Difference between revisions of "2011 UNCO Math Contest II Problems"
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The largest integer <math>n</math> so that <math>3^n</math> evenly divides <math>9! = 1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9</math> is <math>n = 4</math>. Determine the largest integer <math>n</math> so that <math>3^n</math> evenly divides <math>85! = 1\cdot 2\cdot 3\cdot 4\cdots 84\cdot 85</math>. | The largest integer <math>n</math> so that <math>3^n</math> evenly divides <math>9! = 1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9</math> is <math>n = 4</math>. Determine the largest integer <math>n</math> so that <math>3^n</math> evenly divides <math>85! = 1\cdot 2\cdot 3\cdot 4\cdots 84\cdot 85</math>. | ||
− | [[2011 | + | [[2011 UNCO Math Contest II Problems/Problem 1|Solution]] |
==Problem 2== | ==Problem 2== | ||
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As an example, <math>20</math> <math>\underline{can}</math> be so expressed since <math>20 = 2 + 6 + 2\cdot 6</math>. | As an example, <math>20</math> <math>\underline{can}</math> be so expressed since <math>20 = 2 + 6 + 2\cdot 6</math>. | ||
− | [[2011 | + | [[2011 UNCO Math Contest II Problems/Problem 2|Solution]] |
==Problem 3== | ==Problem 3== | ||
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draw(P--(P+R-D2)--R,black); | draw(P--(P+R-D2)--R,black); | ||
</asy> | </asy> | ||
− | [[2011 | + | [[2011 UNCO Math Contest II Problems/Problem 3|Solution]] |
==Problem 4== | ==Problem 4== | ||
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(b) Find all numbers that are in both <math>A</math> and <math>B</math>. | (b) Find all numbers that are in both <math>A</math> and <math>B</math>. | ||
− | [[2011 | + | [[2011 UNCO Math Contest II Problems/Problem 4|Solution]] |
==Problem 5== | ==Problem 5== | ||
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</asy> | </asy> | ||
− | [[2011 | + | [[2011 UNCO Math Contest II Problems/Problem 5|Solution]] |
==Problem 6== | ==Problem 6== | ||
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What is the remainder when <math>1! + 2! + 3! + ?+ 2011!</math> is divided by <math>18</math>? | What is the remainder when <math>1! + 2! + 3! + ?+ 2011!</math> is divided by <math>18</math>? | ||
− | [[2011 | + | [[2011 UNCO Math Contest II Problems/Problem 6|Solution]] |
==Problem 7== | ==Problem 7== | ||
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term in an odd numbered position is one more that the previous term. | term in an odd numbered position is one more that the previous term. | ||
− | [[2011 | + | [[2011 UNCO Math Contest II Problems/Problem 7|Solution]] |
==Problem 8== | ==Problem 8== | ||
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as the sum of two squares. | as the sum of two squares. | ||
− | [[2011 | + | [[2011 UNCO Math Contest II Problems/Problem 8|Solution]] |
==Problem 9== | ==Problem 9== | ||
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(b) Determine a closed formula for T(n). | (b) Determine a closed formula for T(n). | ||
− | [[2011 | + | [[2011 UNCO Math Contest II Problems/Problem 9|Solution]] |
==Problem 10== | ==Problem 10== | ||
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and <math>7</math>, replace them with <math>47</math>. You now have two <math>47</math>’s in this case but that’s OK. | and <math>7</math>, replace them with <math>47</math>. You now have two <math>47</math>’s in this case but that’s OK. | ||
− | [[2011 | + | [[2011 UNCO Math Contest II Problems/Problem 10|Solution]] |
==Problem 11== | ==Problem 11== | ||
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Tie breaker – Generalize problem #2, and prove your statement. | Tie breaker – Generalize problem #2, and prove your statement. | ||
− | [[2011 | + | [[2011 UNCO Math Contest II Problems/Problem 11|Solution]] |
+ | |||
+ | == See Also == | ||
+ | {{UNCO Math Contest box|year=2011|n=II|before=[[2010 UNCO Math Contest II]]|after=[[2012 UNCO Math Contest II]]}} |
Latest revision as of 22:10, 7 November 2014
University of Northern Colorado MATHEMATICS CONTEST FINAL ROUND January 29, 2011 For Colorado Students Grades 7-12
• , read as n factorial, is computed as
• The factorials are
• The square integers are
Contents
Problem 1
The largest integer so that
evenly divides
is
. Determine the largest integer
so that
evenly divides
.
Problem 2
Let and
be positive integers. List all the integers in the set
that
be written in the form
.
As an example,
be so expressed since
.
Problem 3
The two congruent rectangles shown have
dimensions in. by
in. What is the area
of the shaded overlap region?
Solution
Problem 4
Let be the set of all positive squares plus
and
be the set of all positive squares plus
.
(a) What is the smallest number in both and
?
(b) Find all numbers that are in both and
.
Problem 5
Determine the area of the square ,
with the given lengths along a zigzag
line connecting
and
.
Problem 6
What is the remainder when is divided by
?
Problem 7
What is the of the first
terms of the sequence
that appeared
on the First Round? Recall that a term in an even numbered position is twice the previous term, while a
term in an odd numbered position is one more that the previous term.
Problem 8
The integer can be expressed as a sum of two squares as
.
(a) Express as the sum of two squares.
(b) Express the product as the sum of two squares.
(c) Prove that the product of two sums of two squares, , can be represented
as the sum of two squares.
Problem 9
Let be the number of ways of selecting three distinct numbers from
so that they are
the lengths of the sides of a triangle. As an example,
; the only possibilities are
,
and
.
(a) Determine a recursion for T(n).
(b) Determine a closed formula for T(n).
Problem 10
The integers are written on the blackboard. Select any two, call them
and
and replace
these two with the one number
. Continue doing this until only one number remains and
explain, with proof, what happens. Also explain with proof what happens in general as you replace
with
. As an example, if you select
and
you replace them with
. If you select
and
, replace them with
. You now have two
’s in this case but that’s OK.
Problem 11
Tie breaker – Generalize problem #2, and prove your statement.
See Also
2011 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
Preceded by 2010 UNCO Math Contest II |
Followed by 2012 UNCO Math Contest II | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNCO Math Contest Problems and Solutions |