Difference between revisions of "MIE 2016"
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==Day 2== | ==Day 2== | ||
===Problem 1=== | ===Problem 1=== | ||
− | Let <math>M</math> be a | + | Let <math>M</math> be a 2x2 real matrix . Define a function <math>f(x)</math> that each element of the matrix moves to the next position in clockwise direction, in other words, if <math>M=\begin{pmatrix}a&b\\c&d\end{pmatrix}</math>, we have <math>f(M)=\begin{pmatrix}c&a\\d&b\end{pmatrix}</math>. Find all 2x2 real symmetric matrixes such that <math>M^2=f(M)</math>. |
{{stub}} | {{stub}} |
Revision as of 18:15, 7 January 2018
Note: Anyone that solve any of the problems can post your solutions.
Contents
Day 1
Problem 1
Choose the correct answer.
(a)
(b)
(c)
(d)
(e)
Problem 2
The following system has integer solutions. We can say that:
(a)
(b)
(c)
(d)
(e)
Problem 3
Let and
be complex numbers such that
is a pure imaginary number and
. For any values of
and
that satisfies these conditions we have:
(a)
(b)
(c)
(d)
(e)
Problem 4
In the expansion of
the independent term (in other words, the term without ) is equal to
. With
being a real number such that
and
, the value of
is:
(a)
(b)
(c)
(d)
(e)
Problem 5
Compute , knowing that
.
(a)
(b)
(c)
(d)
(e)
Problem 6
Let be
with
. We know that
. The sum of the values of
that satisfies this condition is:
(a)
(b)
(c)
(d)
(e)
Note: is the determinant of the matrix
.
Problem 7
The product of the real roots of the following equation is equal to:
(a)
(b)
(c)
(d)
(e)
Problem 8
Let . The minimum value of
is in the interval:
(a)
(b)
(c)
(d)
(e)
Problem 9
Let ,
and
be complex numbers that satisfies the following system:
Compute .
(a)
(b)
(c)
(d)
(e)
Problem 10
A hexagon is divided into 6 equilateral triangles. How many ways can we put the numbers from 1 to 6 in each triangle, without repetition, such that the sum of the numbers of three adjacent triangles is always a multiple of 3? Solutions obtained by rotation or reflection are differents, thus the following figures represent two distinct solutions.
(a)
(b)
(c)
(d)
(e)
Problem 11
Let be an arithmetic progression and
, an geometric progression of integer terms, of ratio
and
, respectively, where
and
are positive integers, with
and
. We also know that
and
. The value of
is:
(a)
(b)
(c)
(d)
(e)
Day 2
Problem 1
Let be a 2x2 real matrix . Define a function
that each element of the matrix moves to the next position in clockwise direction, in other words, if
, we have
. Find all 2x2 real symmetric matrixes such that
.
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