Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2015 UNCO Math Contest II Problems"

(Created page with "Twenty-third Annual UNC Math Contest Final Round January 31, 2015 Three hours; no electronic devices. Answers must be justified to receive full credit. We hope you enjoy thinking...")
 
m
 
(10 intermediate revisions by 3 users not shown)
Line 1: Line 1:
−
Twenty-third Annual UNC Math Contest Final Round January 31, 2015
+
Twenty-third Annual UNC Math Contest Final Round January 31, 2015.
 +
 
 
Three hours; no electronic devices. Answers must be justified to receive full credit.
 
Three hours; no electronic devices. Answers must be justified to receive full credit.
−
We hope you enjoy thinking about these problems, but you are not expected to
 
−
do them all.
 
  
−
The positive integers are <math>1, 2, 3, 4, \ldots </math>
+
We hope you enjoy thinking about these problems, but you are not expected to do them all.
−
A polynomial is quadratic if its highest power term has power two.
+
 
 +
The positive integers are <math>1, 2, 3, 4, \ldots </math> A polynomial is quadratic if its highest power term has power two.
  
 
==Problem 1==
 
==Problem 1==
Line 15: Line 15:
 
==Problem 2==
 
==Problem 2==
  
−
Find the area of the shaded region. The outer circle
+
<asy>
−
has radius <math>3</math>. The shaded region is outlined by half circles
+
filldraw(circle((0,0),3),white);
−
whose radii are <math>1</math> and <math>2</math> and whose centers lie on
+
filldraw(arc((-1,0),2,0,180)--cycle,grey);
−
the dashed diameter of the big circle.
+
filldraw(arc((-2,0),1,0,180)--cycle,white);
 +
filldraw(arc((1,0),2,180,360)--cycle,grey);
 +
filldraw(arc((2,0),1,180,360)--cycle,white);
 +
</asy>
 +
 
 +
Find the area of the shaded region. The outer circle has radius <math>3</math>. The shaded region is outlined by half circles whose radii are <math>1</math> and <math>2</math> and whose centers lie on the dashed diameter of the big circle.
  
 
[[2015 UNCO Math Contest II Problems/Problem 2|Solution]]
 
[[2015 UNCO Math Contest II Problems/Problem 2|Solution]]
Line 30: Line 35:
 
==Problem 4==
 
==Problem 4==
  
−
Tarantulas <math>A, B,</math> and <math>C</math> start together at the same time and race straight along a <math>100</math> foot
+
Tarantulas <math>A, B,</math> and <math>C</math> start together at the same time and race straight along a <math>100</math> foot path, each running at a constant speed the whole distance.  
−
path, each running at a constant speed the whole distance. When <math>A</math> reaches the end, <math>B</math> still
+
When <math>A</math> reaches the end, <math>B</math> still has <math>10</math> feet more to run. When <math>B</math> reaches the end, <math>C</math> has <math>20</math> feet more to run.  
−
has <math>10</math> feet more to run. When <math>B</math> reaches the end, <math>C</math> has <math>20</math> feet more to run. How many more
+
How many more feet does Tarantula <math>C</math> have to run when Tarantula <math>A</math> reaches the end?
−
feet does Tarantula <math>C</math> have to run when Tarantula <math>A</math> reaches the end?
 
  
  
Line 40: Line 44:
 
==Problem 5==
 
==Problem 5==
  
−
A termite nest has the shape of an irregular polyhedron.
+
<asy>
−
The bottom face is a quadrilateral. The top face
+
pair a1=(0,0),b1=(1,0),c1=(3,0),d1=(4,0),e1=(4.5,-.25),f1=(4.25,-2),g1=(3.75,-2.25),h1=(1.5,-2);
−
is another polygon. The sides comprise <math>9</math> triangles, <math>6</math>
+
pair i1=(.75,-3),j1=(1.25,-3),k1=(3.25,-3),l1=(4,-2.75);
−
quadrilaterals, and <math>1</math> pentagon. The nest has <math>10</math> vertices
+
draw(a1--b1--c1--d1--e1--f1--d1--g1--f1,dot);
−
on its sides and bottom, not counting the several
+
draw(b1--h1--c1--g1--h1,dot);
−
around the top face. How many edges does the top face
+
draw(a1--i1--j1--h1,dot);
−
have?
+
draw(j1--k1--l1--f1,dot);
−
You may use Euler’s polyhedral identity, which says that on a convex polyhedron the number
+
draw(k1--g1,dot);
−
of faces plus the number of vertices is two more than the number of edges. (A vertex is a corner
+
</asy>
−
point and an edge is a line segment along which two faces meet.)
+
 
