MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea Winter Program Practice Test
2021 Korea Winter Program Practice Test
2021 Korea Winter Program Practice Test
Part of
Korea Winter Program Practice Test
Subcontests
(8)
6
2
Hide problems
Imaginary sequence
Is there exist a sequence
a
0
,
a
1
,
a
2
,
⋯
a_0,a_1,a_2,\cdots
a
0
,
a
1
,
a
2
,
⋯
consisting of non-zero integers that satisfies the following condition?Condition: For all integers
n
n
n
(
≥
2020
\ge 2020
≥
2020
), equation
a
n
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
0
=
0
a_n x^n+a_{n-1}x^{n-1}+\cdots +a_0=0
a
n
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
0
=
0
has a real root with its absolute value larger than
2.001
2.001
2.001
.
Another cute geo
The acute triangle
A
B
C
ABC
A
BC
satisfies
A
B
‾
<
B
C
‾
<
C
A
‾
\overline {AB}<\overline {BC}<\overline {CA}
A
B
<
BC
<
C
A
. Denote the foot of perpendicular from
A
,
B
,
C
A,B,C
A
,
B
,
C
to opposing sides as
D
,
E
,
F
D,E,F
D
,
E
,
F
. Let
P
P
P
a foot of perpendicular from
F
F
F
to
D
E
DE
D
E
, and
Q
(
≠
F
)
Q(\neq F)
Q
(
=
F
)
a intersection point of line
F
P
FP
FP
and circumcircle of
B
D
F
BDF
B
D
F
. Prove that
∠
P
B
Q
=
∠
P
A
D
\angle PBQ=\angle PAD
∠
PBQ
=
∠
P
A
D
.
7
2
Hide problems
Spreading numbers
For all integers
x
,
y
x,y
x
,
y
, a non-negative integer
f
(
x
,
y
)
f(x,y)
f
(
x
,
y
)
is written on the point
(
x
,
y
)
(x,y)
(
x
,
y
)
on the coordinate plane. Initially,
f
(
0
,
0
)
=
4
f(0,0) = 4
f
(
0
,
0
)
=
4
and the value written on all remaining points is
0
0
0
. For integers
n
,
m
n, m
n
,
m
that satisfies
f
(
n
,
m
)
≥
2
f(n,m) \ge 2
f
(
n
,
m
)
≥
2
, define '[color=#9a00ff]Seehang' as the act of reducing
f
(
n
,
m
)
f(n,m)
f
(
n
,
m
)
by
1
1
1
, selecting 3 of
f
(
n
,
m
+
1
)
,
f
(
n
,
m
−
1
)
,
f
(
n
+
1
,
m
)
,
f
(
n
−
1
,
m
)
f(n,m+1), f(n,m-1), f(n+1,m), f(n-1,m)
f
(
n
,
m
+
1
)
,
f
(
n
,
m
−
1
)
,
f
(
n
+
1
,
m
)
,
f
(
n
−
1
,
m
)
and increasing them by 1. Prove that after a finite number of '[color=#0f0][color=#9a00ff]Seehang's, it cannot be
f
(
n
,
m
)
≤
1
f(n,m)\le 1
f
(
n
,
m
)
≤
1
for all integers
n
,
m
n,m
n
,
m
.
Laivirt algebra
Find all pair of constants
(
a
,
b
)
(a,b)
(
a
,
b
)
such that there exists real-coefficient polynomial
p
(
x
)
p(x)
p
(
x
)
and
q
(
x
)
q(x)
q
(
x
)
that satisfies the condition below.Condition:
∀
x
∈
R
,
\forall x\in \mathbb R,
∀
x
∈
R
,
p
(
x
2
)
q
(
x
+
1
)
−
p
(
x
+
1
)
q
(
x
2
)
=
x
2
+
a
x
+
b
p(x^2)q(x+1)-p(x+1)q(x^2)=x^2+ax+b
p
(
x
2
)
q
(
x
+
1
)
−
p
(
x
+
1
)
q
(
x
2
)
=
x
2
+
a
x
+
b
5
2
Hide problems
nice geometry
E
,
F
E,F
E
,
F
are points on
A
B
,
A
C
AB,AC
A
B
,
A
C
that satisfies
(
B
,
E
,
F
,
C
)
(B,E,F,C)
(
B
,
E
,
F
,
C
)
cyclic.
