MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea Summer Program Practice Test
2024 Korea Summer Program Practice Test
2024 Korea Summer Program Practice Test
Part of
Korea Summer Program Practice Test
Subcontests
(8)
8
2
Hide problems
(Original version) Same number of divisors
For a positive integer
n
n
n
, let
τ
(
n
)
\tau(n)
τ
(
n
)
denote the number of positive divisors of
n
n
n
. Determine whether there exists a positive integer triple
a
,
b
,
c
a, b, c
a
,
b
,
c
such that there are exactly
1012
1012
1012
positive integers
K
K
K
not greater than
2024
2024
2024
that satisfies the following: the equation
τ
(
x
)
=
τ
(
y
)
=
τ
(
z
)
=
τ
(
a
x
+
b
y
+
c
z
)
=
K
\tau(x) = \tau(y) = \tau(z) = \tau(ax + by + cz) = K
τ
(
x
)
=
τ
(
y
)
=
τ
(
z
)
=
τ
(
a
x
+
b
y
+
cz
)
=
K
holds for some positive integers
x
,
y
,
z
x,y,z
x
,
y
,
z
.
(Easy version) Same number of divisors
For a positive integer
n
n
n
, let
τ
(
n
)
\tau(n)
τ
(
n
)
denote the number of positive divisors of
n
n
n
. Determine all positive integers
K
K
K
such that the equation
τ
(
x
)
=
τ
(
y
)
=
τ
(
z
)
=
τ
(
2
x
+
3
y
+
3
z
)
=
K
\tau(x) = \tau(y) = \tau(z) = \tau(2x + 3y + 3z) = K
τ
(
x
)
=
τ
(
y
)
=
τ
(
z
)
=
τ
(
2
x
+
3
y
+
3
z
)
=
K
holds for some positive integers
x
,
y
,
z
x,y,z
x
,
y
,
z
.
7
1
Hide problems
TA asks a math question to the students during a mock test?
2024
2024
2024
people attended a party. Eunson, the host of the party, wanted to make the participant shake hands in pairs. As a professional daydreamer, Eunsun wondered which would be greater: the number of ways each person could shake hands with
4
4
4
others or the number of ways each person could shake hands with
3
3
3
others. Solve Eunsun's peculiar question.
6
2
Hide problems
(Original version) Infinite sequence with a weird inequality
Find all possible values of
C
∈
R
C\in \mathbb R
C
∈
R
such that there exists a real sequence
{
a
n
}
n
=
1
∞
\{a_n\}_{n=1}^\infty
{
a
n
}
n
=
1
∞
such that
a
n
a
n
+
1
2
≥
a
n
+
2
4
+
C
a_na_{n+1}^2\ge a_{n+2}^4 +C
a
n
a
n
+
1
2
≥
a
n
+
2
4
+
C
for all
n
≥
1
n\ge 1
n
≥
1
.
(Easy version) Infinite sequence with a weird inequality
Does there exist a real sequence
{
a
n
}
n
=
1
∞
\{a_n\}_{n=1}^\infty
{
a
n
}
n
=
1
∞
such that
a
n
a
n
+
1
≥
a
n
+
2
2
+
1
a_na_{n+1}\ge a_{n+2}^2 +1
a
n
a
n
+
1
≥
a
n
+
2
2
+
1
for all
n
≥
1
n\ge 1
n
≥
1
?
5
2
Hide problems
Many midpoints and parallel lines
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral such that
∠
A
,
∠
B
,
∠
C
\angle A, \angle B, \angle C
∠
A
,
∠
B
,
∠
C
are acute.
A
B
AB
A
B
and
C
D
CD
C
D
meet at
E
E
E
and
B
C
,
D
A
BC,DA
BC
,
D
A
meet at
F
F
F
. Let
K
,
L
,
M
,
N
K,L,M,N
K
,
L
,
M
,
N
be the midpoints of
A
B
,
B
C
,
C
D
,
D
A
AB,BC,CD,DA
A
B
,
BC
,
C
D
,
D
A
repectively.
K
M
KM
K
M
meets
B
C
,
D
A
BC,DA
BC
,
D
A
at
X
X
X
and
Y
Y
Y
, and
L
N
LN
L
N
meets
A
B
,
C
D
AB,CD
A
B
,
C
D
at
Z
Z
Z
and
W
W
W
. Prove that the line passing
E
E
E
and the midpoint of
Z
W
ZW
Z
W
is parallel to the line passing
F
F
F
and the midpoint of
X
Y
XY
X
Y
.
