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Difference between revisions of "2017 UNM-PNM Statewide High School Mathematics Contest II Problems"

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==Problem 3==
 
==Problem 3==
  
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Let <math>f(x) =\frac{x}{ x−1}</math> and <math>g(x) =\frac{x}{3x−1}</math>.  
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Let <math>f(x)=\frac{x}{x-1}</math> and <math>g(x)=\frac{x}{3x-1}</math>.  
  
 
(a) Determine <math>f\circ g(x)</math> and <math>g\circ f(x)</math>.  
 
(a) Determine <math>f\circ g(x)</math> and <math>g\circ f(x)</math>.  
  
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(b) Denote <math>\underbracket{h\circ h\circ \cdot \circ h}_text{n times} := h^n</math>.  
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(b) Denote <math>\underbrace{h\circ h\circ \cdot \circ h}_\text{n times} := h^n</math>.  
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Determine all the functions in the set <math>S = \{H | H = (g\circ f)n\circ g</math> or <math>H = (f\circ g)n\circ f</math> for some <math>n</math> a whole number<math>\}</math>.  
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Determine all the functions in the set <math>S = \{H | H = (g\circ f)^n\circ g</math> or <math>H = (f\circ g)^n\circ f</math> for some <math>n</math> a whole number<math>\}</math>.  
  
 
[[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 3|Solution]]
 
[[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 3|Solution]]

Revision as of 02:52, 15 January 2019

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

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

See Also

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