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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1904 Eotvos Mathematical Competition
1904 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
2
1
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ni of pos. integer solutions of x_1 + 2x_2 + 3x_3 + ... + nx_n = a
If a is a natural number, show that the number of positive integral solutions of the indeterminate equation
x
1
+
2
x
2
+
3
x
3
+
.
.
.
+
n
x
n
=
a
(
1
)
x_1 + 2x_2 + 3x_3 + ... + nx_n = a \ \ (1)
x
1
+
2
x
2
+
3
x
3
+
...
+
n
x
n
=
a
(
1
)
is equal to the number of non-negative integral solutions of
y
1
+
2
y
2
+
3
y
3
+
.
.
.
+
n
y
n
=
a
−
n
(
n
+
1
)
2
(
2
)
y_1 + 2y_2 + 3y_3 + ... + ny_n = a - \frac{n(n + 1)}{2} \ \ (2)
y
1
+
2
y
2
+
3
y
3
+
...
+
n
y
n
=
a
−
2
n
(
n
+
1
)
(
2
)
[By a solution of equation (1), we mean a set of numbers
{
x
1
,
x
2
,
.
.
.
,
x
n
}
\{x_1, x_2,..., x_n\}
{
x
1
,
x
2
,
...
,
x
n
}
which satisfies equation (1)].
3
1
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A_1P > OP, A_2P > OP, B_1P > OP , B_2P > OP, rectangle
Let
A
1
A
2
A_1A_2
A
1
A
2
and
B
1
B
2
B_1B_2
B
1
B
2
be the diagonals of a rectangle, and let
O
O
O
be its center. Find and construct the set of all points
P
P
P
that satisfy simultaneously the four inequaliies:
A
1
P
>
O
P
,
A
2
P
>
O
P
,
B
1
P
>
O
P
,
B
2
P
>
O
P
.
A_1P > OP , \\A_2P > OP, \ \ B_1P > OP , \ \ B_2P > OP.
A
1
P
>
OP
,
A
2
P
>
OP
,
B
1
P
>
OP
,
B
2
P
>
OP
.
1
1
Hide problems
equiangular cyclic pentagon is regular
Prove that, if a pentagon (five-sided polygon) inscribed in a circle has equal angles, then its sides are equal.