Partitioning into epic sets
Call a set
{
a
,
b
,
c
,
d
}
\{a,b,c,d\}
{
a
,
b
,
c
,
d
}
epic
if for any four different positive integers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
, there is a unique way to select three of them to form the sides of a triangle. Find all positive integers
n
n
n
such that
{
1
,
2
,
…
,
4
n
}
\{1, 2, \ldots, 4n\}
{
1
,
2
,
…
,
4
n
}
can be partitioned into
n
n
n
disjoint epic sets.
4
1
Hide problems
(Original version) Drawing non-intersecting diagonals on dominos
Find all pairs of positive integers
(
m
,
n
)
(m,n)
(
m
,
n
)
such that one can partition a
m
×
n
m\times n
m
×
n
board with
1
×
2
1\times 2
1
×
2
or
2
×
1
2\times 1
2
×
1
dominoes and draw one of the diagonals on each of the dominos so that none of the diagonals share endpoints.
3
2
Hide problems
Prime divisors of a sequence are 12k+1
Define the sequence
{
a
n
}
n
=
1
∞
\{a_n\}_{n=1}^\infty
{
a
n
}
n
=
1
∞
as a_1 = a_2 = 1, a_{n+2} = 14a_{n+1} - a_n \; (n \geq 1) Prove that if
p
p
p
is prime and there exists a positive integer
n
n
n
such that
a
n
p
\frac{a_n}p
p
a
n
is an integer, then
p
−
1
12
\frac{p-1}{12}
12
p
−
1
is also an integer.
(Easy version) Drawing non-intersecting diagonals on dominos
Find all pairs of positive integers
n
n
n
such that one can partition a
n
×
(
n
+
1
)
n\times (n+1)
n
×
(
n
+
1
)
board with
1
×
2
1\times 2
1
×
2
or
2
×
1
2\times 1
2
×
1
dominoes and draw one of the diagonals on each of the dominos so that none of the diagonals share endpoints.
2
2
Hide problems
Same ratio gives coaxial circles
Let
A
B
C
D
ABCD
A
BC
D
be a quadtrilateral with no parallel sides. The diagonals intersect at
E
E
E
, and
P
,
Q
P, Q
P
,
Q
are points on sides
A
B
,
C
D
AB, CD
A
B
,
C
D
respectively such that
A
P
P
B
=
C
Q
Q
D
\frac{AP}{PB} = \frac{CQ}{QD}
PB
A
P
=
Q
D
CQ
.
P
Q
PQ
PQ
meet
A
C
AC
A
C
and
B
D
BD
B
D
at
R
,
S
R,S
R
,
S
. Prove that
(
E
A
B
)
,
(
E
C
D
)
,
(
E
R
S
)
(EAB),(ECD),(ERS)
(
E
A
B
)
,
(
EC
D
)
,
(
ERS
)
all meet a point other than
E
E
E
.
Integer sequence with divisibility condition
Find all integer sequences
a
1
,
a
2
,
…
,
a
2024
a_1 , a_2 , \ldots , a_{2024}
a
1
,
a
2
,
…
,
a
2024
such that
1
≤
a
i
≤
2024
1\le a_i \le 2024
1
≤
a
i
≤
2024
for
1
≤
i
≤
2024
1\le i\le 2024
1
≤
i
≤
2024
and
i
+
j
∣
i
a
i
−
j
a
j
i+j|ia_i-ja_j
i
+
j
∣
i
a
i
−
j
a
j
for each pair
1
≤
i
,
j
≤
2024
1\le i,j \le 2024
1
≤
i
,
j
≤
2024
.
1
2
Hide problems
P(P(x))-x is irreducible
Find all polynomials
P
P
P
with integer coefficients such that
P
(
P
(
x
)
)
−
x
P(P(x))-x
P
(
P
(
x
))
−
x
is irreducible over
Z
[
x
]
\mathbb{Z}[x]
Z
[
x
]
.
Just a FE
Find all
f
:
R
→
R
f:\mathbb{R}\rightarrow \mathbb{R}
f
:
R
→
R
such that the equation
f
(
x
2
+
y
f
(
x
)
)
=
(
1
−
x
)
f
(
y
−
x
)
f(x^2+yf(x))=(1-x)f(y-x)
f
(
x
2
+
y
f
(
x
))
=
(
1
−
x
)
f
(
y
−
x
)
holds for all
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
.