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Difference between revisions of "2014 UNM-PNM Statewide High School Mathematics Contest II Problems"

(Created page with "UNM - PNM STATEWIDE MATHEMATICS CONTEST XLVI. February 1, 2014. Second Round. Three Hours ==Problem 1== Four siblings BRYAN, BARRY, SARAH and SHANA are having their names monog...")
 
m
Line 19: Line 19:
  
 
Two people, call them <math>A</math> and <math>B</math>, are having a discussion about the ages of <math>B</math>’s children.
 
Two people, call them <math>A</math> and <math>B</math>, are having a discussion about the ages of <math>B</math>’s children.
 +
 
A: “What are the ages, in years only, of your four children?”
 
A: “What are the ages, in years only, of your four children?”
 +
 
B: “The product of their ages is <math>72</math>.”
 
B: “The product of their ages is <math>72</math>.”
 +
 
A: “Not enough information.”
 
A: “Not enough information.”
 +
 
B: “The sum of their ages equals your eldest daughter’s age.”
 
B: “The sum of their ages equals your eldest daughter’s age.”
 +
 
A: “Still not enough information.”
 
A: “Still not enough information.”
−
B: “My oldest child who is at least a year older than her siblings took the AMC 8 for
+
 
−
the first time this year.”
+
B: “My oldest child who is at least a year older than her siblings took the AMC 8 for the first time this year.”
 +
 
 
A: “Still not enough information.”
 
A: “Still not enough information.”
 +
 
B: “My youngest child is my only son.”
 
B: “My youngest child is my only son.”
 +
 
A: “Now I know their ages..”
 
A: “Now I know their ages..”
 +
 
What are their ages?
 
What are their ages?
  
Line 53: Line 62:
 
How many triples <math>(x, y, z)</math> of rational numbers satisfy the following system of equations?
 
How many triples <math>(x, y, z)</math> of rational numbers satisfy the following system of equations?
 
<cmath>\begin{align*}  
 
<cmath>\begin{align*}  
−
x + y + z &= 0
+
x + y + z &= 0\\
−
xyz + 4z &= 0
+
xyz + 4z &= 0\\
−
xy + xz + yz + 2y &= 0  
+
xy + xz + yz + 2y &= 0 \\
 
\end{align*}</cmath>
 
\end{align*}</cmath>
  
Line 64: Line 73:
 
Let <math>k</math> be a natural number. Show that the sum of the <math>k^{th}</math> powers of the first <math>n</math> positive
 
Let <math>k</math> be a natural number. Show that the sum of the <math>k^{th}</math> powers of the first <math>n</math> positive
 
integers is a polynomial of degree <math>k + 1</math>, i.e.,
 
integers is a polynomial of degree <math>k + 1</math>, i.e.,
−
<math>1^k + 2^k + 3^k + · · · + n^k = p_{k+1}(n)</math>,
+
<math>1^k + 2^k + 3^k + \cdots + n^k = p_{k+1}(n)</math>,
 
where <math>p_{k+1}(t)</math> is a polynomial of degree <math>k + 1</math>. For example, for <math>k = 1</math> we have
 
where <math>p_{k+1}(t)</math> is a polynomial of degree <math>k + 1</math>. For example, for <math>k = 1</math> we have
−
<cmath>1 + 2 + · · · + n = \sum_{j=1}^{n} �j = \frac{n(n+1)}{2}=\tfrac{1}{2}n^2+\tfrac{1}{2}n</cmath>
+
 
 +
<cmath>1 + 2 +\cdots + n = \sum_{j=1}^{n} {j} = \frac{n(n+1)}{2}=\tfrac{1}{2}n^2+\tfrac{1}{2}n</cmath>
 +
 
 
hence <math>p_2(t) = \tfrac{1}{2} t^2 + \tfrac{1}{2}t.</math>
 
hence <math>p_2(t) = \tfrac{1}{2} t^2 + \tfrac{1}{2}t.</math>
  
Line 73: Line 84:
 
==Problem 8==
 
==Problem 8==
  
−
A certain country uses bills of denominations equivalent to <math>textdollar{15}</math> and <math>textdollar{44}</math>. The ATM
+
A certain country uses bills of denominations equivalent to <math>\textdollar{15}</math> and <math>\textdollar{44}</math>. The ATM
 
machines in this country can give at a single withdrawer any amount you request as long
 
machines in this country can give at a single withdrawer any amount you request as long
−
as both bills are used. Show that you can withdraw <math>textdollar{x}</math> if and only if you cannot withdraw
+
as both bills are used. Show that you can withdraw <math>\textdollar{x}</math> if and only if you cannot withdraw
−
<math>textdollar{y}</math>, where x + y = <math>textdollar{719}</math>.
+
<math>\textdollar{y}</math>, where x + y = <math>\textdollar{719}</math>.
  
