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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2012 Korea Junior Math Olympiad
2012 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
7
1
Hide problems
max of \frac{(\sqrt{s_1x_1} +...+\sqrt{s_5x_5})^2}{a_1x_1+...+a_5x_5}
If all
x
k
x_k
x
k
(
k
=
1
,
2
,
3
,
4
,
5
)
k = 1, 2, 3, 4, 5)
k
=
1
,
2
,
3
,
4
,
5
)
are positive reals, and
{
a
1
,
a
2
,
a
3
,
a
4
,
a
5
}
=
{
1
,
2
,
3
,
4
,
5
}
\{a_1,a_2, a_3, a_4, a_5\} = \{1, 2,3 , 4, 5\}
{
a
1
,
a
2
,
a
3
,
a
4
,
a
5
}
=
{
1
,
2
,
3
,
4
,
5
}
, find the maximum of
(
s
1
x
1
+
s
2
x
2
+
s
3
x
3
+
s
4
x
4
+
s
5
x
5
)
2
a
1
x
1
+
a
2
x
2
+
a
3
x
3
+
a
4
x
4
+
a
5
x
5
\frac{(\sqrt{s_1x_1} +\sqrt{s_2x_2}+\sqrt{s_3x_3}+\sqrt{s_4x_4}+\sqrt{s_5x_5})^2}{a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 + a_5x_5}
a
1
x
1
+
a
2
x
2
+
a
3
x
3
+
a
4
x
4
+
a
5
x
5
(
s
1
x
1
+
s
2
x
2
+
s
3
x
3
+
s
4
x
4
+
s
5
x
5
)
2
(
s
k
=
a
1
+
a
2
+
.
.
.
+
a
k
s_k = a_1 + a_2 +... + a_k
s
k
=
a
1
+
a
2
+
...
+
a
k
)
6
1
Hide problems
number theory with sequence, related to prime p=2k+3
p
>
3
p > 3
p
>
3
is a prime number such that
p
∣
2
p
−
1
−
1
p|2^{p-1} - 1
p
∣
2
p
−
1
−
1
and
p
∤
2
x
−
1
p \nmid 2^x - 1
p
∤
2
x
−
1
for
x
=
1
,
2
,
.
.
.
,
p
−
2
x = 1, 2,...,p-2
x
=
1
,
2
,
...
,
p
−
2
. Let
p
=
2
k
+
3
p = 2k + 3
p
=
2
k
+
3
. Now we define sequence
{
a
n
}
\{a_n\}
{
a
n
}
as
a
i
=
a
i
+
k
=
2
i
(
1
≤
i
≤
k
)
,
a
j
+
2
k
=
a
j
a
j
+
k
(
j
≤
1
)
a_i = a_{i+k} = 2^i \,\, (1 \le i \le k ), \,\,\,\, a_{j+2k} = a_ja_{j+k} \,\, (j \le 1)
a
i
=
a
i
+
k
=
2
i
(
1
≤
i
≤
k
)
,
a
j
+
2
k
=
a
j
a
j
+
k
(
j
≤
1
)
Prove that there exist
2
k
2k
2
k
consecutive terms of sequence
a
x
+
1
,
a
x
+
2
,
.
.
.
,
a
x
+
2
k
a_{x+1},a_{x+2},..., a_{x+2k}
a
x
+
1
,
a
x
+
2
,
...
,
a
x
+
2
k
such that
a
x
+
i
≢
a
x
+
j
a_{x+i } \not\equiv a_{x+j}
a
x
+
i
≡
a
x
+
j
(mod
p
p
p
) for all
1
≤
i
<
j
≤
2
k
1 \le i < j \le 2k
1
≤
i
<
j
≤
2
k
.
8
1
Hide problems
n students picking n cards with numbers 1-n, probanility to have same no
Let there be
n
n
n
students, numbered
1
1
1
through
n
n
n
. Let there be
n
n
n
cards with numbers
1
1
1
through
n
n
n
written on them. Each student picks a card from the stack, and two students are called a pair if they pick each other's number. Let the probability that there are no pairs be
p
n
p_n
p
n
. Prove that
p
n
−
p
n
−
1
=
0
p_n - p_{n-1}=0
p
n
−
p
n
−
1
=
0
if
n
n
n
is odd, and prove that
p
n
−
p
n
−
1
=
1
(
−
2
)
k
k
1
−
k
p_n - p_{n-1}= \frac{1}{(-2)^kk^{1-k}}
p
n
−
p
n
−
1
=
(
−
2
)
k
k
1
−
k
1
if
n
=
2
k
n = 2k
n
=
2
k
.
