2017 UNM-PNM Statewide High School Mathematics Contest II Problems
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 $x_{n+1} =x_n- \frac{(x_n)^3 − x_n}{3{x_n}^2 −1} , n = 0,1,2,\ldots$ (Error compiling LaTeX. Unknown error_msg)
(a) What are the roots of ?
(b) Study the behavior of the iteration when x0 > 1/√3 to conclude that the sequence 0,x
approaches the same root as long as you choose
. It may be helpful to start with the case
.
(c) Assume $−\alpha < x_0 < \alpha$ (Error compiling LaTeX. Unknown error_msg). For what number does the sequence always approach
?
(d) For $x_0 \element (\alpha,\frac{1}{\sqrt{3}})$ (Error compiling LaTeX. Unknown error_msg) the sequence may approach either of the roots $±x∗. Can you find an (implicit) expression that can be used to determine limits ai and ai+1 such that if x0 ∈ (ai,ai+1) then the sequence approaches \pn{x^∗}. Hint:$ (Error compiling LaTeX. Unknown error_msg)a_1 = \frac{1}{\sqrt{3}}, a_i > a_{i+1}a_i
\frac{1}{\sqrt{5}}
i$ becomes large.