 +
A termite nest has the shape of an irregular polyhedron. The bottom face is a quadrilateral. The top face is another polygon.  
 +
The sides comprise <math>9</math> triangles, <math>6</math> quadrilaterals, and <math>1</math> pentagon. The nest has <math>10</math> vertices on its sides and bottom,  
 +
not counting the several around the top face. How many edges does the top face have?
 +
 
 +
You may use Euler’s polyhedral identity, which says that on a convex polyhedron the number of faces plus the number of vertices is two more than the number of edges.
 +
(A vertex is a corner point and an edge is a line segment along which two faces meet.)
  
 
[[2015 UNCO Math Contest II Problems/Problem 5|Solution]]
 
[[2015 UNCO Math Contest II Problems/Problem 5|Solution]]
Line 55: Line 65:
 
==Problem 6==
 
==Problem 6==
  
−
How many ordered pairs <math>(n,m)</math> of positive integers satisfying <math>m < n \le 50</math> have the property
+
How many ordered pairs <math>(n,m)</math> of positive integers satisfying <math>m < n \le 50</math> have the property that their product <math>mn</math> is less than <math>2015</math>?
−
that their product <math>mn</math> is less than <math>2015</math>?
 
  
 
[[2015 UNCO Math Contest II Problems/Problem 6|Solution]]
 
[[2015 UNCO Math Contest II Problems/Problem 6|Solution]]
Line 67: Line 76:
 
face along an edge, and any colors may meet at vertices. Describe your polyhedron and also
 
face along an edge, and any colors may meet at vertices. Describe your polyhedron and also
 
describe how to assign colors to the faces.
 
describe how to assign colors to the faces.
 +
 
(b) Show that if the faces of a polyhedron are colored in such a way that each face is either blue
 
(b) Show that if the faces of a polyhedron are colored in such a way that each face is either blue
 
or gold and no two gold faces meet along an edge, and if the polyhedron contains a sphere
 
or gold and no two gold faces meet along an edge, and if the polyhedron contains a sphere
Line 76: Line 86:
 
==Problem 8==
 
==Problem 8==
  
−
A garden urn contains 18 colored beetles: 6 red beetles, numbered from 1 to 6, and 12
+
A garden urn contains <math>18</math> colored beetles: <math>6</math> red beetles, numbered from <math>1</math> to <math>6</math>, and <math>12</math>
−
yellow beetles, numbered from 1 to 12. Beetles wander out of the urn in random order, one
+
yellow beetles, numbered from <math>1</math> to <math>12</math>. Beetles wander out of the urn in random order, one
 
at a time, without any going back in. What is the probability that the sequence of numbers
 
at a time, without any going back in. What is the probability that the sequence of numbers
 
on the first four beetles to wander out is steadily increasing, that is, that the number on each
 
on the first four beetles to wander out is steadily increasing, that is, that the number on each
Line 88: Line 98:
 
==Problem 9==
 
==Problem 9==
  
−
Starting at the node in the center of the diagram,
+
<asy>
−
an orb spider moves along its web. It is permissible
+
pair A=dir(90),B=dir(210),C=dir(330),O=(0,0);
−
for the spider to backtrack as often as it likes, in either
+
draw(circle(O,1),black);
−
direction, on segments it has previously travelled. On
+
draw(O--A--B--O--B--C--O--C--A,dot);
−
each move, the spider moves along one of the segments
+
 
−
(curved or straight) to some adjacent node that is different
+
</asy>
−
from the node that it currently occupies.
+
 
 +
Starting at the node in the center of the diagram, an orb spider moves along its web. It is permissible for the spider to backtrack as often as it likes,  
 +
in either direction, on segments it has previously traveled. On each move, the spider moves along one of the segments (curved or straight)
 +
to some adjacent node that is different from the node that it currently occupies.
 +
 
 
(a) How many different five-move paths start at the center node and end at the center node?
 
(a) How many different five-move paths start at the center node and end at the center node?
 +
 
(b) How many different seven-move paths start at the center node and end at the center node?
 
(b) How many different seven-move paths start at the center node and end at the center node?
  
