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 "2008 UNCO Math Contest II Problems"

(Created page with "UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST FINAL ROUND February 2, 2008. For Colorado Students Grades 7-12. ==Problem 1== Determine the number of <math>3 \times 3</ma...")
 
m (→‎Problem 6)
Line 93: Line 93:
 
dot((2,5));dot((8,0));dot((2+sqrt(128),9));
 
dot((2,5));dot((8,0));dot((2+sqrt(128),9));
 
MP("A",(2,5),W);MP("P",(8,0),S);MP("B",(2+sqrt(128),9),E);MP("L",(14,0),S);
 
MP("A",(2,5),W);MP("P",(8,0),S);MP("B",(2+sqrt(128),9),E);MP("L",(14,0),S);
 +
MP("5",(2,2.5),W);MP("12",(2+sqrt(128)/2,7),N);MP("9",(2+sqrt(128),4.5),E);
 
</asy>
 
</asy>
  

Revision as of 21:47, 19 October 2014

UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST FINAL ROUND February 2, 2008.

For Colorado Students Grades 7-12.

Problem 1

Determine the number of $3 \times 3$ square arrays whose row and column sums are equal to $2$, using $0, 1, 2$ as entries. Entries may be repeated, and not all of $0, 1, 2$ need be used as the two examples show.

\[\begin{tabular}{c c c c c c c c c} 1 & 1 & 0 & & & & 0 & 1 & 1 \\ 0 & 0 & 2 & & & & 1 & 1 & 0 \\ 1 & 1 & 0 & & & & 1 & 0 & 1 \\ \end{tabular}\]

Solution

Problem 2

Let $S = {a,b,c,d}$ be a set of four positive integers. If pairs of distinct elements of $S$ are added, the following six sums are obtained: $5, 10, 11, 13, 14, 19.$ Determine the values of $a, b, c$, and $d.$ [Hint: there are two possibilities.]

Solution

Problem 3

3. A rectangle is inscribed in a square creating four isosceles right triangles. If the total area of these four triangles is $200$, what is the length of the diagonal of the rectangle?

[asy] draw((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); draw((0,2/3)--(2/3,0)--(1,1/3)--(1/3,1)--cycle,black); [/asy]

Solution

Problem 4

4. In the figure there are $8$ line segments drawn from vertex $A$ to the base $BC$ (not counting the segments $AB$ or $AC$).

[asy] for (int x=0;x<11;++x){ draw((5,15)--(x,0),dot); } draw((0,0)--(10,0),black); draw((10/6,5)--(10-10/6,5),black); draw((20/6,10)--(10-20/6,10),black); MP("A",(5,15),N);MP("B",(0,0),W);MP("C",(10,0),E); [/asy]

(a) Determine the total number of triangles of all sizes.

(b) How many triangles are there if there are $n$ lines drawn from $A$ to $n$ interior points on $BC$?

Solution

Problem 5

5. The sum of $400, 3, 500, 800$ and $305$ is $2008$ and the product of these five numbers is $146400000000 = 1464 \times 10^8.$

(a) Determine the largest number which is the product of positive integers whose sum is $2008$.

(b) Determine the largest number which is the product of positive integers whose sum is $n$.

Solution

Problem 6

6. Points $A$ and $B$ are on the same side of line $L$ in the plane. $A$ is $5$ units away from $L, B$ is $9$ units away from $L$. The distance between $A$ and $B$ is $12$. For all points $P$ on $L$ what is the smallest value of the sum $AP + PB$ of the distances from $A$ to $P$ and from $P$ to $B$ ?

[asy] draw((0,0)--(15,0),arrow=Arrow()); draw((15,0)--(0,0),arrow=Arrow()); draw((2,0)--(2,5)--(2+sqrt(128),9)--(2+sqrt(128),0),black); draw((2,5)--(8,0)--(2+sqrt(128),9),dashed); dot((2,5));dot((8,0));dot((2+sqrt(128),9)); MP("A",(2,5),W);MP("P",(8,0),S);MP("B",(2+sqrt(128),9),E);MP("L",(14,0),S); MP("5",(2,2.5),W);MP("12",(2+sqrt(128)/2,7),N);MP("9",(2+sqrt(128),4.5),E); [/asy]

Solution

Problem 7

Determine the value of $a$ so that the following fraction reduces to a quotient of two linear expressions: \[\frac{x^3+(a-10)x^2-x+(a-6)}{x^3+(a-6)x^2-x+(a-10)}\]

Solution

Problem 8

Triangle $ABC$ has integer side lengths. One side is twice the length of a second side.

[asy] draw((0,0)--(185/16,sqrt(225-(185/16)^2))--(40,0)--cycle,black); MP("A",(0,0),W);MP("C",(185/16,sqrt(225-(185/16)^2)),N);MP("B",(40,0),E); [/asy]

(a) If the third side has length $40$ what is the greatest possible perimeter?

(b) If the third side has length $n$ what is the greatest possible perimeter?

(c) Now suppose one side is three times the length of a second side and the third side has length of $40$. What is the maximum perimeter?

(d) Generalize

Solution

Problem 9

Let $C_n = 1+10 +10^2 + \cdots + 10^{n-1}.$

(a) Prove that $9C_n = 10^n -1.$

(b) Prove that $(3C_3+ 2)^2 =112225.$

(c) Prove that each term in the following sequence is a perfect square: \[25, 1225, 112225, 11122225, 1111222225,\ldots\]

Solution

Problem 10

Let $f(n,2)$ be the number of ways of splitting $2n$ people into $n$ groups, each of size $2$. As an example,

the $4$ people $A, B, C, D$ can be split into $3$ groups: $\fbox{AB} \ \fbox{CD} ; \fbox{AC} \ \fbox{BD} ;$ and $\fbox{AD} \ \fbox{BC}.$

Hence $f(2,2)= 3.$

(a) Compute $f(3,2)$ and $f(4,2).$

(b) Conjecture a formula for $f(n,2).$

(c) Let $f(n,3)$ be the number of ways of splitting $\left \{1, 2, 3,\ldots ,3n \right \}$ into $n$ subsets of size $3$. Compute $f(2,3),f(3,3)$ and conjecture a formula for $f(n,3).$

Solution

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2008_UNCO_Math_Contest_II_Problems&oldid=65290"