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Problems
Contests
National and Regional Contests
Japan Contests
Japan MO Finals
1999 Japan MO Finals
1999 Japan MO Finals
Part of
Japan MO Finals
Subcontests
(5)
2
1
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Finding x such that 3^n||x^3+17
Let
f
(
x
)
=
x
3
+
17
f(x)=x^3+17
f
(
x
)
=
x
3
+
17
. Prove that for every integer
n
≥
2
n\ge 2
n
≥
2
there exists a natural number
x
x
x
for which
f
(
x
)
f(x)
f
(
x
)
is divisible by
3
n
3^n
3
n
but not by
3
n
+
1
3^{n+1}
3
n
+
1
.
1
1
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Placing stones on a 1999 x 1999 board
One can place a stone on each of the squares of a
1999
×
1999
1999\times 1999
1999
×
1999
board. Find the minimum number of stones that must be placed so that, for any blank square on the board, the total number of stones placed in the corresponding row and column is at least
1999
1999
1999
.
5
1
Hide problems
Find range of the max/min of diagonals in ABCDEF
All sides of a convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
are
1
1
1
. Let
M
,
m
M,m
M
,
m
be the maximum and minimum possible values of three diagonals
A
D
,
B
E
,
C
F
AD,BE,CF
A
D
,
BE
,
CF
. Find the range of
M
M
M
,
m
m
m
.
3
1
Hide problems
Weights
From a group of
2
n
+
1
2n+1
2
n
+
1
weights, if we remove any weight, the remaining
2
n
2n
2
n
, can be divided in two groups of
n
n
n
elements, such that they have the same total weight. Prove all weights are equal.
4
1
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Function
Prove that the polynomial
f
(
x
)
=
(
x
2
+
1
)
(
x
2
+
2
2
)
⋯
(
x
2
+
n
2
)
+
1
f(x)=(x^2+1)(x^2+2^2)\cdots (x^2+n^2)+1
f
(
x
)
=
(
x
2
+
1
)
(
x
2
+
2
2
)
⋯
(
x
2
+
n
2
)
+
1
cannot be expressed as a product of two polynomials with integer coefficients with degree greater than
1
1
1
.