DISINI 06-07

UNIVERSITY OF NORTHERN COLORADO

MATHEMATICS CONTEST

First Round

For all Colorado Students Grades 7-12

November 4, 2006

• An arithmetic progression is a sequence of the form a_, a₊d_{, a+2}d_{, …} • n _{factorial is computed as} n_!=n₍n₋₁₎₍n₋₂₎K_3⋅2⋅1.

• An isosceles triangle has two sides with equal length. • The positive odd integers are 1, 3, 5, 7, 9, 11, 13,

…

.

1. In the 4 by 4 square the boxes can be filled in with each of the numbers 1, 2, 3, 4 used

exactly once in each row and column. What is A?

2. Determine the sum of the first 500 digits of the unending decimal expansion for K

2307692 .

13 3

= .

3. The length of each leg of an isosceles triangle is 2x−1_{. If the base is} 5x₋7 _determine all possible integer values ofx_.

4. The odd number 7 can be expressed as16₋9₌42 ₋32_{, a difference of two squares. Express each}

of the following odd integers as the difference of two squares: (a) 17 (b) 83

5. The three roots of the cubic equation 9x3 ₋36x2 ₊cx₋16₌0 _{are in arithmetic progression.} Determine the value of c _{and the three roots.}

6. The perimeter of the triangle is 24 in., and its area is 8 2

in. . What is the exact

area of the inscribed circle? [That is, express the area as a fractional multiple ofπ].

7. Let f(x)=5x4 −6x3 −3x2 +8x+2. Determine coefficients a_, b_, c_, d _and e _{so that}

( )

x =a+b(x−2)+c(x−2)2 +d(x−2)3 +e(x−2)4.

f

8. Determine all n _{for which} n n y

x + factors. [As a reminder, x3 + y3 factors as )

)(

(x+y x2 −xy+ y2 , and x4 −y4 factors as (x2 −y2)(x2 + y2)].

9. An International Conference on Global Warming has 5 diplomats from the US, 3 diplomats from Russia and 4 diplomats from China. These 12 diplomats are to be seated at the head table in a single row.

Determine the number of possible seating arrangements if the diplomats from each country must be seated together as a group. Express your answer using the n_{! notation.}

10.A square P1P2P3P4 is drawn in the coordinate plane with P₁ at (1,0) and P3 at (3,0). Let Pn denote the point (xn,yn). Compute the numerical value of the following product of complex numbers: (x₁+iy₁)(x₂ +iy₂)(x₃ +iy₃)(x₄ +iy₄).

11.A quaternary “number” is an arrangement of digits, each of which is 0, 1, 2, 3. Some examples: 12203, 01130, 22222, 00031

(a) How many 5-digit quaternary numbers are there?

BRIEF SOLUTIONS TO FIRST ROUND

NOVEMBER 2006

1. _{A = 3; by trial and error.}

2. _{Sum = 2246 ; There are 83 blocks of} 2₊3₊0₊7₊6₊9₌27_{, with} 2₊3 _{left over.}

83_×27₊5₌2246_.

3. x _{= 2, 3, or 4;} 2x₋1₊2x₋1_>5x₋7 _gives 5_>x_. 2x₋1₊5x₋7_>2x₋1_gives x_>1.

4. _(a) 17₌92 ₋82 _(b) 83₌422₋412

5. _{2 3, 43, 6 3 and} c _{= 44; 36 9=4=(}a₊d₎₊a₊₍a₋d₎₌₃a _and a ₌4 3 _{is one root.}

(

) (

)

(

2 2

)

(

2

)

9 16 3 4 9

16 ₌ a₊d a a₋d ₌aa ₋d _{= −}d

− and d ₌2 3_{. So the roots are} a+d_, a_, a-d_{, or}

6 3_, 4 3_, 2 3_{. Substituting any root into the cubic gives} c_=44.

6. _π

9 4 =

A ; label the sides as a_, b_, c_{. Then} 8 2 1

2 1

2 1

= +

+ br cr

ar _{, where} r _{is the radius. Using} a₊b₊c₌₂₄, you

have ( ) 8

2 + + =

c b a r

or 24_r=16_, 3 2 =

r . Then 

     =       =

9 4

3 2 2

π π

A .

