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2008 UNCO Math Contest II Problems

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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); [/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

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