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Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2010 Kurschak Competition
2010 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
Hide problems
Complete residue system of a[i]b[j]
For what positive integers
n
n
n
and
k
k
k
do there exits integers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots,a_n
a
1
,
a
2
,
…
,
a
n
and
b
1
,
b
2
,
…
,
b
k
b_1,b_2,\dots,b_k
b
1
,
b
2
,
…
,
b
k
such that the products
a
i
b
j
a_ib_j
a
i
b
j
(
1
≤
i
≤
n
,
1
≤
j
≤
k
1\le i\le n,1\le j\le k
1
≤
i
≤
n
,
1
≤
j
≤
k
) give pairwise different residues modulo
n
k
nk
nk
?
2
1
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Concurrence of three radical axes
Consider a triangle
A
B
C
ABC
A
BC
, with the points
A
1
A_1
A
1
,
A
2
A_2
A
2
on side
B
C
BC
BC
,
B
1
,
B
2
∈
A
C
‾
B_1,B_2\in\overline{AC}
B
1
,
B
2
∈
A
C
,
C
1
,
C
2
∈
A
B
‾
C_1,C_2\in\overline{AB}
C
1
,
C
2
∈
A
B
such that
A
C
1
<
A
C
2
AC_1<AC_2
A
C
1
<
A
C
2
,
B
A
1
<
B
A
2
BA_1<BA_2
B
A
1
<
B
A
2
,
C
B
1
<
C
B
2
CB_1<CB_2
C
B
1
<
C
B
2
. Let the circles
A
B
1
C
1
AB_1C_1
A
B
1
C
1
and
A
B
2
C
2
AB_2C_2
A
B
2
C
2
meet at
A
A
A
and
A
∗
A^*
A
∗
. Similarly, let the circles
B
C
1
A
1
BC_1A_1
B
C
1
A
1
and
B
C
2
A
2
BC_2A_2
B
C
2
A
2
intersect at
B
∗
≠
B
B^*\neq B
B
∗
=
B
, let
C
A
1
B
1
CA_1B_1
C
A
1
B
1
and
C
A
2
B
2
CA_2B_2
C
A
2
B
2
intersect at
C
∗
≠
C
C^*\neq C
C
∗
=
C
. Prove that the lines
A
A
∗
AA^*
A
A
∗
,
B
B
∗
BB^*
B
B
∗
,
C
C
∗
CC^*
C
C
∗
are concurrent.
1
1
Hide problems
Keys and chests
We have
n
n
n
keys, each of them belonging to exactly one of
n
n
n
locked chests. Our goal is to decide which key opens which chest. In one try we may choose a key and a chest, and check whether the chest can be opened with the key. Find the minimal number
p
(
n
)
p(n)
p
(
n
)
with the property that using
p
(
n
)
p(n)
p
(
n
)
tries, we can surely discover which key belongs to which chest.