Line 102: Line 117:
 
==Problem 10==
 
==Problem 10==
  
−
(a) You want to arrange 8 biologists of 8 different
+
<cmath>\begin{tabular}[t]{|c|c|c|c|}\hline
−
heights in two rows for a photograph. Each row must
+
& & & \\\hline
−
have 4 biologists. Height must increase from left to
+
& & & \\\hline
−
right in each row. Each person in back must be taller
+
\end{tabular}</cmath>
−
than the person directly in front of him. How many
 
−
different arrangements are possible?
 
−
(b) You arrange 12 biologists of 12 different heights in two rows of 6, with the same conditions
 
−
on height as in part (a). How many different arrangements are possible? Remember to justify
 
−
your answers.
 
−
(c) You arrange 2n biologists of 2n different heights in two rows of n, with the same conditions
 
−
on height as in part (a). Give a formula in terms of n for the number of possible arrangements.
 
−
BONUS: You arrange 12 biologists of 12 different heights in three rows of 4, with the same
 
−
conditions on height as in part 10(a) for all three rows. How many different arrangements are
 
−
possible?
 
  
 +
(a) You want to arrange <math>8</math> biologists of <math>8</math> different heights in two rows for a photograph.
 +
Each row must have <math>4</math> biologists. Height must increase from left to right in each row.
 +
Each person in back must be taller than the person directly in front of him. How many different arrangements are possible?
 +
 +
 +
 +
<cmath>\begin{tabular}[t]{|c|c|c|c|c|c|}\hline
 +
& & &  & & \\\hline
 +
& & & & & \\\hline
 +
\end{tabular}</cmath>
 +
 +
(b) You arrange <math>12</math> biologists of <math>12</math> different heights in two rows of <math>6</math>, with the same conditions on height as in part (a).
 +
How many different arrangements are possible? Remember to justify your answers.
 +
 +
 +
 +
(c) You arrange <math>2n</math> biologists of <math>2n</math> different heights in two rows of <math>n</math>, with the same conditions on height as in part (a).
 +
Give a formula in terms of <math>n</math> for the number of possible arrangements.
  
 
[[2015 UNCO Math Contest II Problems/Problem 10|Solution]]
 
[[2015 UNCO Math Contest II Problems/Problem 10|Solution]]
  
−
END OF CONTEST
+
==BONUS==
 +
You arrange <math>12</math> biologists of <math>12</math> different heights in three rows of <math>4</math>, with the same conditions on height as in part 10(a) for all three rows.
 +
How many different arrangements are possible?
 +
 
 +
[[2015 UNCO Math Contest II Problems/BONUS|Solution]]
 +
 
  
 
== See Also ==
 
== See Also ==
 
{{UNCO Math Contest box|year=2015|n=II|before=[[2014 UNCO Math Contest II]]|after=[[2016 UNCO Math Contest II]]}}
 
{{UNCO Math Contest box|year=2015|n=II|before=[[2014 UNCO Math Contest II]]|after=[[2016 UNCO Math Contest II]]}}

Latest revision as of 14:11, 20 August 2020

Twenty-third Annual UNC Math Contest Final Round January 31, 2015.

Three hours; no electronic devices. Answers must be justified to receive full credit.

We hope you enjoy thinking about these problems, but you are not expected to do them all.

The positive integers are $1, 2, 3, 4, \ldots$ A polynomial is quadratic if its highest power term has power two.

Problem 1

The sum of three consecutive integers is $54$. What is the smallest of the three integers?

Solution

Problem 2

[asy] filldraw(circle((0,0),3),white); filldraw(arc((-1,0),2,0,180)--cycle,grey); filldraw(arc((-2,0),1,0,180)--cycle,white); filldraw(arc((1,0),2,180,360)--cycle,grey); filldraw(arc((2,0),1,180,360)--cycle,white); [/asy]

Find the area of the shaded region. The outer circle has radius $3$. The shaded region is outlined by half circles whose radii are $1$ and $2$ and whose centers lie on the dashed diameter of the big circle.

Solution

Problem 3

If P is a polynomial that satisfies $P(x^2 +1) = 5x^4 +7x^2 +19$, then what is $P(x)$? (Hint: $P$ is quadratic.)

Solution

Problem 4

Tarantulas $A, B,$ and $C$ start together at the same time and race straight along a $100$ foot path, each running at a constant speed the whole distance. When $A$ reaches the end, $B$ still has $10$ feet more to run. When $B$ reaches the end, $C$ has $20$ feet more to run. How many more feet does Tarantula $C$ have to run when Tarantula $A$ reaches the end?


