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 $2017^{2017}$ ?

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.