MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2019 Korea Junior Math Olympiad.
2019 Korea Junior Math Olympiad.
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
4
1
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Prove that the sequence is constant
{
a
n
}
\{a_{n}\}
{
a
n
}
is a sequence of natural numbers satisfying the following inequality for all natural number
n
n
n
:
(
a
1
+
⋯
+
a
n
)
(
1
a
1
+
⋯
+
1
a
n
)
≤
n
2
+
2019
(a_{1}+\cdots+a_{n})\left(\frac{1}{a_{1}}+\cdots+\frac{1}{a_{n}}\right)\le{n^{2}}+2019
(
a
1
+
⋯
+
a
n
)
(
a
1
1
+
⋯
+
a
n
1
)
≤
n
2
+
2019
Prove that
{
a
n
}
\{a_{n}\}
{
a
n
}
is constant.
3
1
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Find all pairs of prime numbers (p, q)
Find all pairs of prime numbers
p
,
q
(
p
≤
q
)
p,\,q(p\le q)
p
,
q
(
p
≤
q
)
satisfying the following condition: There exists a natural number
n
n
n
such that
2
n
+
3
n
+
⋯
+
(
2
p
q
−
1
)
n
2^{n}+3^{n}+\cdots+(2pq-1)^{n}
2
n
+
3
n
+
⋯
+
(
2
pq
−
1
)
n
is a multiple of
2
p
q
2pq
2
pq
.
2
1
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Prove angle ADB=3angleBAC if AE=CD
In an acute triangle
A
B
C
ABC
A
BC
, point
D
D
D
is on the segment
A
C
AC
A
C
such that
A
D
‾
=
B
C
‾
\overline{AD}=\overline{BC}
A
D
=
BC
and
A
C
‾
2
−
A
D
‾
2
=
A
C
‾
⋅
A
D
‾
\overline{AC}^2-\overline{AD}^2=\overline{AC}\cdot\overline{AD}
A
C
2
−
A
D
2
=
A
C
⋅
A
D
. The line that is parallel to the bisector of
∠
A
C
B
\angle{ACB}
∠
A
CB
and passes the point
D
D
D
meets the segment
A
B
AB
A
B
at point
E
E
E
. Prove, if
A
E
‾
=
C
D
‾
\overline{AE}=\overline{CD}
A
E
=
C
D
,
∠
A
D
B
=
3
∠
B
A
C
\angle{ADB}=3\angle{BAC}
∠
A
D
B
=
3∠
B
A
C
.
1
1
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Line passing through three points
Each integer coordinates are colored with one color and at least 5 colors are used to color every integer coordinates. Two integer coordinates
(
x
,
y
)
(x, y)
(
x
,
y
)
and
(
z
,
w
)
(z, w)
(
z
,
w
)
are colored in the same color if
x
−
z
x-z
x
−
z
and
y
−
w
y-w
y
−
w
are both multiples of 3. Prove that there exists a line that passes through exactly three points when five points with different colors are chosen randomly.
8
1
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Number of world travel modulo $4$
There are two airlines A and B and finitely many airports. For each pair of airports, there is exactly one airline among A and B whose flights operates in both directions. Each airline plans to develop world travel packages which pass each airport exactly once using only its flights. Let
a
a
a
and
b
b
b
be the number of possible packages which belongs to A and B respectively. Prove that
a
−
b
a-b
a
−
b
is a multiple of
4
4
4
. The official statement of the problem has been changed. The above is the form which appeared during the contest. Now the condition 'the number of airports is no less than 4'is attached. Cite the following link. https://artofproblemsolving.com/community/c6h2923697p26140823
7
1
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A common ratio related to an acute triangle
Let
O
O
O
be the circumcenter of an acute triangle
A
B
C
ABC
A
BC
. Let
D
D
D
be the intersection of the bisector of the angle
A
A
A
and
B
C
BC
BC
. Suppose that
∠
O
D
C
=
2
∠
D
A
O
\angle ODC = 2 \angle DAO
∠
O
D
C
=
2∠
D
A
O
. The circumcircle of
A
B
D
ABD
A
B
D
meets the line segment
O
A
OA
O
A
and the line
O
D
OD
O
D
at
E
(
≠
A
,
O
)
E (\neq A,O)
E
(
=
A
,
O
)
, and
F
(
≠
D
)
F(\neq D)
F
(
=
D
)
, respectively. Let
X
X
X
be the intersection of the line
D
E
DE
D
E
and the line segment
A
C
AC
A
C
. Let
Y
Y
Y
be the intersection of the bisector of the angle
B
A
F
BAF
B
A
F
and the segment
B
E
BE
BE
. Prove that
A
Y
‾
B
Y
‾
=
E
X
‾
E
O
‾
\frac{\overline{AY}}{\overline{BY}}= \frac{\overline{EX}}{\overline{EO}}
B
Y
A
Y
=
EO
EX
.
6
1
Hide problems
A strange functional equation
Find all functions
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
which satisfies the followings. (Note that
R
\mathbb{R}
R
stands for the set of all real numbers) (1) For each real numbers
x
x
x
,
y
y
y
, the equality
f
(
x
+
f
(
x
)
+
x
y
)
=
2
f
(
x
)
+
x
f
(
y
)
f(x+f(x)+xy) = 2f(x)+xf(y)
f
(
x
+
f
(
x
)
+
x
y
)
=
2
f
(
x
)
+
x
f
(
y
)
holds. (2) For every real number
z
z
z
, there exists
x
x
x
such that
f
(
x
)
=
z
f(x) = z
f
(
x
)
=
z
.
5
1
Hide problems
Common prime divisor of a certain polynomial
For prime number
p
p
p
, prove that there are integers
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
such that for every integer
n
n
n
, the expression
n
4
+
1
−
(
n
2
+
a
n
+
b
)
(
n
2
+
c
n
+
d
)
n^4+1-\left( n^2+an+b \right) \left(n^2+cn+d \right)
n
4
+
1
−
(
n
2
+
an
+
b
)
(
n
2
+
c
n
+
d
)
is a multiple of
p
p
p
.