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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch IMO TST
2015 Dutch IMO TST
2015 Dutch IMO TST
Part of
Dutch IMO TST
Subcontests
(5)
5
1
Hide problems
D_n as the largest integer that is a divisor of a^n + (a + 1)^n + (a + 2)^n
For a positive integer
n
n
n
, we de ne
D
n
D_n
D
n
as the largest integer that is a divisor of
a
n
+
(
a
+
1
)
n
+
(
a
+
2
)
n
a^n + (a + 1)^n + (a + 2)^n
a
n
+
(
a
+
1
)
n
+
(
a
+
2
)
n
for all positive integers
a
a
a
. 1. Show that for all positive integers
n
n
n
, the number
D
n
D_n
D
n
is of the form
3
k
3^k
3
k
with
k
≥
0
k \ge 0
k
≥
0
an integer. 2. Show that for all integers
k
≥
0
k \ge 0
k
≥
0
there exists a positive integer n such that
D
n
=
3
k
D_n = 3^k
D
n
=
3
k
.
1
2
Hide problems
moving a pawn on grid points (x,y) to (x\pm a,y \pm a) or (x\pm b,y \pm b)
Let
a
a
a
and
b
b
b
be two positive integers satifying
g
c
d
(
a
,
b
)
=
1
gcd(a, b) = 1
g
c
d
(
a
,
b
)
=
1
. Consider a pawn standing on the grid point
(
x
,
y
)
(x, y)
(
x
,
y
)
. A step of type A consists of moving the pawn to one of the following grid points:
(
x
+
a
,
y
+
a
)
,
(
x
+
a
,
y
−
a
)
,
(
x
−
a
,
y
+
a
)
(x+a, y+a),(x+a,y-a), (x-a, y + a)
(
x
+
a
,
y
+
a
)
,
(
x
+
a
,
y
−
a
)
,
(
x
−
a
,
y
+
a
)
or
(
x
−
a
,
y
−
a
)
(x - a, y - a)
(
x
−
a
,
y
−
a
)
. A step of type B consists of moving the pawn to
(
x
+
b
,
y
+
b
)
,
(
x
+
b
,
y
−
b
)
,
(
x
−
b
,
y
+
b
)
(x + b,y + b),(x + b,y - b), (x - b,y + b)
(
x
+
b
,
y
+
b
)
,
(
x
+
b
,
y
−
b
)
,
(
x
−
b
,
y
+
b
)
or
(
x
−
b
,
y
−
b
)
(x - b,y - b)
(
x
−
b
,
y
−
b
)
. Now put a pawn on
(
0
,
0
)
(0, 0)
(
0
,
0
)
. You can make a ( nite) number of steps, alternatingly of type A and type B, starting with a step of type A. You can make an even or odd number of steps, i.e., the last step could be of either type A or type B. Determine the set of all grid points
(
x
,
y
)
(x,y)
(
x
,
y
)
that you can reach with such a series of steps.
<ADB = < EDC iff MA = MC, ABCD with <A=<C=90^o given
In a quadrilateral
A
B
C
D
ABCD
A
BC
D
we have
∠
A
=
∠
C
=
9
0
o
\angle A = \angle C = 90^o
∠
A
=
∠
C
=
9
0
o
. Let
E
E
E
be a point in the interior of
A
B
C
D
ABCD
A
BC
D
. Let
M
M
M
be the midpoint of
B
E
BE
BE
. Prove that
∠
A
D
B
=
∠
E
D
C
\angle ADB = \angle EDC
∠
A
D
B
=
∠
E
D
C
if and only if
∣
M
A
∣
=
∣
M
C
∣
|MA| = |MC|
∣
M
A
∣
=
∣
MC
∣
.