  

Revision as of 23:22, 7 November 2014

UNM - PNM STATEWIDE MATHEMATICS CONTEST XLVI. February 1, 2014. Second Round. Three Hours

Problem 1

Four siblings BRYAN, BARRY, SARAH and SHANA are having their names monogrammed on their towels. Different letters may cost different amounts to monogram. If it costs $\textdollar{21}$ to monogram BRYAN, $\textdollar{25}$ to monogram BARRY and $\textdollar{18}$ to monogram SARAH, how much does it cost to monogram SHANA?

Solution

Problem 2

If $f(x) = x^3 + 6x^2 + 12x + 6$, solve the equation $f(f(f(x))) = 0.$

Solution

Problem 3

Two people, call them $A$ and $B$, are having a discussion about the ages of $B$’s children.

A: “What are the ages, in years only, of your four children?”

B: “The product of their ages is $72$.”

A: “Not enough information.”

B: “The sum of their ages equals your eldest daughter’s age.”

A: “Still not enough information.”

B: “My oldest child who is at least a year older than her siblings took the AMC 8 for the first time this year.”

A: “Still not enough information.”

B: “My youngest child is my only son.”

A: “Now I know their ages..”

What are their ages?

Solution

Problem 4

Find the smallest and largest possible distances between the centers of two circles of radius $1$ such that there is an equilateral triangle of side of length $1$ with two vertices on one of the circles and the third vertex on the second circle.

Solution

Problem 5

$5^n$ is written on the blackboard. The sum of its digits is calculated. Then the sum of the digits of the result is calculated and so on until we have a single digit. If $n = 2014$, what is this digit?

Solution

Problem 6

How many triples $(x, y, z)$ of rational numbers satisfy the following system of equations? \begin{align*} x + y + z &= 0\\ xyz + 4z &= 0\\ xy + xz + yz + 2y &= 0 \\ \end{align*}

Solution

Problem 7

Let $k$ be a natural number. Show that the sum of the $k^{th}$ powers of the first $n$ positive integers is a polynomial of degree $k + 1$, i.e., $1^k + 2^k + 3^k + \cdots + n^k = p_{k+1}(n)$, where $p_{k+1}(t)$ is a polynomial of degree $k + 1$. For example, for $k = 1$ we have

\[1 + 2 +\cdots + n = \sum_{j=1}^{n} {j} = \frac{n(n+1)}{2}=\tfrac{1}{2}n^2+\tfrac{1}{2}n\]

hence $p_2(t) = \tfrac{1}{2} t^2 + \tfrac{1}{2}t.$

Solution

Problem 8

A certain country uses bills of denominations equivalent to $\textdollar{15}$ and $\textdollar{44}$. The ATM machines in this country can give at a single withdrawer any amount you request as long as both bills are used. Show that you can withdraw $\textdollar{x}$ if and only if you cannot withdraw $\textdollar{y}$, where x + y = $\textdollar{719}$.


Solution

Problem 9

Suppose that $f$ is a mapping of the plane into itself such that the vertices of every equilateral triangle of side one are mapped onto the vertices of a congruent triangle. Show that the map $f$ is distance preserving, i.e., $d(p, q) = d(f(p), f(q))$ for all points $p$ and $q$ in the plane, where $d(x, y)$ denotes the distance between the points $x$ and $y$ in the plane. In other words, if any two points that are one unit apart are mapped to points that are one unit apart, then any two points are mapped to two points that are the same distance as their pre-images.

Solution

Problem 10

Given a sheet in the shape of a rhombus whose side is $2$ meters long and one of its angles is $60^{\circ}$ what is the maximum area that can be cut out of the sheet if we are allowed to cut two discs.

Solution

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