MathDB
Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
1999 German National Olympiad
1999 German National Olympiad
Part of
German National Olympiad
Subcontests
(7)
6b
1
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4^m + 5^n is a perfect square
Determine all pairs (
m
,
n
m,n
m
,
n
) of natural numbers for which
4
m
+
5
n
4^m + 5^n
4
m
+
5
n
is a perfect square.
6a
1
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isosceles right-angled triangle is divided into m acute-angled triangles.
Suppose that an isosceles right-angled triangle is divided into
m
m
m
acute-angled triangles. Find the smallest possible
m
m
m
for which this is possible.
5
1
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|x-y|+|y-z|+|z-x| \le a \sqrt{x^2 +y^2 +z^2}
Consider the following inequality for real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
:
∣
x
−
y
∣
+
∣
y
−
z
∣
+
∣
z
−
x
∣
≤
a
x
2
+
y
2
+
z
2
|x-y|+|y-z|+|z-x| \le a \sqrt{x^2 +y^2 +z^2}
∣
x
−
y
∣
+
∣
y
−
z
∣
+
∣
z
−
x
∣
≤
a
x
2
+
y
2
+
z
2
. (a) Prove that the inequality is valid for
a
=
2
2
a = 2\sqrt2
a
=
2
2
(b) Assuming that
x
,
y
,
z
x,y,z
x
,
y
,
z
are nonnegative, show that the inequality is also valid for
a
=
2
a = 2
a
=
2
.
4
1
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convex polygon P is placed inside a unit square Q, perimeter P <=4
A convex polygon
P
P
P
is placed inside a unit square
Q
Q
Q
. Prove that the perimeter of
P
P
P
does not exceed
4
4
4
.
3
1
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A =\frac{a+c}{2\frac{b+d}{2} , A = \sqrt{(p-a)(p-b)(p-c)(p-d)} , area quadril.
A mathematician investigates methods of finding area of a convex quadrilateral obtains the following formula for the area
A
A
A
of a quadrilateral with consecutive sides
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
:
A
=
a
+
c
2
b
+
d
2
A =\frac{a+c}{2}\frac{b+d}{2}
A
=
2
a
+
c
2
b
+
d
(1) and
A
=
(
p
−
a
)
(
p
−
b
)
(
p
−
c
)
(
p
−
d
)
A = \sqrt{(p-a)(p-b)(p-c)(p-d)}
A
=
(
p
−
a
)
(
p
−
b
)
(
p
−
c
)
(
p
−
d
)
(2) where
p
=
(
a
+
b
+
c
+
d
)
/
2
p = (a+b+c+d)/2
p
=
(
a
+
b
+
c
+
d
)
/2
. However, these formulas are not valid for all convex quadrilaterals. Prove that (1) holds if and only if the quadrilateral is a rectangle, while (2) holds if and only if the quadrilateral is cyclic.
2
1
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1+\frac{x}{2} -\frac{x^2}{8} \le \sqrt{1+x} \le 1+\frac{x}{2}
Determine all real numbers
x
x
x
for which
1
+
x
2
−
x
2
8
≤
1
+
x
≤
1
+
x
2
1+\frac{x}{2} -\frac{x^2}{8} \le \sqrt{1+x} \le 1+\frac{x}{2}
1
+
2
x
−
8
x
2
≤
1
+
x
≤
1
+
2
x
1
1
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x^2 +xy+y^2 = 97 diophantine
Find all
x
,
y
x,y
x
,
y
which satisfy the equality
x
2
+
x
y
+
y
2
=
97
x^2 +xy+y^2 = 97
x
2
+
x
y
+
y
2
=
97
, when
x
,
y
x,y
x
,
y
are a) natural numbers, b) integers