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

(→‎Problem 3)
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==Problem 4==
 
==Problem 4==
  
 +
Find a second-degree polynomial with integer coefficients, <math>p(x) = ax^2 + bx + c</math>, such that <math>p(1),p(3),p(5)</math>, and <math>p(7)</math> are perfect squares, but <math>p(2)</math> is not.
  
 
[[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 4|Solution]]
 
[[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 4|Solution]]
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==Problem 5==
 
==Problem 5==
  
 +
Find all real triples <math>(x,y,z)</math> which are solutions to the system:
 +
 +
<math>x^3 + x^2y + x^2z = 40</math>
 +
 +
<math>y^3 + y^2x + y^2z = 90</math>
 +
 +
<math>z^3 + z^2x + z^2y = 250</math>
  
 
[[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5|Solution]]
 
[[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5|Solution]]
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==Problem 6==
 
==Problem 6==
  
 +
There are <math>12</math> stacks of <math>12</math> coins. Each of the coins in <math>11</math> of the <math>12</math> stacks weighs <math>10</math> grams each. Suppose the coins in the remaining stack each weigh <math>9.9</math> 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.
  
 
[[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 6|Solution]]
 
[[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 6|Solution]]
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==Problem 7==
 
==Problem 7==
  
 +
Find a formula for
 +
<math>sum_{k=0}^{</math>\lceil \frac{n}{4} \rceil } binom{n}{4k}<math> for any natural number </math>n$.
  
 
[[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 7|Solution]]
 
[[2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 7|Solution]]

Revision as of 03:01, 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

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 $sum_{k=0}^{$ (Error compiling LaTeX. Unknown error_msg)\lceil \frac{n}{4} \rceil } binom{n}{4k}$for any natural number$n$.

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

See Also

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