4
1
Hide problems
n students shaking hands
There are
n
n
n
students
A
1
,
A
2
,
.
.
.
,
A
n
A_1,A_2,...,A_n
A
1
,
A
2
,
...
,
A
n
and some of them shaked hands with each other. (
A
i
A_i
A
i
and
A
−
j
A-j
A
−
j
can shake hands more than one time.) Let the student
A
i
A_i
A
i
shaked hands
d
i
d_i
d
i
times. Suppose
d
1
+
d
2
+
.
.
.
+
d
n
>
0
d_1 + d_2 +... + d_n > 0
d
1
+
d
2
+
...
+
d
n
>
0
. Prove that there exist
1
≤
i
<
j
≤
n
1 \le i < j \le n
1
≤
i
<
j
≤
n
satisfying the following conditions: (a) Two students
A
i
A_i
A
i
and
A
j
A_j
A
j
shaked hands each other. (b)
(
d
1
+
d
2
+
.
.
.
+
d
n
)
2
n
2
≤
d
i
d
j
\frac{(d_1 + d_2 +... + d_n)^2}{n^2}\le d_id_j
n
2
(
d
1
+
d
2
+
...
+
d
n
)
2
≤
d
i
d
j
5
1
Hide problems
AL = AD iff <KCE = <ALE , starting with a cyclic ABCD
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral inscirbed in a circle
O
O
O
(
A
B
>
A
D
AB> AD
A
B
>
A
D
), and let
E
E
E
be a point on segment
A
B
AB
A
B
such that
A
E
=
A
D
AE = AD
A
E
=
A
D
. Let
A
C
∩
D
E
=
F
AC \cap DE = F
A
C
∩
D
E
=
F
, and
D
E
∩
O
=
K
(
≠
D
)
DE \cap O = K(\ne D)
D
E
∩
O
=
K
(
=
D
)
. The tangent to the circle passing through
C
,
F
,
E
C,F,E
C
,
F
,
E
at
E
E
E
hits
A
K
AK
A
K
at
L
L
L
. Prove that
A
L
=
A
D
AL = AD
A
L
=
A
D
if and only if
∠
K
C
E
=
∠
A
L
E
\angle KCE = \angle ALE
∠
K
CE
=
∠
A
L
E
.
2
1
Hide problems
concyclic, starting with an inscribed pentagon
A pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
is inscribed in a circle
O
O
O
, and satis fies
∠
A
=
9
0
o
,
A
B
=
C
D
\angle A = 90^o, AB = CD
∠
A
=
9
0
o
,
A
B
=
C
D
. Let
F
F
F
be a point on segment
A
E
AE
A
E
. Let
B
F
BF
BF
hit
O
O
O
again at
J
(
≠
B
)
J(\ne B)
J
(
=
B
)
,
C
E
∩
D
J
=
K
CE \cap DJ = K
CE
∩
D
J
=
K
,
B
D
∩
F
K
=
L
BD\cap FK = L
B
D
∩
F
K
=
L
. Prove that
B
,
L
,
E
,
F
B,L,E,F
B
,
L
,
E
,
F
are cyclic.
3
1
Hide problems
Find l,m,n that satisfy the equation
Find all
l
,
m
,
n
∈
N
l,m,n \in\mathbb{N}
l
,
m
,
n
∈
N
that satisfies the equation
5
l
4
3
m
+
1
=
n
3
5^l43^m+1=n^3
5
l
4
3
m
+
1
=
n
3
1
1
Hide problems
An easy symmetric inequality
Prove the following inequality where positive reals
a
a
a
,
b
b
b
,
c
c
c
satisfies
a
b
+
b
c
+
c
a
=
1
ab+bc+ca=1
ab
+
b
c
+
c
a
=
1
.
a
+
b
a
b
(
1
−
a
b
)
+
b
+
c
b
c
(
1
−
b
c
)
+
c
+
a
c
a
(
1
−
c
a
)
≤
2
a
b
c
\frac{a+b}{\sqrt{ab(1-ab)}} + \frac{b+c}{\sqrt{bc(1-bc)}} + \frac{c+a}{\sqrt{ca(1-ca)}} \le \frac{\sqrt{2}}{abc}
ab
(
1
−
ab
)
a
+
b
+
b
c
(
1
−
b
c
)
b
+
c
+
c
a
(
1
−
c
a
)
c
+
a
≤
ab
c
2