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Difference between revisions of "2006 UNCO Math Contest II Problems"

(Created page with "UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST FINAL ROUND January 28,2006. For Colorado Students Grades 7-12. ==Problem 1== If a dart is thrown at the <math>6\times 6</...")
 
m (→‎Problem 11)
 
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==Problem 5==
 
==Problem 5==
 +
 +
In the figure <math>BD</math> is parallel to <math>AE</math> and also <math>BF</math> is parallel to <math>DE</math>. The area of the larger triangle <math>ACE</math> is <math>128</math>.
 +
The area of the trapezoid <math>BDEA</math> is <math>78</math>. Determine the area of triangle <math>ABF</math>.
 +
 +
<asy>
 +
draw((0,0)--(1,2)--(4,0)--cycle,black);
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draw((1/2,1)--(2.5,1)--(2,0),black);
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MP("A",(4,0),SE);MP("C",(1,2),N);MP("E",(0,0),SW);
 +
MP("D",(.5,1),W);MP("B",(2.5,1),NE);MP("F",(2,0),S);
 +
</asy>
  
 
[[2006 UNCO Math Contest II Problems/Problem 5|Solution]]
 
[[2006 UNCO Math Contest II Problems/Problem 5|Solution]]
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==Problem 6==
 
==Problem 6==
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 +
The sum of all of the positive integer divisors of <math>6^2=36</math> is <math>1+2+3+4+6+9+12+18+36=91</math>
 +
 +
(a) Determine a nice closed formula (i.e. without dots or the summation symbol) for the sum of all positive divisors of <math>6^n</math>.
 +
 +
(b) Repeat for <math>12^n</math>.
 +
 +
(c) Generalize.
  
 
[[2006 UNCO Math Contest II Problems/Problem 6|Solution]]
 
[[2006 UNCO Math Contest II Problems/Problem 6|Solution]]
  
−
==Problem 7
+
 
 +
 
 +
==Problem 7==
 +
 
 +
The five digits <math>a,b,c,d</math> and <math>e</math> of <math>55225</math> are such that <math>a=b=e</math> and <math>c=d</math>; in addition, <math>55225=235^2=(235)(235)</math>.
 +
Find another integer <math>m</math> such that <math>m^2</math> is also a five digit number <math>abcde</math> that satisfies <math>a=b=e</math> and <math>c=d</math>.
  
 
[[2006 UNCO Math Contest II Problems/Problem 7|Solution]]
 
[[2006 UNCO Math Contest II Problems/Problem 7|Solution]]
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==Problem 8==
 
==Problem 8==
  
 +
Find all positive integers <math>n</math> such that <math>n^3-12n^2+40n-29</math> is a prime number. For each of your values of <math>n</math> compute
 +
this cubic polynomial showing that it is, in fact, a prime.
  
 
[[2006 UNCO Math Contest II Problems/Problem 8|Solution]]
 
[[2006 UNCO Math Contest II Problems/Problem 8|Solution]]
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==Problem 9==
 
==Problem 9==
  
 +
Determine three positive integers <math>a,b</math> and <math>c</math> that simultaneously satisfy the following three conditions:
 +
 +
(i) <math>a<b<c</math>
 +
 +
(ii) Each of <math>a+b,a+c</math> and <math>b+c</math> is the square of an integer, and
 +
 +
(iii) <math>c</math> is as small as is possible.
  
 
[[2006 UNCO Math Contest II Problems/Problem 9|Solution]]
 
[[2006 UNCO Math Contest II Problems/Problem 9|Solution]]
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==Problem 10==
 
==Problem 10==
  
 +
How many triples of positive integers <math>a,b</math> and <math>c</math> are there with <math>a<b<c</math> and <math>a+b+c=401</math>.
  
 
[[2006 UNCO Math Contest II Problems/Problem 10|Solution]]
 
[[2006 UNCO Math Contest II Problems/Problem 10|Solution]]
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==Problem 11==
 
==Problem 11==
  
 +
Call the figure below a "<math>4</math>-tableau" shape. Determine the number of rectangles of all sizes contained within this shape.
 +
Note that a square is considered a rectangle, and a <math>2\times 1</math> rectangle is considered different from a <math>1\times 2</math>.
 +
Express your answer as a binomial coefficient and explain the significance of your expression. Generalize, with proof, to an "<math>n</math>-tableau" shape.
 +
 +
<asy>
 +
for(int j=0;j<5;++j){
 +
draw((0,j)--(min(j+1,4),j),black);
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draw((j,max(0,j-1))--(j,4),black);
 +
 +
}
 +
filldraw((2,2)--(2,3)--(1,3)--(1,2)--cycle,blue);
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filldraw((2,2)--(3,2)--(3,3)--(2,3)--cycle,blue);
 +
</asy>
  
 
[[2006 UNCO Math Contest II Problems/Problem 11|Solution]]
 
[[2006 UNCO Math Contest II Problems/Problem 11|Solution]]
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 +
== See Also ==
 +
{{UNCO Math Contest box|year=2006|n=II|before=[[2005 UNCO Math Contest II]]|after=[[2007 UNCO Math Contest II]]}}

Latest revision as of 22:15, 7 November 2014

UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST FINAL ROUND January 28,2006.

