MIE 2016
Note: Anyone that solve any of the problems can post your solutions.
Contents
Day 1
Problem 1
Choose the correct answer.
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Problem 2
The following system has integer solutions. We can say that:
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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:
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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:
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Problem 5
Compute , knowing that .
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Problem 6
Let be with . We know that . The sum of the values of that satisfies this condition is:
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Note: is the determinant of the matrix .
Problem 7
The product of the real roots of the following equation is equal to:
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Problem 8
Let . The minimum value of is in the interval:
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Problem 9
Let , and be complex numbers that satisfies the following system:
Compute .
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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.
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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:
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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 .
Problem 2
Solve the inequation, where .
Problem 3
Solve the system, where .
Problem 4
Classify the following system as determined, possible indetermined and impossible according to the real values of .
Problem 5
Let the complex numbers and , such that . Find the value of , and , knowing that they are positive integers.
Problem 6
A triangle has its vertex at the origin of the cartesian system, its centroid is the point and its circumcenter is the point . Determine:
- The equation of the circumcircle of ;
- The coordinates of the vertices and .
Problem 7
If , then compute .
Problem 8
Let .
- How many function from to have exactly 2 elements in its image set?
- Between the 256 functions from to , we draw the function and , knowing that can have repetition. What's the probability of be a constant function?
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