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Difference between revisions of "2017 UNCO Math Contest II Problems"

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==Problem 1==
 
==Problem 1==
 +
 +
<asy>
 +
pair A=24*dir(40),B=15*dir(220);
 +
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pair C1=(0,0),C2=(52,0);
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draw(circle(C1,24),black);
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draw(circle(C2,15),black);
 +
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draw(C1--C2,dot);
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draw(A--(B+C2));
 +
</asy>
 +
 
A circle has radius 24, a second circle has radius 15, and the centers of the two circles are 52
 
A circle has radius 24, a second circle has radius 15, and the centers of the two circles are 52
 
units apart. A line tangent to both circles crosses the line connecting the two centers at a point
 
units apart. A line tangent to both circles crosses the line connecting the two centers at a point
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==Problem 6==
 
==Problem 6==
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 +
<asy>
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pair A=dir(72),B=dir(144),C=dir(216),D=dir(288),E=dir(360),O=(0,0);
 +
draw(A--B--C--D--E--A);
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pair AB1=(A+2*B)/3,AB2=(A+B)/2,AB3=(2*A+B)/3;
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draw(C--AB1--C--AB2--C--AB3);
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pair BC1=(B+2*C)/3,BC2=(B+C)/2,BC3=(2*B+C)/3;
 +
draw(D--BC1--D--BC2--D--BC3);
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pair CD1=(C+2*D)/3,CD2=(C+D)/2,CD3=(2*C+D)/3;
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draw(E--CD1--E--CD2--E--CD3);
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pair DE1=(D+2*E)/3,DE2=(D+E)/2,DE3=(2*D+E)/3;
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draw(A--DE1--A--DE2--A--DE3);
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pair EA1=(E+2*A)/3,EA2=(E+A)/2,EA3=(2*E+A)/3;
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draw(B--EA1--B--EA2--B--EA3);
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</asy>
 +
 
  The Spider's Divider  
 
  The Spider's Divider  
  
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==Problem 8==
 
==Problem 8==
  
−
For what integer n does <math>x^2</math> − <math>x</math> + <math>n</math> divide into <math>x^{13}</math> − <math>233x</math> − <math>144</math> with no remainder? That is,
+
For what integer <math>n</math> does <math>x^2 - x + n</math> divide into <math>x^{13} - 233x - 44</math> with no remainder? That is,
−
for what integer n is the first polynomial a factor of the second one? As always, justify your
+
for what integer <math>n</math> is the first polynomial a factor of the second one? As always, justify your
 
answer.
 
answer.
  
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==Problem 9==
 
==Problem 9==
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<asy>
 +
 +
pair A=dir(0),B=dir(80),C=dir(160),D=dir(310),O=(0,0);
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draw(circle(O,1),black);
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draw(A--B--C--D--A--C--B--D);
 +
 +
</asy>
  
 
Suppose n points on the circumference of a circle
 
Suppose n points on the circumference of a circle
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  Powerless Progressions  
 
  Powerless Progressions  
  
−
Find an infinite sequence of integers a1, a2, a3, . . . that has all of
+
Find an infinite sequence of integers <math>a_1, a_2, a_3, \ldots</math>  that has all of
 
these properties:
 
these properties:
−
(1) an = c + dn with c and d the same for all n = 1, 2, 3, . . .
+
 
−
(2) c and d are positive integers, and
+
(1) <math>a_n = c + dn</math> with c and d the same for all <math>n = 1, 2, 3, \ldots</math>
−
(3) no number in the sequence is the rth power of any integer, for any power r = 2, 3, 4, . . .
+
 
 +
(2) <math>c</math> and <math>d</math> are positive integers, and
 +
 
 +
(3) no number in the sequence is the <math>r^{th}</math> power of any integer, for any power <math>r = 2, 3, 4, \ldots</math>
 +
 
 
Reminder: Justify answers. In particular, for maximum credit, make it clear in your presentation
 
Reminder: Justify answers. In particular, for maximum credit, make it clear in your presentation
 
that your sequence possesses the third property.
 
that your sequence possesses the third property.
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  Divide and Conquer  
 