7. _{a = 38, b = 84, c = 81, d = 34, e = 5; Let x−2= y; Then x= y+2 and}

( )

_x = _f

(

_y+2

)

=5

(

_y+2

)

4 −6

(

_y+2

)

3 −3

(

_y+2

)

2 +8

(

_y+2

)

+2=

f

5

(

x−2

)

4+34

(

x−2

)

3+81

(

x−2

)

2+84

(

x−2

)

+38=

(

2

)

34

(

2

)

81

(

2

)

84

(

2

)

38 5

38 64 81

34 5

2 3

4

2 3

4

+ − + − + − + −

= + + +

+

x x

x x

y y

y y

8. For all n _{having an odd factor; or} n_≠2k_{; For example x}6 _+y6 ₌

(

_x2 _+y2

)(

_x4_−x2_y2 _+y4

)

9. 6_⋅5!_⋅3!_⋅4! _{; The US diplomats can be arranged in a group in} 5! _{ways. Same for the other groups.}

But then the three groups can be permuted around in 3!₌6 _ways.

10. 15; Label the points as P₁ =1, _{P =2+}i

2 , P3 =3, P₄ =2−i and multiply:

P₁P₂P₃P_{4 =}1_⋅

(

2₊i

)

_⋅3_⋅

(

2₋i

)

₌3

(

4₊1

)

₌15_.

11. (a) 45_{; There are four choices for each of five spots.}

(b) 45₋35_{; Take all from part (a) and subtract those where the digit 3 fails to appear.}

UNIVERSITY OF NORTHERN COLORADO

MATHEMATICS CONTEST

FINAL ROUND

For Colorado Students Grades 7-12

February 3, 2007

• The sequence of Fibonacci numbers is 1, 1, 2, 3, 5, 8, 13, 21, K _. • The positive odd integers are 1, 3, 5, 7, 9, 11, 13, … .

• A regular decagon is a 10-sided figure all of whose sides are congruent.

_________________________________________________________________________________

1. Express the following sum as a whole number:

2007 2006

2005 12

11 10 9 8 7 6 5 4 3 2

1_{+ − + + − + + − + + − +L+ + − .}

2. In Grants Pass, Oregon 4 5 _{of the men are married to} 3 7 _{of the women. What fraction of the} adult population is married? Give a possible generalization.

3. State the general rule illustrated here and prove it:

4. If x _{is a primitive cube root of one (this means that} x3 ₌1 _butx_{≠1) compute the value of}

2007 2007

2006

2006 1 1

x x

x

x _{+ + + .}

5. Ten different playing cards have the numbers 1, 1, 2, 2, 3, 3, 4, 4, 5, 5 written on them as shown. Three cards are selected at random without replacement. What is the

probability that the sum of the

numbers on the three cards is divisible by 7?

6. (a) Demonstrate that every odd number ₂n+_{1 can be expressed as a difference of two squares.}

(b) Demonstrate which even numbers can be expressed as a difference of two squares.

7. (a) Express the infinite sum = + + ₂ + ₃ +_L 3

1

3 1

3 1 1

S _{as a reduced fraction.}

(b) Express the infinite sum = + + + + +_L

3125 5

625 3

125 2

25 1

5 1

T as a (reduced) fraction. Here

the denominators are powers of 5 and the numerators 1, 1, 2, 3, 5, K _{are the Fibonacci numbers}

n

F _where

2

1 −

− +

= n n

n F F

F _.

8. A regular decagon P1P2P3KP10 is drawn

in the coordinate plane with P₁ at (2,0)

and P6 at (8,0). If Pn denotes the point

(x_n,y_n), compute the numerical value of the following product of complex numbers:

(

x1 +iy1

)(

x2 +iy2

)(

x3 +iy3

) (

L x10 +iy10

)

where i = −₁ as usual.

9. A circle is inscribed in an equilateral triangle whose side length is 2. Then another circle is inscribed externally tangent to the first circle but inside the triangle as shown. And then another, and another. If this process continues

forever what is the total area of all the circles? Express your answer as an exact multiple of π (and not as a decimal approximation).

