of products of all numbers in k-th row is even
Let there be a
n
×
n
n\times n
n
×
n
board. Write down
0
0
0
or
1
1
1
in all
n
2
n^2
n
2
squares. For
1
≤
k
≤
n
1 \le k \le n
1
≤
k
≤
n
, let
A
k
A_k
A
k
be the product of all numbers in the
k
k
k
th row. How many ways are there to write down the numbers so that
A
1
+
A
2
+
.
.
.
+
A
n
A_1 + A_2 + ... + A_n
A
1
+
A
2
+
...
+
A
n
is even?
4
1
Hide problems
a_{n+4} = (a_n + 2a_{n+1} + 3a_{n+2} + 4a_{n+3}$) mod 9
Let there be a sequence
a
n
a_n
a
n
such that
a
1
=
2
,
a
2
=
0
,
a
3
=
1
,
a
4
=
0
a_1 = 2,a_2 = 0, a_3 = 1, a_4 = 0
a
1
=
2
,
a
2
=
0
,
a
3
=
1
,
a
4
=
0
, and for
n
≥
1
,
a
n
+
4
n \ge 1, a_{n+4}
n
≥
1
,
a
n
+
4
is the remainder when
a
n
+
2
a
n
+
1
+
3
a
n
+
2
+
4
a
n
+
3
a_n + 2a_{n+1} + 3a_{n+2} + 4a_{n+3}
a
n
+
2
a
n
+
1
+
3
a
n
+
2
+
4
a
n
+
3
is divided by
9
9
9
. Prove that there are no positive integer
k
k
k
such that
a
k
=
0
,
a
k
+
1
=
1
,
a
k
+
2
=
0
,
a
k
+
3
=
2.
a_k = 0, a_{k+1} = 1, a_{k+2} = 0,a_{k+3} = 2.
a
k
=
0
,
a
k
+
1
=
1
,
a
k
+
2
=
0
,
a
k
+
3
=
2.
8
1
Hide problems
no of longest path in a rectangle with vertices (0, 0), (0, 2), (n,0), (n, 2)
In a rectangle with vertices
(
0
,
0
)
,
(
0
,
2
)
,
(
n
,
0
)
,
(
n
,
2
)
(0, 0), (0, 2), (n,0),(n, 2)
(
0
,
0
)
,
(
0
,
2
)
,
(
n
,
0
)
,
(
n
,
2
)
, (
n
n
n
is a positive integer) find the number of longest paths starting from
(
0
,
0
)
(0, 0)
(
0
,
0
)
and arriving at
(
n
,
2
)
(n, 2)
(
n
,
2
)
which satis fy the following:
∙
\bullet
∙
At each movement, you can move right, up, left, down by
1
1
1
.
∙
\bullet
∙
You cannot visit a point you visited before.
∙
\bullet
∙
You cannot move outside the rectangle.
3
1
Hide problems
incenter lies on circumircle so the other incenter is collinear with points
In an acute triangle
△
A
B
C
\triangle ABC
△
A
BC
, let there be point
D
D
D
on segment
A
C
,
E
AC, E
A
C
,
E
on segment
A
B
AB
A
B
such that
∠
A
D
E
=
∠
A
B
C
\angle ADE = \angle ABC
∠
A
D
E
=
∠
A
BC
. Let the bisector of
∠
A
\angle A
∠
A
hit
B
C
BC
BC
at
K
K
K
. Let the foot of the perpendicular from
K
K
K
to
D
E
DE
D
E
be
P
P
P
, and the foot of the perpendicular from
A
A
A
to
D
E
DE
D
E
be
L
L
L
. Let
Q
Q
Q
be the midpoint of
A
L
AL
A
L
. If the incenter of
△
A
B
C
\triangle ABC
△
A
BC
lies on the circumcircle of
△
A
D
E
\triangle ADE
△
A
D
E
, prove that
P
,
Q
P,Q
P
,
Q
and the incenter of
△
A
D
E
\triangle ADE
△
A
D
E
are collinear.
7
1
Hide problems
collinear wanted, given intersections and perpendulars of cyclic ABCD
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic convex quadrilateral. Let
E
E
E
be the intersection of lines
A
B
,
C
D
AB,CD
A
B
,
C
D
.
P
P
P
is the intersection of line passing
B
B
B
and perpendicular to
A
C
AC
A
C
, and line passing
C
C
C
and perpendicular to
B
D
BD
B
D
.
Q
Q
Q
is the intersection of line passing
D
D
D
and perpendicular to
A
C
AC
A
C
, and line passing
A
A
A
and perpendicular to
B
D
BD
B
D
. Prove that three points
E
,
P
,
Q
E, P,Q
E
,
P
,
Q
are collinear.
5
1
Hide problems
sin^2 x + sin^2 y + sin^2 z < 1 if tan x + tan y + tan z = 2
If reals
x
,
y
,
z
x, y, z
x
,
y
,
z
satises
t
a
n
x
+
t
a
n
y
+
t
a
n
z
=
2
tan x + tan y + tan z = 2
t
an
x
+
t
an
y
+
t
an
z
=
2
and
0
<
x
,
y
,
z
<
π
2
.
0 < x, y,z < \frac{\pi}{2}.
0
<
x
,
y
,
z
<
2
π
.
Prove that
s
i
n
2
x
+
s
i
n
2
y
+
s
i
n
2
z
<
1.
sin^2 x + sin^2 y + sin^2 z < 1.
s
i
n
2
x
+
s
i
n
2
y
+
s
i
n
2
z
<
1.
6
1
Hide problems
Prove that the number is compound
Let
n
∈
N
n\in\mathbb{N}
n
∈
N
and
p
p
p
is the odd prime number. Define the sequence
a
n
a_n
a
n
such that
a
1
=
p
n
+
1
a_1=pn+1
a
1
=
p
n
+
1
and
a
k
+
1
=
n
a
k
+
1
a_{k+1}=na_k+1
a
k
+
1
=
n
a
k
+
1
for all
k
∈
N
k \in \mathbb{N}
k
∈
N
. Prove that
a
p
−
1
a_{p-1}
a
p
−
1
is compound number.