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2017 UNM-PNM Statewide High School Mathematics Contest II Problems

UNM - PNM STATEWIDE MATHEMATICS CONTEST XLIX. February 4, 2017. Second Round. Three Hours

Problem 1

What are the last two digits of $2017^{2017}$?

Solution

Problem 2

Suppose $A, R, S$, and $T$ all denote distinct digits from $1$ to $9$. If $\sqrt{STARS} = SAT$, what are $A, R, S$, and $T$?

Solution

Problem 3

Let $f(x)=\frac{x}{x-1}$ and $g(x)=\frac{x}{3x-1}$.

(a) Determine $f\circ g(x)$ and $g\circ f(x)$.

(b) Denote $\underbrace{h\circ h\circ \cdot \circ h}_\text{n times} := h^n$. Determine all the functions in the set $S = \{H | H = (g\circ f)^n\circ g$ or $H = (f\circ g)^n\circ f$ for some $n$ a whole number$\}$.

Solution

Problem 4

Find a second-degree polynomial with integer coefficients, $p(x) = ax^2 + bx + c$, such that $p(1),p(3),p(5)$, and $p(7)$ are perfect squares, but $p(2)$ is not.

Solution

Problem 5

Find all real triples $(x,y,z)$ which are solutions to the system:

$x^3 + x^2y + x^2z = 40$

$y^3 + y^2x + y^2z = 90$

$z^3 + z^2x + z^2y = 250$

Solution

Problem 6

There are $12$ stacks of $12$ coins. Each of the coins in $11$ of the $12$ stacks weighs $10$ grams each. Suppose the coins in the remaining stack each weigh $9.9$ 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.

Solution

Problem 7

Find a formula for $\sum_{k=0}^{\lfloor \frac{n}{4} \rfloor } \binom{n}{4k}$ for any natural number $n$.

Solution

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 $ABC$ be a right triangle with right angle at $C$. Suppose $AC = 12$ and $BC = 5$ and $CX$ is the diameter of a semicircle, where $X$ lies on $AC$ and the semicircle is tangent to side $AB$. Find the radius of the semicircle.

Solution

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 $100$ triangles, on average, takes $10$ 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?

Solution

Problem 10

Newton’s method applied to the equation $f(x) = x^3-x$ takes the form of the iteration $x_{n+1} =x_n-\frac{(x_n)^3-x_n}{3{x_n}^2-1} , n = 0,1,2,\ldots$

(a) What are the roots of $f(x) = 0$?

(b) Study the behavior of the iteration when $x_0 > \frac{1}{\sqrt{3}}$ to conclude that the sequence $\{x_0,x_1,\ldots\}$ approaches the same root as long as you choose $x_0 > \frac{1}{\sqrt{3}}$. It may be helpful to start with the case $x_0>1$.

(c) Assume $-\alpha < x_0 < \alpha$. For what number $\alpha$ does the sequence always approach $0$?

(d) For $x_0 \in(\alpha,\frac{1}{\sqrt{3}})$ the sequence may approach either of the roots $\pm x^{*}$. Can you find an (implicit) expression that can be used to determine limits $a_i$ and $a_{i+1}$ such that if $x0 \in (a_i,a_{i+1})$ then the sequence approaches $(-1)^i{x^{*}}$. Hint:$a_1 = \frac{1}{\sqrt{3}}, a_i > a_{i+1}$ and $a_i$ approaches $\frac{1}{\sqrt{5}}$ when $i$ becomes large.

Solution

See Also

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