D
D
D
is the intersection of
B
C
BC
BC
and the perpendicular bisecter of
E
F
EF
EF
, and
B
′
,
C
′
B',C'
B
′
,
C
′
are the reflections of
B
,
C
B,C
B
,
C
on
A
D
AD
A
D
.
X
X
X
is a point on the circumcircle of
△
B
E
C
′
\triangle{BEC'}
△
BE
C
′
that
A
B
AB
A
B
is perpendicular to
B
X
BX
BX
,and
Y
Y
Y
is a point on the circumcircle of
△
C
F
B
′
\triangle{CFB'}
△
CF
B
′
that
A
C
AC
A
C
is perpendicular to
C
Y
CY
C
Y
. Show that
D
X
=
D
Y
DX=DY
D
X
=
D
Y
.
Permutation consisting of A&B
For positive integers
k
k
k
and
n
n
n
, express the number of permutation
P
=
x
1
x
2
.
.
.
x
2
n
P=x_1x_2...x_{2n}
P
=
x
1
x
2
...
x
2
n
consisting of
A
A
A
and
B
B
B
that satisfies all three of the following conditions, using
k
k
k
and
n
n
n
.
(
i
)
(i)
(
i
)
A
,
B
A, B
A
,
B
appear exactly
n
n
n
times respectively in
P
P
P
.
(
i
i
)
(ii)
(
ii
)
For each
1
≤
i
≤
n
1\le i\le n
1
≤
i
≤
n
, if we denote the number of
A
A
A
in
x
1
,
x
2
,
.
.
.
,
x
i
x_1,x_2,...,x_i
x
1
,
x
2
,
...
,
x
i
as
a
i
a_i
a
i
,
,
,
then
∣
2
a
i
−
i
∣
≤
1
\mid 2a_i -i\mid \le 1
∣
2
a
i
−
i
∣≤
1
.
(
i
i
i
)
(iii)
(
iii
)
A
B
AB
A
B
appears exactly
k
k
k
times in
P
P
P
. (For example,
A
B
AB
A
B
appears 3 times in
A
B
B
A
B
A
A
B
ABBABAAB
A
BB
A
B
AA
B
)
1
2
Hide problems
number theory
Does there exist such infinite set of positive integers
S
S
S
that satisfies the condition below? *for all
a
,
b
a,b
a
,
b
in
S
S
S
, there exists an odd integer
k
k
k
that
a
a
a
divides
b
k
+
1
b^k+1
b
k
+
1
.
A nice trip
There is a group of more than three airports. For any two airports
A
,
B
A, B
A
,
B
belonging to this group, if there is an aircraft from
A
A
A
to
B
B
B
, there is an aircraft from
B
B
B
to
A
A
A
. For a list of different airports
A
0
,
A
1
,
.
.
.
A
n
A_0,A_1,...A_n
A
0
,
A
1
,
...
A
n
, define this list as a '[color=#00f]route' if there is an aircraft from
A
i
A_i
A
i
to
A
i
+
1
A_{i+1}
A
i
+
1
for each
i
=
0
,
1
,
.
.
.
,
n
−
1
i=0,1,...,n-1
i
=
0
,
1
,
...
,
n
−
1
. Also, define the beginning of this [color=#00f]route as
A
0
A_0
A
0
, the end as
A
n
A_n
A
n
, and the length as
n
n
n
. (
n
∈
N
n\in \mathbb N
n
∈
N
)
Now, let's say that for any three different pairs of airports
(
A
,
B
,
C
)
(A,B,C)
(
A
,
B
,
C
)
, there is always a [color=#00f]route
P
P
P
that satisfies the following condition.
Condition:
P
P
P
begins with
A
A
A
and ends with
B
B
B
, and does not include
C
C
C
.
When the length of the longest of the existing [color=#00f]routes is
M
M
M
(
≥
2
\ge 2
≥
2
), prove that any two [color=#00f]routes of length
M
M
M
contain at least two different airports simultaneously.
