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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1996 Dutch Mathematical Olympiad
1996 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
5
1
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1/x+1/y=1/z , gcd (x,y,z)=1 then (x + y) is a perfect square
For the positive integers
x
,
y
x , y
x
,
y
and
z
z
z
apply
1
x
+
1
y
=
1
z
\frac{1}{x}+\frac{1}{y}=\frac{1}{z}
x
1
+
y
1
=
z
1
. Prove that if the three numbers
x
,
y
,
x , y,
x
,
y
,
and
z
z
z
have no common divisor greater than
1
1
1
,
x
+
y
x + y
x
+
y
is the square of an integer.
4
1
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rectangle and tangent wanted, starting with 3 tangent circles in pairs
A line
l
l
l
intersects the segment
A
B
AB
A
B
perpendicular to
C
C
C
. Three circles are drawn successively with
A
B
,
A
C
AB, AC
A
B
,
A
C
and
B
C
BC
BC
as the diameter. The largest circle intersects
l
l
l
in
D
D
D
. The segments
D
A
DA
D
A
and
D
B
DB
D
B
still intersect the two smaller circles in
E
E
E
and
F
F
F
. a. Prove that quadrilateral
C
F
D
E
CFDE
CF
D
E
is a rectangle. b. Prove that the line through
E
E
E
and
F
F
F
touches the circles with diameters
A
C
AC
A
C
and
B
C
BC
BC
in
E
E
E
and
F
F
F
.[asy] unitsize (2.5 cm);pair A, B, C, D, E, F, O;O = (0,0); A = (-1,0); B = (1,0); C = (-0.3,0); D = intersectionpoint(C--(C + (0,1)), Circle(O,1)); E = (C + reflect(A,D)*(C))/2; F = (C + reflect(B,D)*(C))/2;draw(Circle(O,1)); draw(Circle((A + C)/2, abs(A - C)/2)); draw(Circle((B + C)/2, abs(B - C)/2)); draw(A--B); draw(interp(C,D,-0.4)--D); draw(A--D--B);dot("
A
A
A
", A, W); dot("
B
B
B
", B, dir(0)); dot("
C
C
C
", C, SE); dot("
D
D
D
", D, NW); dot("
E
E
E
", E, SE); dot("
F
F
F
", F, SW); [/asy]
3
1
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max horses in a chessboard
What is the largest number of horses that you can put on a chessboard without there being two horses that can beat each other? a. Describe an arrangement with that maximum number. b. Prove that a larger number is not possible.(A chessboard consists of
8
×
8
8 \times 8
8
×
8
spaces and a horse jumps from one field to another field according to the line "two squares vertically and one squared horizontally" or "one square vertically and two squares horizontally")[asy] unitsize (0.5 cm);int i, j;for (i = 0; i <= 7; ++i) { for (j = 0; j <= 7; ++j) { if ((i + j) % 2 == 0) { if ((i - 2)^2 + (j - 3)^2 == 5) { fill(shift((i,j))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red); } else { fill(shift((i,j))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.8)); } } }}for (i = 0; i <= 8; ++i) { draw((i,0)--(i,8)); draw((0,i)--(8,i)); }label("
a
a
a
", (0.5,-0.5), fontsize(10)); label("
b
b
b
", (1.5,-0.5), fontsize(10)); label("
c
c
c
", (2.5,-0.5), fontsize(10)); label("
d
d
d
", (3.5,-0.5), fontsize(10)); label("
e
e
e
", (4.5,-0.5), fontsize(10)); label("
f
f
f
", (5.5,-0.5), fontsize(10)); label("
g
g
g
", (6.5,-0.5), fontsize(10)); label("
h
h
h
", (7.5,-0.5), fontsize(10)); label("
1
1
1
", (-0.5,0.5), fontsize(10)); label("
2
2
2
", (-0.5,1.5), fontsize(10)); label("
3
3
3
", (-0.5,2.5), fontsize(10)); label("
4
4
4
", (-0.5,3.5), fontsize(10)); label("
5
5
5
", (-0.5,4.5), fontsize(10)); label("
6
6
6
", (-0.5,5.5), fontsize(10)); label("
7
7
7
", (-0.5,6.5), fontsize(10)); label("
8
8
8
", (-0.5,7.5), fontsize(10)); label("
P
P
P
", (2.5,3.5), fontsize(10)); [/asy]
2
1
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(m^2 + n) and (n^2 + m) are both perfect squares
Investigate whether for two positive integers
m
m
m
and
n
n
n
the numbers
m
2
+
n
m^2 + n
m
2
+
n
and
n
2
+
m
n^2 + m
n
2
+
m
can be both squares of integers.
1
1
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number of triangles with integer angles
How many different (non similar) triangles are there whose angles have an integer number of degrees?