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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1991 Spain Mathematical Olympiad
1991 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(6)
2
1
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row of matrix A is linear combination of the rows in S with integer coefficients
Given two distinct elements
a
,
b
∈
{
−
1
,
0
,
1
}
a,b \in \{-1,0,1\}
a
,
b
∈
{
−
1
,
0
,
1
}
, consider the matrix
A
A
A
. Find a subset
S
S
S
of the set of the rows of
A
A
A
, of minimum size, such that every other row of
A
A
A
is a linear combination of the rows in
S
S
S
with integer coefficients.
3
1
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roots of the polynomial x^3 −ux^2+vx−w are sides of triangle, conditions
What condition must be satisfied by the coefficients
u
,
v
,
w
u,v,w
u
,
v
,
w
if the roots of the polynomial
x
3
−
u
x
2
+
v
x
−
w
x^3 -ux^2+vx-w
x
3
−
u
x
2
+
vx
−
w
are the sides of a triangle
1
1
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set of all segments of integer lengths whose endpoints are lattice points
In the coordinate plane, consider the set of all segments of integer lengths whose endpoints have integer coordinates. Prove that no two of these segments form an angle of
4
5
o
45^o
4
5
o
. Are there such segments in coordinate space?
5
1
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Σ s(m), from m=1 to 2^k, where s(n) is the sum of binary digits of n
For a positive integer
n
n
n
, let
s
(
n
)
s(n)
s
(
n
)
denote the sum of the binary digits of
n
n
n
. Find the sum
s
(
1
)
+
s
(
2
)
+
s
(
3
)
+
.
.
.
+
s
(
2
k
)
s(1)+s(2)+s(3)+...+s(2^k)
s
(
1
)
+
s
(
2
)
+
s
(
3
)
+
...
+
s
(
2
k
)
for each positive integer
k
k
k
.
4
1
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angle chasing with the incircle
The incircle of
A
B
C
ABC
A
BC
touches the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
A
′
,
B
′
,
C
′
A' ,B' ,C'
A
′
,
B
′
,
C
′
respectively. The line
A
′
C
′
A' C'
A
′
C
′
meets the angle bisector of
∠
A
\angle A
∠
A
at
D
D
D
. Find
∠
A
D
C
\angle ADC
∠
A
D
C
.
6
1
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spain 1991
Find the integer part of
1
1
+
1
2
+
1
3
+
.
.
.
+
1
1000
\frac{1}{\sqrt1}+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}+...+\frac{1}{\sqrt{1000}}
1
1
+
2
1
+
3
1
+
...
+
1000
1