3
2
Hide problems
inequality
n
≥
2
n\ge2
n
≥
2
is a given positive integer.
i
≤
a
i
≤
n
i\leq a_i \leq n
i
≤
a
i
≤
n
satisfies for all
1
≤
i
≤
n
1\leq i\leq n
1
≤
i
≤
n
, and
S
i
S_i
S
i
is defined as
a
1
+
a
2
+
.
.
.
+
a
i
(
S
0
=
0
)
a_1+a_2+...+a_i(S_0=0)
a
1
+
a
2
+
...
+
a
i
(
S
0
=
0
)
. Show that there exists such
1
≤
k
≤
n
1\leq k\leq n
1
≤
k
≤
n
that satisfies
a
k
2
+
S
n
−
k
<
2
S
n
−
n
(
n
+
1
)
2
a_k^2+S_{n-k}<2S_n-\frac{n(n+1)}{2}
a
k
2
+
S
n
−
k
<
2
S
n
−
2
n
(
n
+
1
)
.
R,D,F is collinear
The acute triangle
A
B
C
ABC
A
BC
satisfies
A
B
‾
<
B
C
‾
<
C
A
‾
\overline {AB}<\overline {BC}<\overline {CA}
A
B
<
BC
<
C
A
. Let
H
H
H
a orthocenter of
A
B
C
ABC
A
BC
,
D
D
D
a intersection point of
A
H
AH
A
H
and
B
C
BC
BC
,
E
E
E
a intersection point of
B
H
BH
B
H
and
A
C
AC
A
C
, and
M
M
M
a midpoint of segment
B
C
BC
BC
. A circle with center
E
E
E
and radius
A
E
AE
A
E
intersects the segment
A
C
AC
A
C
at point
F
F
F
(
≠
A
\neq A
=
A
), and circumcircle of triangle
B
F
C
BFC
BFC
intersects the segment
A
M
AM
A
M
at point
S
S
S
. Let
P
P
P
(
≠
D
\neq D
=
D
),
Q
Q
Q
(
≠
F
\neq F
=
F
) a intersection point of circumcircle of triangle
A
S
D
ASD
A
S
D
and
D
F
DF
D
F
, circumcircle of triangle
A
S
F
ASF
A
SF
and
D
F
DF
D
F
respectively. Also, define
R
R
R
as a intersection point of circumcircles of triangle
A
H
Q
AHQ
A
H
Q
and
A
E
P
AEP
A
EP
. Prove that
R
R
R
lies on line
D
F
DF
D
F
.
8
2
Hide problems
number theory& polynomial
P
P
P
is an monic integer coefficient polynomial which has no integer roots. deg
P
=
n
P=n
P
=
n
and define
A
A
A
:
=
:=
:=
{
v
2
(
P
(
m
)
)
∣
m
∈
Z
,
v
2
(
P
(
m
)
)
≥
1
v_2(P(m))|m\in Z, v_2(P(m)) \ge 1
v
2
(
P
(
m
))
∣
m
∈
Z
,
v
2
(
P
(
m
))
≥
1
}. If
∣
A
∣
=
n
|A|=n
∣
A
∣
=
n
, show that all of the elements of
A
A
A
is smaller than
3
2
n
2
\frac{3}{2}n^2
2
3
n
2
.
Periodic function from N to R
For function
f
:
Z
+
→
R
f:\mathbb Z^+ \to \mathbb R
f
:
Z
+
→
R
and coprime positive integers
p
,
q
p,q
p
,
q
; define
f
p
,
f
q
f_p,f_q
f
p
,
f
q
as
f
p
(
x
)
=
f
(
p
x
)
−
f
(
x
)
,
f
q
(
x
)
=
f
(
q
x
)
−
f
(
x
)
(
x
∈
Z
+
)
f_p(x)=f(px)-f(x), f_q(x)=f(qx)-f(x) \space \space (x\in\mathbb Z^+)
f
p
(
x
)
=
f
(
p
x
)
−
f
(
x
)
,
f
q
(
x
)
=
f
(
q
x
)
−
f
(
x
)
(
x
∈
Z
+
)
f
f
f
satisfies following conditions.
