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Rioplatense Mathematical Olympiad, Level 3
2005 Rioplatense Mathematical Olympiad, Level 3
2005 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(3)
2
2
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Equal angles in trapezoid where sum of the bases = diagonal
In trapezoid
A
B
C
D
ABCD
A
BC
D
, the sum of the lengths of the bases
A
B
AB
A
B
and
C
D
CD
C
D
is equal to the length of the diagonal
B
D
BD
B
D
. Let
M
M
M
denote the midpoint of
B
C
BC
BC
, and let
E
E
E
denote the reflection of
C
C
C
about the line
D
M
DM
D
M
. Prove that
∠
A
E
B
=
∠
A
C
D
\angle AEB=\angle ACD
∠
A
EB
=
∠
A
C
D
.
Worst approximation to 100 by summing subsequences
Consider all finite sequences of positive real numbers each of whose terms is at most
3
3
3
and the sum of whose terms is more than
100
100
100
. For each such sequence, let
S
S
S
denote the sum of the subsequence whose sum is the closest to
100
100
100
, and define the defect of this sequence to be the value
∣
S
−
100
∣
|S-100|
∣
S
−
100∣
. Find the maximum possible value of the defect.
1
2
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Numbers of the form n = k + 2[sqrt(k)] + 2 where k in N
Find all numbers
n
n
n
that can be expressed in the form
n
=
k
+
2
⌊
k
⌋
+
2
n=k+2\lfloor\sqrt{k}\rfloor+2
n
=
k
+
2
⌊
k
⌋
+
2
for some nonnegative integer
k
k
k
.
Circumradius ineq PA/(AB*AC) + PB/(BC*BA) + PC/(CA*CB) > 1/R
Let
P
P
P
be a point inside triangle
A
B
C
ABC
A
BC
and let
R
R
R
denote the circumradius of triangle
A
B
C
ABC
A
BC
. Prove that
P
A
A
B
⋅
A
C
+
P
B
B
C
⋅
B
A
+
P
C
C
A
⋅
C
B
≥
1
R
.
\frac{PA}{AB\cdot AC}+\frac{PB}{BC\cdot BA}+\frac{PC}{CA\cdot CB}\ge\frac{1}{R}.
A
B
⋅
A
C
P
A
+
BC
⋅
B
A
PB
+
C
A
⋅
CB
PC
≥
R
1
.
3
2
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Integer divisible by numbers obtained by deleting its digits
Find the largest positive integer
n
n
n
not divisible by
10
10
10
which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of
n
n
n
. (Note:
n
n
n
is written in the usual base ten notation.)
rioplatense(2005/6)
Let
k
k
k
be a positive integer. Show that for all
n
>
k
n>k
n
>
k
there exist convex figures
F
1
,
…
,
F
n
F_{1},\ldots, F_{n}
F
1
,
…
,
F
n
and
F
F
F
such that there doesn't exist a subset of
k
k
k
elements from
F
1
,
.
.
.
,
F
n
F_{1},..., F_{n}
F
1
,
...
,
F
n
and
F
F
F
is covered for this elements, but
F
F
F
is covered for every subset of
k
+
1
k+1
k
+
1
elements from
F
1
,
F
2
,
.
.
.
.
.
,
F
n
F_{1}, F_{2},....., F_{n}
F
1
,
F
2
,
.....
,
F
n
.