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Difference between revisions of "2010 UNCO Math Contest II Problems"

m (moved 2010 UNC Math Contest II Problems to 2010 UNCO Math Contest II Problems: disambiguation of University of Northern Colorado with University of North Carolina)
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the original number.
 
the original number.
  
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==Problem 2==
 
==Problem 2==
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</asy>
 
</asy>
  
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==Problem 3==
 
==Problem 3==
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What is the smallest possible value of the sum <math>r+s+t</math>?
 
What is the smallest possible value of the sum <math>r+s+t</math>?
  
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==Problem 4==
 
==Problem 4==
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Factor <math>n^4+2n^3+2n^2+2n+1</math> completely.
 
Factor <math>n^4+2n^3+2n^2+2n+1</math> completely.
  
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==Problem 5==
 
==Problem 5==
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(b) Generalize this to an <math>N \times N</math> grid.
 
(b) Generalize this to an <math>N \times N</math> grid.
  
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==Problem 6==
 
==Problem 6==
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three times <math>C</math>, what is the number <math>A</math>?
 
three times <math>C</math>, what is the number <math>A</math>?
  
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==Problem 7==
 
==Problem 7==
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What is <math>R</math>?
 
What is <math>R</math>?
  
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==Problem 8==
 
==Problem 8==
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express your answer. Generalize this result.
 
express your answer. Generalize this result.
  
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==Problem 9==
 
==Problem 9==
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(b) Find integers <math>A, B, C</math> and <math>D</math> so that <math>A^3+B^4+C^5=3^D.</math>
 
(b) Find integers <math>A, B, C</math> and <math>D</math> so that <math>A^3+B^4+C^5=3^D.</math>
  
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==Problem 10==
 
==Problem 10==
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Since there is no subset of size <math>4</math> satisfying these conditions, the answer for <math>n=5</math> is <math>3</math>.
 
Since there is no subset of size <math>4</math> satisfying these conditions, the answer for <math>n=5</math> is <math>3</math>.
  
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==Problem 11==
 
==Problem 11==
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Revision as of 20:34, 19 October 2014

University of Northern Colorado MATHEMATICS CONTEST FINAL ROUND January 30, 2010.

For Colorado Students Grades 7-12.

• The ten digits are $0, 1, 2, 3, 4, 5, 6, 7, 8, 9.$

• The positive integers are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,\cdots$

• The prime numbers are $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, \cdots$

Problem 1

Find a $3$-digit integer less than $200$ where each digit is odd and the sum of the cubes of the digits is the original number.

Solution

Problem 2

The rectangle has dimensions $67 \times 75$. The diagonal $AB$ is divided into five segments of equal length. Find the total area of the shaded regions.

[asy] pair A,B,C,D; A==(0,0);B=(75,0);C=(75,67);D=(0,67); draw(A--B--C--D--cycle,black); filldraw(A--(D+.2(B-D))--C--(D+.4(B-D))--cycle,blue); filldraw(A--(D+.6(B-D))--C--(D+.8(B-D))--cycle,blue); draw(B--D,black); dot(D+.2(B-D));dot(D+.4(B-D));dot(D+.6(B-D));dot(D+.8(B-D)); MP("A",D,NW);MP("B",B,SE); [/asy]

Solution

Problem 3

Suppose $r, s$, and $t$ are three different positive integers and that their product is $48$, i.e., $rst=48.$ What is the smallest possible value of the sum $r+s+t$?

Solution

Problem 4

Factor $n^4+2n^3+2n^2+2n+1$ completely.

Solution

Problem 5

(a) In the $4 \times 4$ grid shown, four coins are randomly placed in different squares. What is the probability that no two coins lie in the same row or column?

$\begin{tabular}{|c|c|c|c|} \hline &&& \\ \hline &&& \\ \hline &&& \\ \hline &&& \\ \hline \end{tabular}$

(b) Generalize this to an $N \times N$ grid.

Solution

Problem 6

$A$ is a $4$-digit number $abcd$. $B$ is a $5$-digit number formed by augmenting $A$ with a $3$ on the right, i.e. $B=abcd3$.

$C$ is another $5$-digit number formed by placing a $2$ on the left $A$, i.e. $C=2abcd$. If $B$ is three times $C$, what is the number $A$?

Solution

Problem 7

$P$ and $Q$ are each $2$-digit $\underline{prime}$ numbers (like $73$ and $19$), and all four digits are different. The sum $P+Q$ is a $2$-digit number made up of two more different digits ($P+Q$ is not necessarily prime). Further, the difference $P-Q$ consists of yet two more different digits (again, $P-Q$ is not necessarily prime). The number $R$ is a two digit $\underline{prime}$ number which uses the remaining two digits. What is $R$?

Solution

Problem 8

Simplify $(3^1+1)(3^2+1)(3^4+1)(3^8+1)(3^{16}+1)\cdots (3^{1024}+1)$, using exponential notation to express your answer. Generalize this result.

Solution

Problem 9

(a) Find integers $A, B$, and $C$ so that $A^3+B^4=C^5.$ Express your answers in exponential form.

(b) Find integers $A, B, C$ and $D$ so that $A^3+B^4+C^5=3^D.$

Solution

Problem 10

Let $S=\left\{1,2,3,\cdots ,n\right\}$ where $n \ge 4$. What is the maximum number of elements in a subset $A$ of $S$, which has at least three elements, such that $a+b>c$ for all $a, b, c$ in $A$? As an example, the subset $A=\left \{2,3,4\right \}$ of $S=\left\{1,2,3,4,5 \right\}$ has the property that the sum of any two elements is strictly bigger than the third element, but the subset $\left \{2,3,4,5 \right \}$ does not since $2+3$ is $\underline{not}$ greater than $5$. Since there is no subset of size $4$ satisfying these conditions, the answer for $n=5$ is $3$.

Solution

Problem 11

(a) The $3 \times 3$ square grid has $9$ dots equally spaced. How many squares (of all sizes) can you make using four of these dots as vertices? Two examples are shown.

[asy] for (int x=0; x<3; ++x) {for (int y=0; y<3; ++y) {dot((x,y));dot((x+4,y));} } draw((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); draw((4,1)--(5,0)--(6,1)--(5,2)--cycle,black); draw((3,-1)--(3,3),dashed); [/asy]

(b) How many for a $4 \times 4$?

(c) How many for a $5 \times 5$?

(d) How many for an $(N+1) \times (N+1)$ grid of dots?


Solution

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