4
2
Hide problems
each of numbers 1-2014 colored red and blue, half each, max sum 2 balls
Each of the numbers
1
1
1
up to and including
2014
2014
2014
has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number
k
k
k
of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of
k
k
k
that can be obtained.
perpendicular wanted, intersecting circles and parallel given
Let
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
be circles - with respective centres
O
1
O_1
O
1
and
O
2
O_2
O
2
- that intersect each other in
A
A
A
and
B
B
B
. The line
O
1
A
O_1A
O
1
A
intersects
Γ
2
\Gamma_2
Γ
2
in
A
A
A
and
C
C
C
and the line
O
2
A
O_2A
O
2
A
intersects
Γ
1
\Gamma_1
Γ
1
in
A
A
A
and
D
D
D
. The line through
B
B
B
parallel to
A
D
AD
A
D
intersects
Γ
1
\Gamma_1
Γ
1
in
B
B
B
and
E
E
E
. Suppose that
O
1
A
O_1A
O
1
A
is parallel to
D
E
DE
D
E
. Show that
C
D
CD
C
D
is perpendicular to
O
2
C
O_2C
O
2
C
.
3
2
Hide problems
c_1 + c_2 + ... + c_k = n^2 - 1, when c_1=a_i if a_i = a_{i-1} etc
Let
n
n
n
be a positive integer. Consider sequences
a
0
,
a
1
,
.
.
.
,
a
k
a_0, a_1, ..., a_k
a
0
,
a
1
,
...
,
a
k
and
b
0
,
b
1
,
,
.
.
,
b
k
b_0, b_1,,..,b_k
b
0
,
b
1
,,
..
,
b
k
such that
a
0
=
b
0
=
1
a_0 = b_0 = 1
a
0
=
b
0
=
1
and
a
k
=
b
k
=
n
a_k = b_k = n
a
k
=
b
k
=
n
and such that for all
i
i
i
such that
1
≤
i
≤
k
1 \le i \le k
1
≤
i
≤
k
, we have that
(
a
i
,
b
i
)
(a_i, b_i)
(
a
i
,
b
i
)
is either equal to
(
1
+
a
i
−
1
,
b
i
−
1
)
(1 + a_{i-1}, b_{i-1})
(
1
+
a
i
−
1
,
b
i
−
1
)
or
(
a
i
−
1
;
1
+
b
i
−
1
)
(a_{i-1}; 1 + b_{i-1})
(
a
i
−
1
;
1
+
b
i
−
1
)
. Consider for
1
≤
i
≤
k
1 \le i \le k
1
≤
i
≤
k
the number
c
i
=
{
a
i
i
f
a
i
=
a
i
−
1
b
i
i
f
b
i
=
b
i
−
1
c_i = \begin{cases} a_i \,\,\, if \,\,\, a_i = a_{i-1} \\ b_i \,\,\, if \,\,\, b_i = b_{i-1}\end{cases}
c
i
=
{
a
i
i
f
a
i
=
a
i
−
1
b
i
i
f
b
i
=
b
i
−
1
Show that
c
1
+
c
2
+
.
.
.
+
c
k
=
n
2
−
1
c_1 + c_2 + ... + c_k = n^2 - 1
c
1
+
c
2
+
...
+
c
k
=
n
2
−
1
.
equilateral given and wanted, IMO TST problem
An equilateral triangle
A
B
C
ABC
A
BC
is given. On the line through
B
B
B
parallel to
A
C
AC
A
C
there is a point
D
D
D
, such that
D
D
D
and
C
C
C
are on the same side of the line
A
B
AB
A
B
. The perpendicular bisector of
C
D
CD
C
D
intersects the line
A
B
AB
A
B
in
E
E
E
. Prove that triangle
C
D
E
CDE
C
D
E
is equilateral.
2
2
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Polynomials
Determine all polynomials P(x) with real coefficients such that [(x + 1)P(x − 1) − (x − 1)P(x)] is a constant polynomial.
system in N*: a_1 + 2a_2 +... + na_n = 6n, 1/a_1+2/a_2+... +n/a_n=2+1/n
Determine all positive integers
n
n
n
for which there exist positive integers
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2, ..., a_n
a
1
,
a
2
,
...
,
a
n
with
a
1
+
2
a
2
+
3
a
3
+
.
.
.
+
n
a
n
=
6
n
a_1 + 2a_2 + 3a_3 +... + na_n = 6n
a
1
+
2
a
2
+
3
a
3
+
...
+
n
a
n
=
6
n
and
1
a
1
+
2
a
2
+
3
a
3
+
.
.
.
+
n
a
n
=
2
+
1
n
\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n
a
1
1
+
a
2
2
+
a
3
3
+
...
+
a
n
n
=
2
+
n
1