Difference between revisions of "2017 UNM-PNM Statewide High School Mathematics Contest II Problems"
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==Problem 8== | ==Problem 8== | ||
+ | <asy> | ||
+ | |||
+ | pair C=(0,0),A=(0,12),B=(5,0); | ||
+ | draw(C--A--B--C,dot); | ||
+ | MP("A",A,N);MP("C",C,SW);MP("B",B,E); | ||
+ | |||
+ | real r=10/3; | ||
+ | pair X=(0,r),Y=X+r*(12/13,5/13); | ||
+ | draw(X--Y,dot); | ||
+ | MP("X",X,W);MP("Y",Y,NE); | ||
+ | |||
+ | draw(circle(X,r)); | ||
+ | |||
+ | MP("12",(0,8),W);MP("5",(2.5,0),S); | ||
+ | |||
+ | </asy> | ||
Let <math>ABC</math> be a right triangle with right angle at <math>C</math>. Suppose <math>AC = 12</math> and <math>BC = 5</math> and <math>CX</math> is the diameter of a semicircle, where <math>X</math> lies on <math>AC</math> and the semicircle is tangent to side <math>AB</math>. Find the radius of the semicircle. | Let <math>ABC</math> be a right triangle with right angle at <math>C</math>. Suppose <math>AC = 12</math> and <math>BC = 5</math> and <math>CX</math> is the diameter of a semicircle, where <math>X</math> lies on <math>AC</math> and the semicircle is tangent to side <math>AB</math>. Find the radius of the semicircle. | ||
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==Problem 10== | ==Problem 10== | ||
− | Newton’s method applied to the equation <math>f(x) = x^3-x</math> takes the form of the iteration <math>x_{n+1} =x_n- \frac{(x_n)^3 | + | Newton’s method applied to the equation <math>f(x) = x^3-x</math> takes the form of the iteration <math>x_{n+1} =x_n-\frac{(x_n)^3-x_n}{3{x_n}^2-1} , n = 0,1,2,\ldots</math> |
(a) What are the roots of <math>f(x) = 0</math>? | (a) What are the roots of <math>f(x) = 0</math>? | ||
− | (b) Study the behavior of the iteration when | + | (b) Study the behavior of the iteration when <math>x_0 > \frac{1}{\sqrt{3}}</math> to conclude that the sequence <math>\{x_0,x_1,\ldots\}</math> approaches the same root as long as you choose <math>x_0 > \frac{1}{\sqrt{3}}</math>. It may be helpful to start with the case <math>x_0>1</math>. |
− | (c) Assume <math> | + | (c) Assume <math>-\alpha < x_0 < \alpha</math>. For what number <math>\alpha</math> does the sequence always approach <math>0</math>? |
− | (d) For <math>x_0 \ | + | (d) For <math>x_0 \in(\alpha,\frac{1}{\sqrt{3}})</math> the sequence may approach either of the roots <math>\pm x^{*}</math>. Can you find an (implicit) expression that can be used to determine limits <math>a_i</math> and <math>a_{i+1}</math> such that if <math>x0 \in (a_i,a_{i+1})</math> then the sequence approaches <math>(-1)^i{x^{*}}</math>. Hint:<math>a_1 = \frac{1}{\sqrt{3}}, a_i > a_{i+1}</math> and <math>a_i</math> approaches <math>\frac{1}{\sqrt{5}}</math> when <math>i</math> becomes large. |
[[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 10|Solution]] | [[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 10|Solution]] | ||
==See Also == | ==See Also == |
Latest revision as of 03:36, 19 January 2019
UNM - PNM STATEWIDE MATHEMATICS CONTEST XLIX. February 4, 2017. Second Round. Three Hours
Contents
Problem 1
What are the last two digits of ?
Problem 2
Suppose , and
all denote distinct digits from
to
. If
, what are
, and
?
Problem 3
Let and
.
(a) Determine and
.
(b) Denote .
Determine all the functions in the set
or
for some
a whole number
.
Problem 4
Find a second-degree polynomial with integer coefficients, , such that
, and
are perfect squares, but
is not.
Problem 5
Find all real triples which are solutions to the system:
Problem 6
There are stacks of
coins. Each of the coins in
of the
stacks weighs
grams each. Suppose the coins in the remaining stack each weigh
grams. You are given one time access to a precise digital scale. Devise a plan to weigh some coins in precisely one weighing to determine which pile has the lighter coins.
Problem 7
Find a formula for
for any natural number
.
Problem 8
Let be a right triangle with right angle at
. Suppose
and
and
is the diameter of a semicircle, where
lies on
and the semicircle is tangent to side
. Find the radius of the semicircle.
Problem 9
Consider a triangulation (mesh) of a polygonal domain like the one in the figure below. (a) Given the vertices of a triangle, devise a strategy for determining if a given point is inside that triangle. (b) Will your strategy work for polygons with more than three sides? (c) After implementing your strategy in an optimally efficient computer code you find that the search for a problem with triangles, on average, takes
seconds. You refine the triangulation by subdividing each of the triangles into smaller triangles by placing a new vertex at the center of gravity of each triangle. On average, how long will it take to find a point in the new mesh?
Problem 10
Newton’s method applied to the equation takes the form of the iteration
(a) What are the roots of ?
(b) Study the behavior of the iteration when to conclude that the sequence
approaches the same root as long as you choose
. It may be helpful to start with the case
.
(c) Assume . For what number
does the sequence always approach
?
(d) For the sequence may approach either of the roots
. Can you find an (implicit) expression that can be used to determine limits
and
such that if
then the sequence approaches
. Hint:
and
approaches
when
becomes large.