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Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1975 Kurschak Competition
1975 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
1
1
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ab^2 (1/(a + c)^2 +1/(a- c)^2 ) = (a -b)
Transform the equation
a
b
2
(
1
(
a
+
c
)
2
+
1
(
a
−
c
)
2
)
=
(
a
−
b
)
ab^2 \left(\frac{1}{(a + c)^2} +\frac{1}{(a- c)^2} \right) = (a -b)
a
b
2
(
(
a
+
c
)
2
1
+
(
a
−
c
)
2
1
)
=
(
a
−
b
)
into a simpler form, given that
a
>
c
≥
0
a > c \ge 0
a
>
c
≥
0
,
b
>
0
b > 0
b
>
0
.
3
1
Hide problems
45 < x_{1000} < 45.1 if x_0 = 5. x_{n+1} = x_n +1/x_n
Let
x
0
=
5
,
x
n
+
1
=
x
n
+
1
x
n
.
x_0 = 5\,\, ,\, \,\,x_{n+1} = x_n +\frac{1}{x_n}.
x
0
=
5
,
x
n
+
1
=
x
n
+
x
n
1
.
Prove that
45
<
x
1000
<
45.1
45 < x_{1000} < 45.1
45
<
x
1000
<
45.1
.
2
1
Hide problems
rhombus and quad/ inscirbed in convex polygon, side of rhombus larger
Prove or disprove: given any quadrilateral inscribed in a convex polygon, we can find a rhombus inscribed in the polygon with side not less than the shortest side of the quadrilateral.