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

(Created page with "University of Northern Colorado MATHEMATICS CONTEST FINAL ROUND January 30, 2010 For Colorado Students Grades 7-12 • The ten digits are <math>0, 1, 2, 3, 4, 5, 6, 7, 8, 9.</ma...")
 
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University of Northern Colorado
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University of Northern Colorado MATHEMATICS CONTEST FINAL ROUND January 30, 2010.
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MATHEMATICS CONTEST
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FINAL ROUND January 30, 2010
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For Colorado Students Grades 7-12.
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For Colorado Students Grades 7-12
 
  
 
• The ten digits are <math>0, 1, 2, 3, 4, 5, 6, 7, 8, 9.</math>
 
• The ten digits are <math>0, 1, 2, 3, 4, 5, 6, 7, 8, 9.</math>
Line 12: Line 11:
 
==Problem 1==
 
==Problem 1==
  
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1. Find a <math>3</math>-digit integer less than <math>200</math> where each digit is odd and the sum of the cubes of the digits is
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Find a <math>3</math>-digit integer less than <math>200</math> where each digit is odd and the sum of the cubes of the digits is
 
the original number.
 
the original number.
  
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[[2010 UNC Math Contest II Problems/Problem 1|Solution]]
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[[2010 UNCO Math Contest II Problems/Problem 1|Solution]]
  
 
==Problem 2==
 
==Problem 2==
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</asy>
 
</asy>
  
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[[2010 UNC Math Contest II Problems/Problem 2|Solution]]
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[[2010 UNCO Math Contest II Problems/Problem 2|Solution]]
  
 
==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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[[2010 UNC Math Contest II Problems/Problem 3|Solution]]
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[[2010 UNCO Math Contest II Problems/Problem 3|Solution]]
  
 
==Problem 4==
 
==Problem 4==
  
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4. Factor <math>n^4+2n^3+2n^2+2n+1</math> completely.
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Factor <math>n^4+2n^3+2n^2+2n+1</math> completely.
  
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[[2010 UNC Math Contest II Problems/Problem 4|Solution]]
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[[2010 UNCO Math Contest II Problems/Problem 4|Solution]]
  
 
==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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[[2010 UNC Math Contest II Problems/Problem 5|Solution]]
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[[2010 UNCO Math Contest II Problems/Problem 5|Solution]]
  
 
==Problem 6==
 
==Problem 6==
  
 
<math>A</math> is a <math>4</math>-digit number <math>abcd</math>. <math>B</math> is a <math>5</math>-digit number formed by augmenting <math>A</math> with a <math>3</math> on the right, i.e.
 
<math>A</math> is a <math>4</math>-digit number <math>abcd</math>. <math>B</math> is a <math>5</math>-digit number formed by augmenting <math>A</math> with a <math>3</math> on the right, i.e.
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<math>B=abcd3</math>. <math>C</math> is another <math>5</math>-digit number formed by placing a <math>2</math> on the left <math>A</math>, i.e. <math>C=2abcd</math>. If <math>B</math> is
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<math>B=abcd3</math>.  
 +
 
 +
<math>C</math> is another <math>5</math>-digit number formed by placing a <math>2</math> on the left <math>A</math>, i.e. <math>C=2abcd</math>. If <math>B</math> is
 
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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[[2010 UNC Math Contest II Problems/Problem 6|Solution]]
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[[2010 UNCO Math Contest II Problems/Problem 6|Solution]]
  
 
==Problem 7==
 
==Problem 7==
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What is <math>R</math>?
 
What is <math>R</math>?
  
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[[2010 UNC Math Contest II Problems/Problem 7|Solution]]
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[[2010 UNCO Math Contest II Problems/Problem 7|Solution]]
  
 
==Problem 8==
 
==Problem 8==
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express your answer. Generalize this result.
 
express your answer. Generalize this result.
  
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[[2010 UNC Math Contest II Problems/Problem 8|Solution]]
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[[2010 UNCO Math Contest II Problems/Problem 8|Solution]]
  
 
==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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[[2010 UNC Math Contest II Problems/Problem 9|Solution]]
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[[2010 UNCO Math Contest II Problems/Problem 9|Solution]]
  
 
==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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[[2010 UNC Math Contest II Problems/Problem 10|Solution]]
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[[2010 UNCO Math Contest II Problems/Problem 10|Solution]]
  
 
==Problem 11==
 
==Problem 11==
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<asy>
 
<asy>
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for(int x=0;x<3;x++){
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for(int y=0;y<3;y++){
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for (int x=0; x<3; ++x)
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dot(x,y);
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{for (int y=0; y<3; ++y)
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dot(x+4,y);
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{dot((x,y));dot((x+4,y));}
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}
 
 
}
 
}
 
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle,black);
 
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle,black);
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draw((5,1)--(6,0)--(7,1)--(6,2)--cycle,black);
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draw((4,1)--(5,0)--(6,1)--(5,2)--cycle,black);
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draw((4,-1)--(4,3),black);
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draw((3,-1)--(3,3),dashed);
 +
 
 
</asy>
 
</asy>
  
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(c) How many for a <math>5 \times 5</math>?
 
(c) How many for a <math>5 \times 5</math>?
  
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(d) How many for an <math>N \times N</math> ( grid of dots?
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(d) How many for an <math>(N+1) \times (N+1)</math> grid of dots?
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[[2010 UNCO Math Contest II Problems/Problem 11|Solution]]
  
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[[2010 UNC Math Contest II Problems/Problem 11|Solution]]
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== See Also ==
 +
{{UNCO Math Contest box|year=2010|n=II|before=[[2009 UNCO Math Contest II]]|after=[[2011 UNCO Math Contest II]]}}

Latest revision as of 21:12, 7 November 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

See Also

2010 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by
2009 UNCO Math Contest II 		Followed by
2011 UNCO Math Contest II
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
All UNCO Math Contest Problems and Solutions
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