10. A quaternary “number” is an arrangement of digits, each of which is 0, 1, 2, 3.

Some examples: 001, 3220, 022113.

(a) How many 6-digit quaternary numbers are there in which each of 0, 1 appear at least once?

(b) How many n_{-digit quaternary numbers are there in which each of 0, 1, 2, appear at least}

once? Test your answer with n₌3_.

Brief Solutions Final Round February 3, 2007

1. 670,338; (1+2−3)+(4+5−6)+(7+8−9)+L+(2005+2006−2007) =0+3+6+L+3⋅668=3(1+2+3+L+668)=3(668)(369)/2

2. 24₄₃; Restated, 15 12

of the men are married to

28 12

of the women. Then

43 24

of the adult population

is married. To generalize, if a b _{of the men are married to} c d _{of the women, then} ca cb_{of the} men are married to ca da _{of the women. The proportion that is married is2}ca

(

cb₊da

)

_.

3. 12₊22₊32_+L+n2₌1_⋅n₊3

(

n₋1

) (

₊5n₋2

)

_+L+

(

2n₋1

)

_⋅1_{. The picture tells the story. For} example, the fourth diagram shows one 4, three 3’s, five 2’s and seven 1’s. Stripping off layers of

1’s also gives 12₊22₊32₊42_.

4. +1; Since x3₌1_, x2006 ₌x2 _and x2007 ₌1_{. The expression becomes}

2

2

2 1 ₂ 1_

    

+ = + +

x x x

x ₌

1 2

=

      −

x x

since x2₊x₊1₌0_{. Or,} 1 2 2 2 1 1 0 1 1

2 3 2 2

2

= + = + + + = + + = +

+ x x

x x x x

x _{. Or,}

1 2 1 2 1

2 1

2

2_{+ + = +x+ =− + =}

x x

x _{since the sum of the vectors} x _and

x 1

is −1.

5. 215_{; There are five ways to achieve a sum divisible by 7; 115 (2 ways), 133 ( 2 ways), 124 (8}

ways), 233 (2 ways), 455 (2 ways). Hence, there are 16 favorable ways out of      

=

3 10

120 _total

choices.

6. (a) ₂n₊₁₌

(

n₊₁

)

2₋n2

(b) Multiples of 4; If x₌₄n_{, 4}n₌

(

n₊₁

)

2₋

(

n₋₁

)

2_{. The even numbers not divisible by 4, namely} 2, 6, 10, 14, K _{cannot be expressed as a product of two even numbers and hence cannot be}

7. (a) 3 2_{; }      + + + + = + + +

= _{2 L 2 L}

3 1 3 1 1 3 1 1 3 1 3 1 1

S _or S S

3 1 1₊ = and 2 3 = S _.

(b) 519_;

T T T T 5 1 1 5 2 5 1 5 1 1 4 5 3 5 2 5 1 5 1 5 5 5 3 5 2 5 1 1 5 4 3 2 4 3 2 4 3 2 + = + + + + = + + + + = + + + + + = L L L

8. 9,706,576; Translate the center of the decagon to the origin. Now the vertices represent the roots of f(x)=x10−310 =0. Since the Pn are each 5 more than the roots of f(x)=0, they would be

the roots of _f

(

_x−5

)

₌0_{or (}x₋₅₎10₋₃10 _{=0. The product then is the constant term, or}

576 , 706 , 9 3

510− 10 = .

9. 8 3_π

; Let_r1, and _r2 be radii of the first and second circles; 3

3

1=

r and the area of the first circle is

3 1 =π

A . From r₁+r₂ =2

(

r₁−r₂

)

, _{r2 =} 3 9 _{andA2 =π} 27_{. Similarly}

27 3

3 1

2

3= r =

r ,

243

3 =π

A . Then the total area = 3 8

9 1 1 3 243 1 27 1 3 1 π π π = − =     + +

+ L .

10. (a) 46₋2_⋅36₊26_{; from the total} 46 _{subtract those that have no} 0

( )

36 _{or no} 1

( )

36 _{. Then add}

back in those that have no 0 and no 1

( )

26 _.

(b) 4n₋3_⋅3n ₊3_⋅2n₋1n