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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1977 Spain Mathematical Olympiad
1977 Spain Mathematical Olympiad
Part of
Spain Mathematical Olympiad
Subcontests
(8)
1
1
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determinant nxn is a multiple of 10
Given the determinant of order
n
n
n
∣
8
3
3
…
3
3
8
3
…
3
3
3
8
…
3
⋮
⋮
⋮
⋱
⋮
3
3
3
…
8
∣
\begin{vmatrix} 8 & 3 & 3 & \dots & 3 \\ 3 & 8 & 3 & \dots & 3 \\ 3 & 3 & 8 & \dots & 3 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 3 & 3 & 3 & \dots & 8 \end{vmatrix}
8
3
3
⋮
3
3
8
3
⋮
3
3
3
8
⋮
3
…
…
…
⋱
…
3
3
3
⋮
8
Calculate its value and determine for which values of
n
n
n
this value is a multiple of
10
10
10
.
2
1
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2x2 matrices form a commutative field
Prove that all square matrices of the form (with
a
,
b
∈
R
a, b \in R
a
,
b
∈
R
),
(
a
b
−
b
a
)
\begin{pmatrix} a & b \\ -b & a \end{pmatrix}
(
a
−
b
b
a
)
form a commutative field
K
K
K
when considering the operations of addition and matrix product. Prove also that if
A
∈
K
A \in K
A
∈
K
is an element of said field, there exist two matrices of
K
K
K
such that the square of each is equal to
A
A
A
.
7
1
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min sum (x-A_i)^2
The numbers
A
1
,
A
2
,
.
.
.
,
A
n
A_1 , A_2 ,... , A_n
A
1
,
A
2
,
...
,
A
n
are given. Prove, without calculating derivatives, that the value of
X
X
X
that minimizes the sum
(
X
−
A
1
)
2
+
(
X
−
A
2
)
2
+
.
.
.
+
(
X
−
A
n
)
2
(X - A_1)^2 + (X -A_2)^2 + ...+ (X - A_n)^2
(
X
−
A
1
)
2
+
(
X
−
A
2
)
2
+
...
+
(
X
−
A
n
)
2
is precisely the arithmetic mean of the given numbers.
8
1
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z_1, z_2, z_3 vetrices of equilateral
Determine a necessary and sufficient condition for the affixes of three complex numbers
z
1
z_1
z
1
,
z
2
z_2
z
2
and
z
3
z_3
z
3
are the vertices of an equilateral triangle.
6
1
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circle tangent to circumcircle
A triangle
A
B
C
ABC
A
BC
is considered, and let
D
D
D
be the intersection point of the angle bisector corresponding to angle
A
A
A
with side
B
C
BC
BC
. Prove that the circumcircle that passes through
A
A
A
and is tangent to line
B
C
BC
BC
at
D
D
D
, it is also tangent to the circle circumscribed around triangle
A
B
C
ABC
A
BC
.
5
1
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steps of stopped mechanical staircase in Metro
Using an escalator to go down to the Metro station and walking with a regular pace, I find that I need
50
50
50
steps to go down. if i come back later to run up it, at a speed
5
5
5
times my previous normal pace, I check that I need
125
125
125
steps to reach the top. How many visible steps does the mechanical staircase have when it is stopped?
3
1
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handshakes in a meeting of 285 people
Prove that in a meeting of
285
285
285
people, at least one of them has given an even number of handshakes (
0
0
0
is considered an even number and corresponds to an assistant who does not shake any hand).
4
1
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sum of the squares of five consecutive integers not a perfect square
Prove that the sum of the squares of five consecutive integers cannot be a perfect square.