  Divide and Conquer  
  
−
(a) How many different factorizations are there of 4096 (which is the
+
(a) How many different factorizations are there of <math>4096</math> (which is <math>2^{12}</math>)
−
twelfth power of 2) in which each factor is either a square or a cube (or both) of an integer and
+
in which each factor is either a square or a cube (or both) of an integer and
−
each factor is greater than one? Regard 4 × 4 × 4 × 8 × 8 and 4 × 8 × 4 × 8 × 4 as the same
+
each factor is greater than one?  
−
factorization: the order in which the factors are written does not matter. Regard the number
+
Regard <math>4 \times 4 \times 4 \times 8 \times 8</math> and <math>4 \times 8 \times 4 \times 8 \times 4</math> as the same factorization: the order in which the factors are written does not matter.  
−
itself, 4096, as one of the factorizations.
+
Regard the number itself, <math>4096</math>, as one of the factorizations.
−
(b) How many different factorizations are there of 46,656 as a product of factors in which each
+
 
 +
(b) How many different factorizations are there of <math>46,656</math> as a product of factors in which each
 
factor is either a square or a cube (or both) of an integer and each factor is greater than one? As
 
factor is either a square or a cube (or both) of an integer and each factor is greater than one? As
 
before, the order in which the factors is written does not matter, and the number itself counts
 
before, the order in which the factors is written does not matter, and the number itself counts
−
as a factorization. Note that <math>46,656</math> = <math>2^6</math> × <math>3^6</math>  
+
as a factorization. Note that <math>46,656</math> = <math>2^6 \times 3^6</math>  
  
 
.
 
.
 
[[2017 UNCO Math Contest II Problems/Problem 11|Solution]]
 
[[2017 UNCO Math Contest II Problems/Problem 11|Solution]]
 +
 +
 +
== See Also ==
 +
{{UNCO Math Contest box|year=2017|n=II|before=[[2016 UNCO Math Contest II]]|after=[[2018 UNCO Math Contest II]]}}

Latest revision as of 14:14, 20 August 2020

Twenty-fifth Annual UNC Math Contest Final Round January 21, 2017

Rules: Three hours; no electronic devices. The positive integers are 1, 2, 3, 4, . . . A prime is an integer strictly greater than one that is evenly divisible by no integers other than itself and 1. The primes are 2, 3, 5, 7, 11, 13, 17, . . .

Problem 1

[asy] pair A=24*dir(40),B=15*dir(220); pair C1=(0,0),C2=(52,0); draw(circle(C1,24),black); draw(circle(C2,15),black); draw(C1--C2,dot); draw(A--(B+C2)); [/asy]

A circle has radius 24, a second circle has radius 15, and the centers of the two circles are 52 units apart. A line tangent to both circles crosses the line connecting the two centers at a point P between the two centers. How much farther is P from the center of the bigger circle than it is from the center of the smaller circle?

Solution

Problem 2

[asy] pair A=dir(60),B=dir(120),C=dir(180),D=dir(240),E=dir(300),F=dir(360),O=(0,0); pair G=(2/sqrt(3))*A,H=(2/sqrt(3))*B,I=(2/sqrt(3))*C,J=(2/sqrt(3))*D,K=(2/sqrt(3))*E,L=(2/sqrt(3))*F; draw(circle(O,1),black); draw(A--B--C--D--E--F--A); draw(G--H--I--J--K--L--G); [/asy]

Find the ratio of the area of a regular hexagon circumscribed around a circle to the area of a regular hexagon inscribed inside the same circle. (A polygon is called regular if all its sides are the same length and all its corner angles have the same measure. A hexagon is a polygon with six sides.)


Solution

Problem 3

Prime Mates

Find the largest 9 digit integer in which no two digits are the same and the sum of each pair of adjacent digits is prime. That is, the sum of the first two digits is prime, the sum of the second and third digits is prime, the sum of the third and fourth digits is prime, and so on.

Solution


Problem 4

Monkey Business

Harold writes an integer; its right-most digit is 4. When Curious George moves that digit to the far left, the new number is four times the integer that Harold wrote. What is the smallest possible positive integer that Harold could have written?

Solution

Problem 5

Double Encryption

(a) Find a substitution code on the seven letters A, B, C, D, E, F, and G that has the property that if you apply it twice in a row (that is, encrypt the encryption), the message ABCDEFG becomes ECBFAGD. Describe your answer by giving the message that results when encryption is applied once to the message ABCDEFG. (b) Find another such code, if there is one.

