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Contests
National and Regional Contests
Korea Contests
Korea Summer Program Practice Test
2023 Korea Summer Program Practice Test
2023 Korea Summer Program Practice Test
Part of
Korea Summer Program Practice Test
Subcontests
(8)
P4
1
Hide problems
Hong travels through towns
In a country there are infinitely many towns and for every pair of towns there is one road connecting them. Initially there are
n
n
n
coin in each city. Every day traveller Hong starts from one town and moves on to another, but if Hong goes from town
A
A
A
to
B
B
B
on the
k
k
k
-th day, he has to send
k
k
k
coins from
B
B
B
to
A
A
A
, and he can no longer use the road connecting
A
A
A
and
B
B
B
. Prove that Hong can't travel for more than
n
+
2
n
2
3
n+2n^\frac{2}{3}
n
+
2
n
3
2
days.
P8
1
Hide problems
Number theory with relatively prime numbers with n
n
n
n
is a natural number larger than
3
3
3
and denote all positive coprime numbers with
n
n
n
as
1
=
b
1
<
b
2
<
⋯
b
k
1= b_1 < b_2 < \cdots b_k
1
=
b
1
<
b
2
<
⋯
b
k
. For a positive integer
m
m
m
which is larger than
3
3
3
and is coprime with
n
n
n
, let
A
A
A
be the set of tuples
(
a
1
,
a
2
,
⋯
a
k
)
(a_1,a_2, \cdots a_k)
(
a
1
,
a
2
,
⋯
a
k
)
satisfying the condition.
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
C
o
n
d
i
t
i
o
n
<
/
s
p
a
n
>
:
For all integers
i
,
0
≤
a
i
<
m
and
a
1
b
1
+
a
2
b
2
+
⋯
a
k
b
k
is a mutiple of
n
<span class='latex-bold'>Condition</span>: \text{For all integers } i, 0 \le a_i < m \text{ and } a_1b_1 + a_2b_2 + \cdots a_kb_k \text{ is a mutiple of } n
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
C
o
n
d
i
t
i
o
n
<
/
s
p
an
>:
For all integers
i
,
0
≤
a
i
<
m
and
a
1
b
1
+
a
2
b
2
+
⋯
a
k
b
k
is a mutiple of
n
For elements of
A
A
A
, show that the difference of number of elements such that
a
1
=
1
a_1 = 1
a
1
=
1
and the number of elements such that
a
2
=
2
a_2 = 2
a
2
=
2
maximum
1
1
1
P6
1
Hide problems
Proving that $AT$ is the radical axis
A
B
<
A
C
AB < AC
A
B
<
A
C
on
△
A
B
C
\triangle ABC
△
A
BC
. The midpoint of arc
B
C
BC
BC
which doesn't include
A
A
A
is
T
T
T
and which includes
A
A
A
is
S
S
S
. On segment
A
B
,
A
C
AB,AC
A
B
,
A
C
,
D
,
E
D,E
D
,
E
exist so that
D
E
DE
D
E
and
B
C
BC
BC
are parallel. The outer angle bisector of
∠
A
B
E
\angle ABE
∠
A
BE
and
∠
A
C
D
\angle ACD
∠
A
C
D
meets
A
S
AS
A
S
at
P
P
P
and
Q
Q
Q
. Prove that the circumcircle of
△
P
B
E
\triangle PBE
△
PBE
and
△
Q
C
D
\triangle QCD
△
QC
D
meets on
A
T
AT
A
T
.
P7
1
Hide problems
Sequence Inequality
Determine the smallest value of
M
M
M
for which for any choice of positive integer
n
n
n
and positive real numbers
x
1
<
x
2
<
…
<
x
n
≤
2023
x_1<x_2<\ldots<x_n \le 2023
x
1
<
x
2
<
…
<
x
n
≤
2023
the inequality
∑
1
≤
i
<
j
≤
n
,
x
j
−
x
i
≥
1
2
i
−
j
≤
M
\sum_{1\le i < j \le n , x_j-x_i \ge 1} 2^{i-j}\le M
1
≤
i
<
j
≤
n
,
x
j
−
x
i
≥
1
∑
2
i
−
j
≤
M
holds.
P5
1
Hide problems
Easy combinatorics in a plane
For a positive integer
n
n
n
,
n
n
n
vertices which have
10000
10000
10000
written on them exist on a plane. For
3
3
3
vertices that are collinear and are written positive numbers on them, denote procedure
P
P
P
as subtracting
1
1
1
from the outer vertices and adding
2023
2023
2023
to the inner vertical. Show that procedure
P
P
P
cannot be repeated infinitely.
P3
1
Hide problems
Nice Geometry
△
A
B
C
\triangle ABC
△
A
BC
is a triangle such that
∠
A
=
6
0
∘
\angle A = 60^{\circ}
∠
A
=
6
0
∘
. The incenter of
△
A
B
C
\triangle ABC
△
A
BC
is
I
I
I
.
A
I
AI
A
I
intersects with
B
C
BC
BC
at
D
D
D
,
B
I
BI
B
I
intersects with
C
A
CA
C
A
at
E
E
E
, and
C
I
CI
C
I
intersects with
A
B
AB
A
B
at
F
F
F
, respectively. Also, the circumcircle of
△
D
E
F
\triangle DEF
△
D
EF
is
ω
\omega
ω
. The tangential line of
ω
\omega
ω
at
E
E
E
and
F
F
F
intersects at
T
T
T
. Show that
∠
B
T
C
≥
6
0
∘
\angle BTC \ge 60^{\circ}
∠
BTC
≥
6
0
∘
P2
1
Hide problems
Normal FE over R to R
Find all functions
f
:
R
→
R
f : \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
f
(
x
)
2
+
∣
y
∣
)
=
x
2
+
f
(
y
)
f(f(x)^2 + |y|) = x^2 + f(y)
f
(
f
(
x
)
2
+
∣
y
∣
)
=
x
2
+
f
(
y
)
P1
1
Hide problems
Corresponding number theory
A natural number
n
n
n
is given. For all integer triplets
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
such that
0
<
∣
a
∣
,
∣
b
∣
,
∣
c
∣
<
2023
0 < |a| , |b|, |c| < 2023
0
<
∣
a
∣
,
∣
b
∣
,
∣
c
∣
<
2023
and satisfying below, show that the product of all possible integer
a
a
a
is a perfect square. (The value of
a
a
a
allows duplication)
(
a
+
n
b
)
(
a
−
n
c
)
+
a
b
c
=
0
(a+nb)(a-nc) + abc = 0
(
a
+
nb
)
(
a
−
n
c
)
+
ab
c
=
0