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International Contests
Rioplatense Mathematical Olympiad, Level 3
2008 Rioplatense Mathematical Olympiad, Level 3
2008 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(3)
3
2
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Find k where n doesn't divide greatest odd divisor of k^n+1
Find all integers
k
≥
2
k\ge 2
k
≥
2
such that for all integers
n
≥
2
n\ge 2
n
≥
2
,
n
n
n
does not divide the greatest odd divisor of
k
n
+
1
k^n+1
k
n
+
1
.
Dividing stones into two groups with almost equal weights
Consider a collection of stones whose total weight is
65
65
65
pounds and each of whose stones is at most
w
w
w
pounds. Find the largest number
w
w
w
for which any such collection of stones can be divided into two groups whose total weights differ by at most one pound.Note: The weights of the stones are not necessarily integers.
2
2
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Assuming every distance between points in n closed intervals
On a line, there are
n
n
n
closed intervals (none of which is a single point) whose union we denote by
S
S
S
. It's known that for every real number
d
d
d
,
0
<
d
≤
1
0<d\le 1
0
<
d
≤
1
, there are two points in
S
S
S
that are a distance
d
d
d
from each other. (a) Show that the sum of the lengths of the
n
n
n
closed intervals is larger than
1
n
\frac{1}{n}
n
1
. (b) Prove that, for each positive integer
n
n
n
, the
1
n
\frac{1}{n}
n
1
in the statement of part (a) cannot be replaced with a larger number.
Perpendicularity in incircle/circumcircle/arc midpt diagram
In triangle
A
B
C
ABC
A
BC
, where
A
B
<
A
C
AB<AC
A
B
<
A
C
, let
X
X
X
,
Y
Y
Y
,
Z
Z
Z
denote the points where the incircle is tangent to
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
, respectively. On the circumcircle of
A
B
C
ABC
A
BC
, let
U
U
U
denote the midpoint of the arc
B
C
BC
BC
that contains the point
A
A
A
. The line
U
X
UX
U
X
meets the circumcircle again at the point
K
K
K
. Let
T
T
T
denote the point of intersection of
A
K
AK
A
K
and
Y
Z
YZ
Y
Z
. Prove that
X
T
XT
XT
is perpendicular to
Y
Z
YZ
Y
Z
.
1
2
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0-1 matrix w/ rows & cols having the same number of 0s or 1s
In each square of a chessboard with
a
a
a
rows and
b
b
b
columns, a
0
0
0
or
1
1
1
is written satisfying the following conditions. [*]If a row and a column intersect in a square with a
0
0
0
, then that row and column have the same number of
0
0
0
s. [*]If a row and a column intersect in a square with a
1
1
1
, then that row and column have the same number of
1
1
1
s. Find all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
for which this is possible.
Partition N so that k, 2k,..., 12k are in different subsets
Can the positive integers be partitioned into
12
12
12
subsets such that for each positive integer
k
k
k
, the numbers
k
,
2
k
,
…
,
12
k
k, 2k,\ldots,12k
k
,
2
k
,
…
,
12
k
belong to different subsets?