For Colorado Students Grades 7-12.


Problem 1

If a dart is thrown at the $6\times 6$ target, what is the probability that it will hit the shaded area?

[asy] filldraw((2,2)--(4,6)--(6,6)--(6,4)--cycle,blue); filldraw((2,2)--(6,2)--(6,1)--cycle,blue); filldraw((2,2)--(0,0)--(0,1)--cycle,blue); filldraw((2,2)--(0,4)--(0,6)--cycle,blue); for(int i=0;i<7;++i){ draw((0,i)--(6,i),black); draw((i,0)--(i,6),black); } dot((2,2));dot((0,4));dot((0,6));dot((4,6));dot((6,6)); dot((6,4));dot((6,2));dot((6,1));dot((0,0));dot((0,1)); [/asy]


Solution

Problem 2

If $a,b$ and $c$ are positive integers, how many integers are strictly between the product $abc$ and $(a+1)(b+1)(c+1)$ ? For example, there are 35 integers strictly between $24=2*3*4$ and $60=3*4*5.$

Solution

Problem 3

The first 14 integers are written in order around a circle.

Starting with 1, every fifth integer is underlined. (That is $1,6,11,2,7,\ldots$). What is the $2006^{th}$ number underlined?

[asy] draw(unitcircle,black); pair A; for(int j=1;j<15;++j){ A=dir(90-(j-1)*(360/14)); MP(string(j),A,A); } [/asy]

Solution

Problem 4

Determine all positive integers $n$ such that $n^2+3$ divides evenly (without remainder) into $n^4-3n^2+10$ ?

Solution

Problem 5

In the figure $BD$ is parallel to $AE$ and also $BF$ is parallel to $DE$. The area of the larger triangle $ACE$ is $128$. The area of the trapezoid $BDEA$ is $78$. Determine the area of triangle $ABF$.

[asy] draw((0,0)--(1,2)--(4,0)--cycle,black); draw((1/2,1)--(2.5,1)--(2,0),black); MP("A",(4,0),SE);MP("C",(1,2),N);MP("E",(0,0),SW); MP("D",(.5,1),W);MP("B",(2.5,1),NE);MP("F",(2,0),S); [/asy]

Solution


Problem 6

The sum of all of the positive integer divisors of $6^2=36$ is $1+2+3+4+6+9+12+18+36=91$

(a) Determine a nice closed formula (i.e. without dots or the summation symbol) for the sum of all positive divisors of $6^n$.

(b) Repeat for $12^n$.

(c) Generalize.

Solution


Problem 7

The five digits $a,b,c,d$ and $e$ of $55225$ are such that $a=b=e$ and $c=d$; in addition, $55225=235^2=(235)(235)$. Find another integer $m$ such that $m^2$ is also a five digit number $abcde$ that satisfies $a=b=e$ and $c=d$.

Solution

Problem 8

Find all positive integers $n$ such that $n^3-12n^2+40n-29$ is a prime number. For each of your values of $n$ compute this cubic polynomial showing that it is, in fact, a prime.

Solution

Problem 9

Determine three positive integers $a,b$ and $c$ that simultaneously satisfy the following three conditions:

(i) $a<b<c$

(ii) Each of $a+b,a+c$ and $b+c$ is the square of an integer, and

(iii) $c$ is as small as is possible.

Solution

Problem 10

How many triples of positive integers $a,b$ and $c$ are there with $a<b<c$ and $a+b+c=401$.

Solution


Problem 11

Call the figure below a "$4$-tableau" shape. Determine the number of rectangles of all sizes contained within this shape. Note that a square is considered a rectangle, and a $2\times 1$ rectangle is considered different from a $1\times 2$. Express your answer as a binomial coefficient and explain the significance of your expression. Generalize, with proof, to an "$n$-tableau" shape.

[asy] for(int j=0;j<5;++j){ draw((0,j)--(min(j+1,4),j),black); draw((j,max(0,j-1))--(j,4),black); } filldraw((2,2)--(2,3)--(1,3)--(1,2)--cycle,blue); filldraw((2,2)--(3,2)--(3,3)--(2,3)--cycle,blue); [/asy]

Solution

See Also

2006 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by
2005 UNCO Math Contest II 		Followed by
2007 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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