Difference between revisions of "2014 UNM-PNM Statewide High School Mathematics Contest II Problems"
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[[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 1|Solution]] | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 1|Solution]] | ||
+ | [[2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 1|Solution]] | ||
+ | [[Category:Beginning Algebra Problems]] | ||
==Problem 2== | ==Problem 2== | ||
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If <math>f(x) = x^3 + 6x^2 + 12x + 6</math>, solve the equation <math>f(f(f(x))) = 0.</math> | If <math>f(x) = x^3 + 6x^2 + 12x + 6</math>, solve the equation <math>f(f(f(x))) = 0.</math> | ||
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 2|Solution]] | + | [[2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 2|Solution]] |
==Problem 3== | ==Problem 3== | ||
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What are their ages? | What are their ages? | ||
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 3|Solution]] | + | [[2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 3|Solution]] |
==Problem 4== | ==Problem 4== | ||
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the circles and the third vertex on the second circle. | the circles and the third vertex on the second circle. | ||
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 4|Solution]] | + | [[2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 4|Solution]] |
==Problem 5== | ==Problem 5== | ||
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is this digit? | is this digit? | ||
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5|Solution]] | + | [[2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5|Solution]] |
==Problem 6== | ==Problem 6== | ||
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\end{align*}</cmath> | \end{align*}</cmath> | ||
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 6|Solution]] | + | [[2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 6|Solution]] |
==Problem 7== | ==Problem 7== | ||
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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> | ||
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 7|Solution]] | + | [[2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 7|Solution]] |
==Problem 8== | ==Problem 8== | ||
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− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 8|Solution]] | + | [[2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 8|Solution]] |
==Problem 9== | ==Problem 9== | ||
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distance as their pre-images. | distance as their pre-images. | ||
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 9|Solution]] | + | [[2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 9|Solution]] |
==Problem 10== | ==Problem 10== | ||
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two discs. | two discs. | ||
− | [[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 10|Solution]] | + | [[2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 10|Solution]] |
+ | |||
+ | ==See Also == |
Latest revision as of 06:07, 29 January 2019
UNM - PNM STATEWIDE MATHEMATICS CONTEST XLVI. February 1, 2014. Second Round. Three Hours
Contents
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 to monogram BRYAN,
to monogram BARRY and
to monogram SARAH,
how much does it cost to monogram SHANA?
Problem 2
If , solve the equation
Problem 3
Two people, call them and
, are having a discussion about the ages of
’s children.
A: “What are the ages, in years only, of your four children?”
B: “The product of their ages is .”
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?
Problem 4
Find the smallest and largest possible distances between the centers of two circles of radius
such that there is an equilateral triangle of side of length
with two vertices on one of
the circles and the third vertex on the second circle.
Problem 5
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
, what
is this digit?
Problem 6
How many triples of rational numbers satisfy the following system of equations?
Problem 7
Let be a natural number. Show that the sum of the
powers of the first
positive
integers is a polynomial of degree
, i.e.,
,
where
is a polynomial of degree
. For example, for
we have
hence
Problem 8
A certain country uses bills of denominations equivalent to and
. 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
if and only if you cannot withdraw
, where x + y =
.
Problem 9
Suppose that 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
is distance preserving, i.e.,
for all points
and
in the plane, where
denotes the distance between the points
and
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.
Problem 10
Given a sheet in the shape of a rhombus whose side is meters long and one of its angles
is
what is the maximum area that can be cut out of the sheet if we are allowed to cut
two discs.