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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2024 Korea Junior Math Olympiad
2024 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
8
1
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Dirty function(not dirty)
f
f
f
is a function from the set of positive integers to the set of all integers that satisfies the following.
⋅
\cdot
⋅
f
(
1
)
=
1
,
f
(
2
)
=
−
1
f(1)=1, f(2)=-1
f
(
1
)
=
1
,
f
(
2
)
=
−
1
⋅
\cdot
⋅
f
(
n
)
+
f
(
n
+
1
)
+
f
(
n
+
2
)
=
f
(
⌊
n
+
2
3
⌋
)
f(n)+f(n+1)+f(n+2)=f(\left\lfloor\frac{n+2}{3}\right\rfloor)
f
(
n
)
+
f
(
n
+
1
)
+
f
(
n
+
2
)
=
f
(
⌊
3
n
+
2
⌋
)
Find the number of positive integers
k
k
k
not exceeding
1000
1000
1000
such that
f
(
3
)
+
f
(
6
)
+
⋯
+
f
(
3
k
−
3
)
+
f
(
3
k
)
=
5
f(3)+f(6)+\cdots+f(3k-3)+f(3k)=5
f
(
3
)
+
f
(
6
)
+
⋯
+
f
(
3
k
−
3
)
+
f
(
3
k
)
=
5
.
7
1
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Korea's unhealthy obession with inequality-combinatorics??
Let
A
k
A_k
A
k
be the number of pairs
(
a
1
,
a
2
,
.
.
.
,
a
2
k
)
(a_1, a_2, ..., a_{2k})
(
a
1
,
a
2
,
...
,
a
2
k
)
for
k
≤
50
k\leq 50
k
≤
50
, where
a
1
,
a
2
,
.
.
.
,
a
2
k
a_1, a_2, ..., a_{2k}
a
1
,
a
2
,
...
,
a
2
k
are all different positive integers that satisfy the following.
⋅
\cdot
⋅
a
1
,
a
2
,
.
.
.
,
a
2
k
≤
100
a_1, a_2, ..., a_{2k} \leq 100
a
1
,
a
2
,
...
,
a
2
k
≤
100
⋅
\cdot
⋅
For an odd number less or equal than
2
k
−
1
2k-1
2
k
−
1
, we have
a
i
>
a
i
+
1
a_i > a_{i+1}
a
i
>
a
i
+
1
⋅
\cdot
⋅
For an even number less or equal than
2
k
−
2
2k-2
2
k
−
2
, we have
a
i
<
a
i
+
1
a_i < a_{i+1}
a
i
<
a
i
+
1
Prove that
A
1
≤
A
2
≤
⋯
≤
A
49
A_1 \leq A_2 \leq \cdots \leq A_{49}
A
1
≤
A
2
≤
⋯
≤
A
49
.
6
1
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2024 KJMO P6 find all pairs
Find all pairs
(
n
,
p
)
(n, p)
(
n
,
p
)
that satisfy the following condition, where
n
n
n
is a positive integer and
p
p
p
is a prime number.Condition)
2
n
−
1
2n-1
2
n
−
1
is a divisor of
p
−
1
p-1
p
−
1
and
p
p
p
is a divisor of
4
n
2
+
7
4n^2+7
4
n
2
+
7
.
5
1
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ANALYTIC GEO?
A
B
C
ABC
A
BC
is a right triangle with
∠
C
\angle C
∠
C
the right angle.
X
X
X
is some point inside
A
B
C
ABC
A
BC
satisfying
C
A
=
A
X
CA=AX
C
A
=
A
X
. Let
D
D
D
be the feet of altitude from
C
C
C
to
A
B
AB
A
B
, and
Y
(
≠
X
)
Y(\neq X)
Y
(
=
X
)
the point of intersection of
D
X
DX
D
X
and the circumcircle of
A
B
X
ABX
A
BX
. Prove that
A
X
=
A
Y
AX=AY
A
X
=
A
Y
.
3
1
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Strange Collinearlity
Acute triangle
A
B
C
ABC
A
BC
satisfies
∠
A
>
∠
C
\angle A > \angle C
∠
A
>
∠
C
. Let
D
,
E
,
F
D, E, F
D
,
E
,
F
be the points that the triangle's incircle intersects with
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
, respectively, and
P
P
P
some point on
A
F
AF
A
F
different from
F
F
F
. The angle bisector of
∠
A
B
C
\angle ABC
∠
A
BC
meets
P
Q
R
PQR
PQR
's circumcircle
O
O
O
at
L
,
R
L, R
L
,
R
.
L
L
L
is the point closer to
B
B
B
than
R
R
R
.
O
O
O
meets
D
F
,
D
R
DF, DR
D
F
,
D
R
at point
Q
(
≠
F
,
L
)
,
S
(
≠
R
)
Q(\neq F, L), S(\neq R)
Q
(
=
F
,
L
)
,
S
(
=
R
)
respectively, and
P
S
PS
PS
hits segment
B
C
BC
BC
at
T
T
T
. Show that
T
,
Q
,
L
T, Q, L
T
,
Q
,
L
are collinear.
2
1
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Double counting? KJMO 2024 antipode
99
99
99
different points
P
1
,
P
2
,
.
.
.
,
P
99
P_1, P_2, ..., P_{99}
P
1
,
P
2
,
...
,
P
99
are marked on circle
O
O
O
. For each
P
i
P_i
P
i
, define
n
i
n_i
n
i
as the number of marked points you encounter starting from
P
i
P_i
P
i
to its antipode, moving clockwise. Prove the following inequality.
n
1
+
n
2
+
⋯
+
n
99
≤
99
⋅
98
2
+
49
=
4900
n_1+n_2+\cdots+n_{99} \leq \frac{99\cdot 98}{2}+49=4900
n
1
+
n
2
+
⋯
+
n
99
≤
2
99
⋅
98
+
49
=
4900
1
1
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Classic NT KJMO
Find the number of positive integer pairs
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
that
1
x
+
1
+
1
y
+
2
+
1
z
+
3
=
11
12
\frac{1}{x+1}+\frac{1}{y+2}+\frac{1}{z+3}=\frac{11}{12}
x
+
1
1
+
y
+
2
1
+
z
+
3
1
=
12
11
4
1
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n! divides a^n + b^n
find all positive integer n such that there exists positive integers (a,b) such that (a^n + b^n)/n! is a positive integer smaller than 101