(
i
)
(i)
(
i
)
for all
r
r
r
that isn't multiple of
p
q
pq
pq
,
f
(
r
)
=
0
f(r)=0
f
(
r
)
=
0
(
i
i
)
(ii)
(
ii
)
∃
m
∈
Z
+
\exists m\in \mathbb Z^+
∃
m
∈
Z
+
s
.
t
.
s.t.
s
.
t
.
∀
x
∈
Z
+
,
f
p
(
x
+
m
)
=
f
p
(
x
)
\forall x\in \mathbb Z^+, f_p(x+m)=f_p(x)
∀
x
∈
Z
+
,
f
p
(
x
+
m
)
=
f
p
(
x
)
and
f
q
(
x
+
m
)
=
f
q
(
x
)
f_q(x+m)=f_q(x)
f
q
(
x
+
m
)
=
f
q
(
x
)
Prove that if
x
≡
y
x\equiv y
x
≡
y
(
m
o
d
m
)
(mod m)
(
m
o
d
m
)
, then
f
(
x
)
=
f
(
y
)
f(x)=f(y)
f
(
x
)
=
f
(
y
)
(
x
,
y
∈
Z
+
x, y\in \mathbb Z^+
x
,
y
∈
Z
+
).
2
2
Hide problems
function
Find all functions
f
:
R
+
→
R
+
f:R^+\rightarrow R^+
f
:
R
+
→
R
+
such that for all positive reals
x
x
x
and
y
y
y
4
f
(
x
+
y
f
(
x
)
)
=
f
(
x
)
f
(
2
y
)
4f(x+yf(x))=f(x)f(2y)
4
f
(
x
+
y
f
(
x
))
=
f
(
x
)
f
(
2
y
)
<XAY >=60
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
=
6
0
∘
\angle A=60^{\circ}
∠
A
=
6
0
∘
. Point
D
,
E
D, E
D
,
E
in lines
A
B
→
,
A
C
→
\overrightarrow{AB}, \overrightarrow{AC}
A
B
,
A
C
respectively satisfies
D
B
=
B
C
=
C
E
DB=BC=CE
D
B
=
BC
=
CE
. (
A
,
B
,
D
A,B,D
A
,
B
,
D
lies on this order, and
A
,
C
,
E
A,C,E
A
,
C
,
E
likewise) Circle with diameter
B
C
BC
BC
and circle with diameter
D
E
DE
D
E
meets at two points
X
,
Y
X, Y
X
,
Y
. Prove that
∠
X
A
Y
≥
6
0
∘
\angle XAY\ge 60^{\circ}
∠
X
A
Y
≥
6
0
∘
4
2
Hide problems
Digging holes
A positive integer
m
(
≥
2
m(\ge 2
m
(
≥
2
) is given. From circle
C
1
C_1
C
1
with a radius 1, construct
C
2
,
C
3
,
C
4
,
.
.
.
C_2, C_3, C_4, ...
C
2
,
C
3
,
C
4
,
...
through following acts: In the
i
i
i
th act, select a circle
P
i
P_i
P
i
inside
C
i
C_i
C
i
with a area
1
m
\frac{1}{m}
m
1
of
C
i
C_i
C
i
. If such circle dosen't exist, the act ends. If not, let
C
i
+
1
C_{i+1}
C
i
+
1
a difference of sets
C
i
−
P
i
C_i -P_i
C
i
−
P
i
. Prove that this act ends within a finite number of times.
Lowest degree of polynomial
Find all
f
(
x
)
∈
Z
(
x
)
f(x)\in \mathbb Z (x)
f
(
x
)
∈
Z
(
x
)
that satisfies the following condition, with the lowest degree. Condition: There exists
g
(
x
)
,
h
(
x
)
∈
Z
(
x
)
g(x),h(x)\in \mathbb Z (x)
g
(
x
)
,
h
(
x
)
∈
Z
(
x
)
such that
f
(
x
)
4
+
2
f
(
x
)
+
2
=
(
x
4
+
2
x
2
+
2
)
g
(
x
)
+
3
h
(
x
)
f(x)^4+2f(x)+2=(x^4+2x^2+2)g(x)+3h(x)
f
(
x
)
4
+
2
f
(
x
)
+
2
=
(
x
4
+
2
x
2
+
2
)
g
(
x
)
+
3
h
(
x
)
.