Difference between revisions of "MIE 2016"
m (→Problem 1) |
(Added more questions) |
||
(8 intermediate revisions by 4 users not shown) | |||
Line 15: | Line 15: | ||
(e) <math>(2\sqrt{2016})^{-1}<\sqrt{2017}-\sqrt{2016}<\sqrt{2016}-\sqrt{2015}</math> | (e) <math>(2\sqrt{2016})^{-1}<\sqrt{2017}-\sqrt{2016}<\sqrt{2016}-\sqrt{2015}</math> | ||
− | + | [[MIE 2016/Problem 1|Solution]] | |
===Problem 2=== | ===Problem 2=== | ||
Line 32: | Line 32: | ||
(e) <math>k\geq8</math> | (e) <math>k\geq8</math> | ||
+ | |||
+ | [[MIE 2016/Day 1/Problem 2|Solution]] | ||
===Problem 3=== | ===Problem 3=== | ||
Line 45: | Line 47: | ||
(e) <math>\text{Re}(Z_1)\geq\text{Im}(Z_2)</math> | (e) <math>\text{Re}(Z_1)\geq\text{Im}(Z_2)</math> | ||
+ | |||
+ | |||
+ | [[MIE 2016/Day 1/Problem 3|Solution]] | ||
===Problem 4=== | ===Problem 4=== | ||
Line 63: | Line 68: | ||
(e) <math>\frac{\pi}{24}</math> | (e) <math>\frac{\pi}{24}</math> | ||
+ | |||
+ | |||
+ | [[MIE 2016/Day 1/Problem 4|Solution]] | ||
===Problem 5=== | ===Problem 5=== | ||
Line 77: | Line 85: | ||
(e) <math>\frac{26}{25}</math> | (e) <math>\frac{26}{25}</math> | ||
+ | |||
+ | |||
+ | [[MIE 2016/Day 1/Problem 5|Solution]] | ||
===Problem 6=== | ===Problem 6=== | ||
Line 140: | Line 151: | ||
(e) <math>325</math> | (e) <math>325</math> | ||
+ | |||
+ | [[MIE 2016/Day 1/Problem 9|Solution]] | ||
===Problem 10=== | ===Problem 10=== | ||
Line 212: | Line 225: | ||
*Between the 256 functions from <math>A</math> to <math>A</math>, we draw the function <math>f</math> and <math>g</math>, knowing that can have repetition. What's the probability of <math>fog</math> be a constant function? | *Between the 256 functions from <math>A</math> to <math>A</math>, we draw the function <math>f</math> and <math>g</math>, knowing that can have repetition. What's the probability of <math>fog</math> be a constant function? | ||
+ | |||
+ | ===Problem 9=== | ||
+ | In a triangle <math>\Delta_{ABC}</math>, let <math>AD</math> be the intern angle bisector such that it's the geometric mean of the <math>BD</math> and <math>DC</math>, and the measure of the median <math>AM</math> is the geometric mean of the sides <math>AB</math> and <math>AC</math>. The points <math>D</math> and <math>M</math> are on the side <math>BC</math> of side <math>a</math>. Compute the sides <math>AB</math> and <math>AC</math> of <math>\Delta_{ABC}</math> em terms of <math>a</math>. | ||
+ | |||
+ | ===Problem 10=== | ||
+ | In a equilateral cone there are two inscribed spheres of radii <math>\frac{\sqrt3-1}{\sqrt3+1}R</math> and <math>R</math>, like in the image. A surface is secant to the cone such that it's tangent to both spheres. Determine in terms of <math>R</math> the largest segment line that there is between the two points of the curve formed by the intersection of the surface with the cone. | ||
+ | |||
+ | [[File:MIE_2016_day_2_problem_10.png]] | ||
{{stub}} | {{stub}} |
Latest revision as of 09:46, 11 January 2018
Note: Anyone that solve any of the problems can post your solutions.
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
.
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?
Problem 9
In a triangle , let
be the intern angle bisector such that it's the geometric mean of the
and
, and the measure of the median
is the geometric mean of the sides
and
. The points
and
are on the side
of side
. Compute the sides
and
of
em terms of
.
Problem 10
In a equilateral cone there are two inscribed spheres of radii and
, like in the image. A surface is secant to the cone such that it's tangent to both spheres. Determine in terms of
the largest segment line that there is between the two points of the curve formed by the intersection of the surface with the cone.
This article is a stub. Help us out by expanding it.