MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea Winter Program Practice Test
2017 Korea Winter Program Practice Test
2017 Korea Winter Program Practice Test
Part of
Korea Winter Program Practice Test
Subcontests
(4)
4
3
Hide problems
Equilateral triangle with vertices near lattices points
For a point
P
P
P
on the plane, denote by
∥
P
∥
\lVert P \rVert
∥
P
∥
the distance to its nearest lattice point. Prove that there exists a real number
L
>
0
L > 0
L
>
0
satisfying the following condition: For every
ℓ
>
L
\ell > L
ℓ
>
L
, there exists an equilateral triangle
A
B
C
ABC
A
BC
with side-length
ℓ
\ell
ℓ
and
∥
A
∥
,
∥
B
∥
,
∥
C
∥
<
1
0
−
2017
\lVert A \rVert, \lVert B \rVert, \lVert C \rVert < 10^{-2017}
∥
A
∥
,
∥
B
∥
,
∥
C
∥
<
1
0
−
2017
.
Bisecting a set into 2-adic squares and non-squares
For a nonzero integer
k
k
k
, denote by
ν
2
(
k
)
\nu_2(k)
ν
2
(
k
)
the maximal nonnegative integer
t
t
t
such that
2
t
∣
k
2^t \mid k
2
t
∣
k
. Given are
n
(
≥
2
)
n (\ge 2)
n
(
≥
2
)
pairwise distinct integers
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
. Show that there exists an integer
x
x
x
, distinct from
a
1
,
…
,
a
n
a_1, \ldots, a_n
a
1
,
…
,
a
n
, such that among
ν
2
(
x
−
a
1
)
,
…
,
ν
2
(
x
−
a
n
)
\nu_2(x - a_1), \ldots, \nu_2(x - a_n)
ν
2
(
x
−
a
1
)
,
…
,
ν
2
(
x
−
a
n
)
there are at least
n
/
4
n/4
n
/4
odd numbers and at least
n
/
4
n/4
n
/4
even numbers.
Inequality of four variables
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be the area of four faces of a tetrahedron, satisfying
a
+
b
+
c
+
d
=
1
a+b+c+d=1
a
+
b
+
c
+
d
=
1
. Show that
a
n
+
b
n
+
c
n
n
+
b
n
+
c
n
+
d
n
n
+
c
n
+
d
n
+
a
n
n
+
d
n
+
a
n
+
b
n
n
≤
1
+
2
n
\sqrt[n]{a^n+b^n+c^n}+\sqrt[n]{b^n+c^n+d^n}+\sqrt[n]{c^n+d^n+a^n}+\sqrt[n]{d^n+a^n+b^n} \le 1+\sqrt[n]{2}
n
a
n
+
b
n
+
c
n
+
n
b
n
+
c
n
+
d
n
+
n
c
n
+
d
n
+
a
n
+
n
d
n
+
a
n
+
b
n
≤
1
+
n
2
holds for all positive integers
n
n
n
.
3
3
Hide problems
Uniform number of distinct roots
Do there exist polynomials
f
(
x
)
f(x)
f
(
x
)
,
g
(
x
)
g(x)
g
(
x
)
with real coefficients and a positive integer
k
k
k
satisfying the following condition? (Here, the equation
x
2
=
0
x^2 = 0
x
2
=
0
is considered to have
1
1
1
distinct real roots. The equation
0
=
0
0 = 0
0
=
0
has infinitely many distinct real roots.) For any real numbers
a
,
b
a, b
a
,
b
with
(
a
,
b
)
≠
(
0
,
0
)
(a,b) \neq (0,0)
(
a
,
b
)
=
(
0
,
0
)
, the number of distinct real roots of
a
f
(
x
)
+
b
g
(
x
)
=
0
a f(x) + b g(x) = 0
a
f
(
x
)
+
b
g
(
x
)
=
0
is
k
k
k
.
Intersection satisfying a symmetric condition
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle with
∠
A
≠
6
0
∘
\angle A \neq 60^\circ
∠
A
=
6
0
∘
. Let
I
B
,
I
C
I_B, I_C
I
B
,
I
C
be the
B
,
C
B, C
B
,
C
-excenters of triangle
A
B
C
ABC
A
BC
, let
B
′
B^\prime
B
′
be the reflection of
B
B
B
with respect to
A
C
AC
A
C
, and let
C
′
C^\prime
C
′
be the reflection of
C
C
C
with respect to
A
B
AB
A
B
. Let
P
P
P
be the intersection of
I
C
B
′
I_C B^\prime
I
C
B
′
and
I
B
C
′
I_B C^\prime
I
B
C
′
. Denote by
P
A
,
P
B
,
P
C
P_A, P_B, P_C
P
A
,
P
B
,
P
C
the reflections of the point
P
P
P
with respect to
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
. Show that the three lines
A
P
A
,
B
P
B
,
C
P
C
A P_A, B P_B, C P_C
A
P
A
,
B
P
B
,
C
P
C
meet at a single point.
Inductive 0/1 sequence
For a number consists of
0
0
0
and
1
1
1
, one can perform the following operation: change all
1
1
1
into
100
100
100
, all
0
0
0
into
1
1
1
. For all nonnegative integer
n
n
n
, let
A
n
A_n
A
n
be the number obtained by performing the operation
n
n
n
times on
1
1
1
(starts with
100
,
10011
,
10011100100
,
…
100,10011,10011100100,\dots
100
,
10011
,
10011100100
,
…
), and
a
n
a_n
a
n
be the
n
n
n
-th digit(from the left side) of
A
n
A_n
A
n
. Prove or disprove that there exists a positive integer
m
m
m
satisfies the following:For every positive integer
l
l
l
, there exists a positive integer
k
≤
m
k\le m
k
≤
m
satisfying
a
l
+
k
+
1
=
a
1
,
a
l
+
k
+
2
=
a
2
,
…
,
a
l
+
k
+
2017
=
a
2017
a_{l+k+1}=a_1,\ a_{l+k+2}=a_2,\ \dots,\ a_{l+k+2017}=a_{2017}
a
l
+
k
+
1
=
a
1
,
a
l
+
k
+
2
=
a
2
,
…
,
a
l
+
k
+
2017
=
a
2017
2
4
Show problems
1
4
Show problems