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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2018 Kürschák Competition
2018 Kürschák Competition
Part of
Kürschák Math Competition
Subcontests
(3)
2
1
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p-length vectors
Given a prime number
p
p
p
and let
v
1
‾
,
v
2
‾
,
…
,
v
n
‾
\overline{v_1},\overline{v_2},\dotsc ,\overline{v_n}
v
1
,
v
2
,
…
,
v
n
be
n
n
n
distinct vectors of length
p
p
p
with integer coordinates in an
R
3
\mathbb{R}^3
R
3
Cartesian coordinate system. Suppose that for any
1
⩽
j
<
k
⩽
n
1\leqslant j<k\leqslant n
1
⩽
j
<
k
⩽
n
, there exists an integer
0
<
ℓ
<
p
0<\ell <p
0
<
ℓ
<
p
such that all three coordinates of
v
j
‾
−
ℓ
⋅
v
k
‾
\overline{v_j} -\ell \cdot \overline{v_k}
v
j
−
ℓ
⋅
v
k
is divisible by
p
p
p
. Prove that
n
⩽
6
n\leqslant 6
n
⩽
6
.
1
1
Hide problems
Seem like some well-known lemma
Given a triangle
A
B
C
ABC
A
BC
with its incircle touching sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
, respectively. Let the median from
A
A
A
intersects
B
1
C
1
B_1C_1
B
1
C
1
at
M
M
M
. Show that
A
1
M
⊥
B
C
A_1M\perp BC
A
1
M
⊥
BC
.
3
1
Hide problems
A beautiful problem from a Hungarian Olympiad
In a village (where only dwarfs live) there are
k
k
k
streets, and there are
k
(
n
−
1
)
+
1
k(n-1)+1
k
(
n
−
1
)
+
1
clubs each containing
n
n
n
dwarfs. A dwarf can be in more than one clubs, and two dwarfs know each other if they live in the same street or they are in the same club (there is a club they are both in). Prove that is it possible to choose
n
n
n
different dwarfs from
n
n
n
different clubs (one dwarf from each club), such that the
n
n
n
dwarfs know each other!