2009 UNCO Math Contest II Problems

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University of Northern Colorado MATHEMATICS CONTEST FINAL ROUND January 31,2009.

Problem 1

How many positive $3$-digit numbers $abc$ are there such that $a+b=c$ For example, $202$ and $178$ have this property but $245$ and $317$ do not.

Problem 2

(a) Let $Q_n=1^n+2^n$. For how many $n$ between $1$ and $100$ inclusive is $Q_n$ a multiple of $5$?

(b) For how many $n$ between $1$ and $100$ inclusive is $R_n=1^n+2^n+3^n+4^n$ a multiple of 5?

Problem 3

An army of ants is organizing a march to the Obama inauguration. If they form columns of $10$ ants there are $8$ left over. If they form columns of $7, 11$ or $13$ ants there are $2$ left over. What is the smallest number of ants that could be in the army?

Problem 5

The two large isosceles right triangles are congruent. If the area of the inscribed square $A$ is $225$ square units, what is the area of the inscribed square $B$? $[asy] draw((0,0)--(1,0)--(0,1)--cycle,black); draw((0,0)--(1/2,0)--(1/2,1/2)--(0,1/2)--cycle,black); MP("A",(1/4,1/8),N); draw((2,0)--(3,0)--(2,1)--cycle,black); draw((2+1/3,0)--(2+2/3,1/3)--(2+1/3,2/3)--(2,1/3)--cycle,black); MP("B",(2+1/3,1/8),N); [/asy]$

Problem 6

Let each of $m$ distinct points on the positive $x$-axis be joined to each of $n$ distinct points on the positive $y$-axis. Assume no three segments are concurrent (except at the axes). Obtain with proof a formula for the number of interior intersection points. The diagram shows that the answer is $3$ when $m=3$ and $n=2.$ $[asy] draw((0,0)--(0,3),arrow=Arrow()); draw((0,0)--(4,0),arrow=Arrow()); for(int x=0;x<4;++x){ for(int y=0;y<3;++y){ D((x,0)--(0,y),black); }} dot(IP((2,0)--(0,1),(1,0)--(0,2))); dot(IP((3,0)--(0,1),(1,0)--(0,2))); dot(IP((3,0)--(0,1),(2,0)--(0,2))); [/asy]$

Problem 7

A polynomial $P(x)$ has a remainder of $4$ when divided by $x+2$ and a remainder of $14$ when divided by $x-3.$ What is the remainder when $P(x)$ is divided by $(x+2)(x-3)$?

Problem 8

Two diagonals are drawn in the trapezoid forming four triangles. The areas of two of the triangles are $9$ and $25$ as shown. What is the total area of the trapezoid? $[asy] draw((0,0)--(20,0)--(2,4)--(14,4)--(0,0),black); draw((0,0)--(2,4)--(14,4)--(20,0),black); MP("25",(9,.25),N);MP("9",(9,2.25),N); [/asy]$

Problem 9

A square is divided into three pieces of equal area by two parallel lines as shown. If the distance between the two parallel lines is $8$ what is the area of the square? $[asy] draw((0,0)--(1,0)--(1,1)--(0,1)--cycle,black); draw((1,0)--(0,2/3),black); draw((1,1/3)--(0,1),black); [/asy]$

Problem 10

Let $S=\left \{1,2,3,\ldots ,n\right \}$. Determine the number of subsets $A$ of $S$ such that $A$ contains at least two elements and such that no two elements of $A$ differ by $1$ when

(a) $n=10$

(b) $n=20$

(c) generalize for any $n$.

Problem 11

If the following triangular array of numbers is continued using the pattern established, how many numbers (not how many digits) would there be in the $100^{th}$ row? As an example, the $5^{th}$ row has $11$ numbers. Use exponent notation to express your answer. \begin{align*} &1 \\ &2 \\ 3\quad &4\quad 5\quad \\ 6\quad 7\quad &8\quad 9\quad 10\quad \\ 11\quad 12\quad 13\quad 14\quad 15\quad &16\quad 17\quad 18\quad 19\quad 20\quad 21\quad \\ 22\quad 23\quad 24\quad 25\quad 26\quad 27\quad 28\quad 29\quad 30\quad 31\quad &32\quad 33\quad 34 \quad 35\quad 36\quad 37\quad 38\quad 39\quad 40\quad 41\quad 42\quad \\ \cdot \quad &\cdot\quad \cdot\quad \\ \end{align*}