Solution

Problem 5

[asy] pair a1=(0,0),b1=(1,0),c1=(3,0),d1=(4,0),e1=(4.5,-.25),f1=(4.25,-2),g1=(3.75,-2.25),h1=(1.5,-2); pair i1=(.75,-3),j1=(1.25,-3),k1=(3.25,-3),l1=(4,-2.75); draw(a1--b1--c1--d1--e1--f1--d1--g1--f1,dot); draw(b1--h1--c1--g1--h1,dot); draw(a1--i1--j1--h1,dot); draw(j1--k1--l1--f1,dot); draw(k1--g1,dot); [/asy]

A termite nest has the shape of an irregular polyhedron. The bottom face is a quadrilateral. The top face is another polygon. The sides comprise $9$ triangles, $6$ quadrilaterals, and $1$ pentagon. The nest has $10$ vertices on its sides and bottom, not counting the several around the top face. How many edges does the top face have?

You may use Euler’s polyhedral identity, which says that on a convex polyhedron the number of faces plus the number of vertices is two more than the number of edges. (A vertex is a corner point and an edge is a line segment along which two faces meet.)

Solution

Problem 6

How many ordered pairs $(n,m)$ of positive integers satisfying $m < n \le 50$ have the property that their product $mn$ is less than $2015$?

Solution

Problem 7

(a) Give an example of a polyhedron whose faces can be colored in such a way that each face is either blue or gold, no two gold faces meet along an edge, and the total area of all the blue faces is half the total area of all the gold faces. A blue face may meet another blue face along an edge, and any colors may meet at vertices. Describe your polyhedron and also describe how to assign colors to the faces.

(b) Show that if the faces of a polyhedron are colored in such a way that each face is either blue or gold and no two gold faces meet along an edge, and if the polyhedron contains a sphere inside it that is tangent to each face, then the total area of all the blue faces is at least as large as the total area of all the gold faces.

Solution

Problem 8

A garden urn contains $18$ colored beetles: $6$ red beetles, numbered from $1$ to $6$, and $12$ yellow beetles, numbered from $1$ to $12$. Beetles wander out of the urn in random order, one at a time, without any going back in. What is the probability that the sequence of numbers on the first four beetles to wander out is steadily increasing, that is, that the number on each beetle to wander out is larger than the number on the beetle before and that no number is repeated? Give your answer as a fraction in lowest terms. You may leave the numerator and denominator in a factored form.

Solution

Problem 9

[asy] pair A=dir(90),B=dir(210),C=dir(330),O=(0,0); draw(circle(O,1),black); draw(O--A--B--O--B--C--O--C--A,dot); [/asy]

Starting at the node in the center of the diagram, an orb spider moves along its web. It is permissible for the spider to backtrack as often as it likes, in either direction, on segments it has previously traveled. On each move, the spider moves along one of the segments (curved or straight) to some adjacent node that is different from the node that it currently occupies.

(a) How many different five-move paths start at the center node and end at the center node?

(b) How many different seven-move paths start at the center node and end at the center node?

Solution

Problem 10

\[\begin{tabular}[t]{|c|c|c|c|}\hline & & & \\\hline & & & \\\hline \end{tabular}\]

(a) You want to arrange $8$ biologists of $8$ different heights in two rows for a photograph. Each row must have $4$ biologists. Height must increase from left to right in each row. Each person in back must be taller than the person directly in front of him. How many different arrangements are possible?


\[\begin{tabular}[t]{|c|c|c|c|c|c|}\hline & & & & & \\\hline & & & & & \\\hline \end{tabular}\]

(b) You arrange $12$ biologists of $12$ different heights in two rows of $6$, with the same conditions on height as in part (a). How many different arrangements are possible? Remember to justify your answers.


(c) You arrange $2n$ biologists of $2n$ different heights in two rows of $n$, with the same conditions on height as in part (a). Give a formula in terms of $n$ for the number of possible arrangements.

Solution

BONUS

You arrange $12$ biologists of $12$ different heights in three rows of $4$, with the same conditions on height as in part 10(a) for all three rows. How many different arrangements are possible?

Solution


See Also

2015 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by
2014 UNCO Math Contest II 		Followed by
2016 UNCO Math Contest II
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
All UNCO Math Contest Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2015_UNCO_Math_Contest_II_Problems&oldid=132240"