# Difference between revisions of "2011 UNCO Math Contest II Problems"

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Let <math>m</math> and <math>n</math> be positive integers. List all the integers in the set | Let <math>m</math> and <math>n</math> be positive integers. List all the integers in the set | ||

− | <math>\left{20 ,21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31\right}</math> that <math>\underline{cannot}</math> be written in the form <math>m+n+m \cdot n</math>. | + | <math>\left\{ 20 ,21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31\right\}</math> that <math>\underline{cannot}</math> be written in the form <math>m+n+m \cdot n</math>. |

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>. | ||

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of the shaded overlap region? | of the shaded overlap region? | ||

<asy> | <asy> | ||

− | filldraw((0,0)--(13,0)--(12,5)--cycle,blue); | + | filldraw((0,0)--(13,0)--(25,5)--(12,5)--cycle,blue); |

pair A1=(0,0),B1=(25,0),C1=(25,5),D1=(0,5); | pair A1=(0,0),B1=(25,0),C1=(25,5),D1=(0,5); | ||

draw(A1--B1--C1--D1--cycle,black); | draw(A1--B1--C1--D1--cycle,black); | ||

− | pair P= | + | pair P,R; |

− | draw(P--(P+R- | + | P=unit(A1-D1)+D1; |

+ | R=unit(C1-D1)+D1; | ||

+ | draw(P--(P+R-D1)--R,black); | ||

pair A2=(0,0),B2=(25/13,-60/13),C2=(25,5),D2=(25-25/13,5+60/13); | pair A2=(0,0),B2=(25/13,-60/13),C2=(25,5),D2=(25-25/13,5+60/13); | ||

draw(A2--B2--C2--D2--cycle,black); | draw(A2--B2--C2--D2--cycle,black); | ||

− | P= | + | P=unit(A2-D2)+D2; |

− | R= | + | R=unit(C2-D2)+D2; |

− | draw(P--(P+R- | + | draw(P--(P+R-D2)--R,black); |

</asy> | </asy> | ||

[[2011 UNC Math Contest II Problems/Problem 3|Solution]] | [[2011 UNC Math Contest II Problems/Problem 3|Solution]] | ||

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==Problem 4== | ==Problem 4== | ||

− | Let <math>A = \left{ 2,5,10,17,\cdots,n^2+1,\cdots\right}</math> be the set of all positive squares plus <math>1</math> and | + | Let <math>A = \left\{ 2,5,10,17,\cdots,n^2+1,\cdots\right\}</math> be the set of all positive squares plus <math>1</math> and |

− | <math>B = \left{101, 104, 109, 116,\cdots,m^2 + 100,\cdots\right}</math> be the set of all positive squares plus <math>100</math>. | + | <math>B = \left\{101, 104, 109, 116,\cdots,m^2 + 100,\cdots\right\}</math> be the set of all positive squares plus <math>100</math>. |

(a) What is the smallest number in both <math>A</math> and <math>B</math>? | (a) What is the smallest number in both <math>A</math> and <math>B</math>? | ||

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line connecting <math>B</math> and <math>D</math>. | line connecting <math>B</math> and <math>D</math>. | ||

<asy> | <asy> | ||

− | pair A=(0,4*sqrt(10)) | + | pair A,B,C,D,E,F,P,R; |

− | + | A=(0,4*sqrt(10)); | |

+ | B=(4*sqrt(10),4*sqrt(10)); | ||

+ | C=(4*sqrt(10),0); | ||

+ | D=(0,0); | ||

+ | E=((9/5)*sqrt(10),(3/5)*sqrt(10)); | ||

+ | F=(sqrt(10),3*sqrt(10)); | ||

draw(A--B--C--D--cycle,black); | draw(A--B--C--D--cycle,black); | ||

draw(D--E--F--B,black); | draw(D--E--F--B,black); | ||

− | + | P=unit(D-E)+E; | |

+ | R=unit(F-E)+E; | ||

draw(P--(P+R-E)--R,black); | draw(P--(P+R-E)--R,black); | ||

− | P= | + | P=unit(B-F)+F; |

+ | R=unit(E-F)+F; | ||

draw(P--(P+R-F)--R,black); | draw(P--(P+R-F)--R,black); | ||

MP("A",A,NW);MP("B",B,NE);MP("C",C,SE);MP("D",D,SW); | MP("A",A,NW);MP("B",B,NE);MP("C",C,SE);MP("D",D,SW); | ||

MP("6",(D/2+E/2),NW);MP("8",(E/2+F/2),NE);MP("10",(F/2+B/2),S); | MP("6",(D/2+E/2),NW);MP("8",(E/2+F/2),NE);MP("10",(F/2+B/2),S); | ||

− | |||

</asy> | </asy> | ||

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==Problem 9== | ==Problem 9== | ||

− | Let <math>T(n)</math> be the number of ways of selecting three distinct numbers from <math>\left{1, 2, 3,\cdots ,n\right}</math> so that they are | + | Let <math>T(n)</math> be the number of ways of selecting three distinct numbers from <math>\left\{1, 2, 3,\cdots ,n\right\}</math> so that they are |

− | the lengths of the sides of a triangle. As an example, <math>T(5) = 3</math>; the only possibilities are <math>2-3-4, 2-4-5</math>, | + | the lengths of the sides of a triangle. As an example, <math>T(5) = 3</math>; the only possibilities are <math>\{2-3-4\},\{ 2-4-5\}</math>, |

− | and <math>3-4-5</math>. | + | and <math>\{3-4-5\}</math>. |

(a) Determine a recursion for T(n). | (a) Determine a recursion for T(n). | ||

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The integers <math>1, 2, 3,\cdots , 50</math> are written on the blackboard. Select any two, call them <math>m</math> and <math>n</math> and replace | The integers <math>1, 2, 3,\cdots , 50</math> are written on the blackboard. Select any two, call them <math>m</math> and <math>n</math> and replace | ||

− | these two with the one number <math>m+n+mn</math> | + | these two with the one number <math>m+n+mn</math>. 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 <math>50</math> | explain, with proof, what happens. Also explain with proof what happens in general as you replace <math>50</math> | ||

with <math>N</math>. As an example, if you select <math>3</math> and <math>17</math> you replace them with <math>3 + 17 + 51 = 71</math>. If you select <math>5</math> | with <math>N</math>. As an example, if you select <math>3</math> and <math>17</math> you replace them with <math>3 + 17 + 51 = 71</math>. If you select <math>5</math> |

## Revision as of 03:06, 18 October 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.