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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch BxMO/EGMO TST
2014 Dutch BxMO/EGMO TST
2014 Dutch BxMO/EGMO TST
Part of
Dutch BxMO/EGMO TST
Subcontests
(5)
5
1
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Game of writing numbers from 1 to n on k pages
Let
n
n
n
be a positive integer. Daniel and Merlijn are playing a game. Daniel has
k
k
k
sheets of paper lying next to each other on a table, where
k
k
k
is a positive integer. On each of the sheets, he writes some of the numbers from
1
1
1
up to
n
n
n
(he is allowed to write no number at all, or all numbers). On the back of each of the sheets, he writes down the remaining numbers. Once Daniel is finished, Merlijn can flip some of the sheets of paper (he is allowed to flip no sheet at all, or all sheets). If Merlijn succeeds in making all of the numbers from
1
1
1
up to n visible at least once, then he wins. Determine the smallest
k
k
k
for which Merlijn can always win, regardless of Daniel’s actions.
4
1
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Divisor of n! that does not have factors from m to n.
Let
m
≥
3
m\ge 3
m
≥
3
and
n
n
n
be positive integers such that
n
>
m
(
m
−
2
)
n>m(m-2)
n
>
m
(
m
−
2
)
. Find the largest positive integer
d
d
d
such that
d
∣
n
!
d\mid n!
d
∣
n
!
and
k
∤
d
k\nmid d
k
∤
d
for all
k
∈
{
m
,
m
+
1
,
…
,
n
}
k\in\{m,m+1,\ldots,n\}
k
∈
{
m
,
m
+
1
,
…
,
n
}
.
3
1
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Length relations and circle tangent to angle bisector.
In triangle
A
B
C
ABC
A
BC
,
I
I
I
is the centre of the incircle. There is a circle tangent to
A
I
AI
A
I
at
I
I
I
which passes through
B
B
B
. This circle intersects
A
B
AB
A
B
once more in
P
P
P
and intersects
B
C
BC
BC
once more in
Q
Q
Q
. The line
Q
I
QI
Q
I
intersects
A
C
AC
A
C
in
R
R
R
. Prove that
∣
A
R
∣
⋅
∣
B
Q
∣
=
∣
P
I
∣
2
|AR|\cdot |BQ|=|P I|^2
∣
A
R
∣
⋅
∣
BQ
∣
=
∣
P
I
∣
2
2
1
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xf(xy)+f(-y)=xf(x)
Find all functions
f
:
R
\
{
0
}
→
R
f:\mathbb{R}\backslash\{0\}\rightarrow\mathbb{R}
f
:
R
\
{
0
}
→
R
for which
x
f
(
x
y
)
+
f
(
−
y
)
=
x
f
(
x
)
xf(xy) + f(-y) = xf(x)
x
f
(
x
y
)
+
f
(
−
y
)
=
x
f
(
x
)
for all non-zero real numbers
x
,
y
x, y
x
,
y
.
1
1
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n^2=a+b, n^3=a^2+b^2
Find all non-negative integer numbers
n
n
n
for which there exists integers
a
a
a
and
b
b
b
such that
n
2
=
a
+
b
n^2=a+b
n
2
=
a
+
b
and
n
3
=
a
2
+
b
2
.
n^3=a^2+b^2.
n
3
=
a
2
+
b
2
.