Solution

Problem 6

[asy] pair A=dir(72),B=dir(144),C=dir(216),D=dir(288),E=dir(360),O=(0,0); draw(A--B--C--D--E--A); pair AB1=(A+2*B)/3,AB2=(A+B)/2,AB3=(2*A+B)/3; draw(C--AB1--C--AB2--C--AB3); pair BC1=(B+2*C)/3,BC2=(B+C)/2,BC3=(2*B+C)/3; draw(D--BC1--D--BC2--D--BC3); pair CD1=(C+2*D)/3,CD2=(C+D)/2,CD3=(2*C+D)/3; draw(E--CD1--E--CD2--E--CD3); pair DE1=(D+2*E)/3,DE2=(D+E)/2,DE3=(2*D+E)/3; draw(A--DE1--A--DE2--A--DE3); pair EA1=(E+2*A)/3,EA2=(E+A)/2,EA3=(2*E+A)/3; draw(B--EA1--B--EA2--B--EA3); [/asy] 

The Spider's Divider

On a regular pentagon, a spider forms segments that connect one endpoint of each side to n different non-vertex points on the side adjacent to the other endpoint of that side, going around clockwise, as shown. Into how many non-overlapping regions do the segments divide the pentagon? Your answer should be a formula involving n. (In the diagram, n = 3 and the pentagon is divided into 61 regions.)

Solution

Problem 7

A box of 48 balls contains balls numbered 1, 2, 3, . . ., 12 in each of four different colors. Without ever looking at any of the balls, you choose balls at random from the box and put them in a bag. (a) If you must be sure that when you finish, the bag contains at least one set of five balls whose numbers are consecutive, then what is the smallest number of balls you can put in the bag? (For example, a set of balls, in any combination of colors, with numbers 3, 4, 5, 6, and 7 is a set of five whose numbers are consecutive.) (b) If instead you must be sure that the bag contains at least one set of five balls all in the same color and with consecutive numbers, then what is the smallest number of balls you can put in the bag? Remember to justify answers for maximum credit.


Solution

Problem 8

For what integer $n$ does $x^2 - x + n$ divide into $x^{13} - 233x - 44$ with no remainder? That is, for what integer $n$ is the first polynomial a factor of the second one? As always, justify your answer.

Solution

Problem 9

[asy] pair A=dir(0),B=dir(80),C=dir(160),D=dir(310),O=(0,0); draw(circle(O,1),black); draw(A--B--C--D--A--C--B--D); [/asy]

Suppose n points on the circumference of a circle are joined by straight line segments in all possible ways and that no point that is not one of the original n points is contained in more than two of the segments. How many triangles are formed by the segments? Count all triangles whose sides lie along the segments, including triangles that overlap with other triangles. For example, for n = 3 there is one triangle and for n = 4 (shown in the diagram) there are 8 triangles.

Solution

Problem 10

Powerless Progressions

Find an infinite sequence of integers $a_1, a_2, a_3, \ldots$ that has all of these properties:

(1) $a_n = c + dn$ with c and d the same for all $n = 1, 2, 3, \ldots$

(2) $c$ and $d$ are positive integers, and

(3) no number in the sequence is the $r^{th}$ power of any integer, for any power $r = 2, 3, 4, \ldots$

Reminder: Justify answers. In particular, for maximum credit, make it clear in your presentation that your sequence possesses the third property.

Solution

Problem 11

Divide and Conquer

(a) How many different factorizations are there of $4096$ (which is $2^{12}$) in which each factor is either a square or a cube (or both) of an integer and each factor is greater than one? Regard $4 \times 4 \times 4 \times 8 \times 8$ and $4 \times 8 \times 4 \times 8 \times 4$ as the same factorization: the order in which the factors are written does not matter. Regard the number itself, $4096$, as one of the factorizations.

(b) How many different factorizations are there of $46,656$ as a product of factors in which each factor is either a square or a cube (or both) of an integer and each factor is greater than one? As before, the order in which the factors is written does not matter, and the number itself counts as a factorization. Note that $46,656$ = $2^6 \times 3^6$

. Solution


See Also

2017 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by
2016 UNCO Math Contest II 		Followed by
2018 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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