1993 UNCO Math Contest II Problems

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UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST FINAL ROUND February 13,1993.

For Colorado Students Grades 7-12.

Problem 1

How many times must one shoot at this target, and which rings must one hit in order to score exactly $100$ points.

[asy] draw(circle((0,0),1),black); draw(circle((0,0),2),black); draw(circle((0,0),3),black); draw(circle((0,0),4),black); draw(circle((0,0),5),black); draw(circle((0,0),6),black); MP("40",(0,0-.3),N); MP("39",(0,1-.1),N); MP("24",(0,2-.1),N); MP("23",(0,3-.1),N); MP("17",(0,4-.1),N); MP("16",(0,5-.1),N); [/asy]

Solution

Problem 2

Determine the digit in the $623^{rd}$ place after the decimal point in the repeating decimal for: \[\frac{1}{9}+\frac{2}{99}+\frac{3}{999}.\]

Solution

Problem 3

A student thinks of four numbers. She adds them in pairs to get the six sums $9,18,21,23,26,35.$ What are the four numbers? There are two different solutions.

Solution

Problem 4

The table gives some of the straight line distances between certain pairs of cities. for example the distance between city $A$ and city $B$ is $34.$ Use the given data to determine the distance between city $A$ and city $C$. (Hint: a problem in the first round was similar in spirit to this one.) \[\begin{tabular}{c|cccc} & A & B & C & D \\ \hline A & 34 & & 16 \\  B & & 42 & \\  C & & & 12\\  D & 30 & & \\  \end{tabular}\]


Solution

Problem 5

A collection of $25$ consecutive positive integers adds to $1000.$ What are the smallest and largest integers in this collection?

Solution

Problem 6

Observe that \begin{align*} 2^2+3^2+6^2 &= 7^2 \\ 3^2+4^2+12^2 &= 13^2 \\ 4^2+5^2+20^2 &= 21^2 \\ \end{align*}

(a) Find integers $x$ and $y$ so that $5^2+6^2+x^2=y^2.$

(b) Conjecture a general rule that is being illustrated here.

(c) Prove your conjecture.


Solution

Problem 7

Choose four numbers by circling exactly one number in each horizontal row, and one number in each vertical column. Compute the product of these four numbers. Explain clearly why the same product results no matter which selection of this type of four numbers you make.

\[\begin{tabular}{|cccc|} \hline 10 & 15 & 35 & 20 \\ 2& 3& 7& 4\\ 8& 12& 28& 16\\ 12& 18& 42& 24\\ \hline \end{tabular}\]

Solution

Problem 8

For what integer value of $n$ is the expression \[\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\cdots +\frac{1}{\sqrt{n}+\sqrt{n+1}}\] equal to $7$ ? (Hint: $(1+\sqrt{2})(1-\sqrt{2})=-1.$)

Solution

Problem 9

Let $P$ be a point inside the rectangle $ABCD$. If $AP=5$ , $BP=11$ and $CP=10$, find the length of $DP$. (Hint: draw helpful vertical and horizontal lines.)

[asy] pair P=(2,2); draw((0,0)--(0,5)--(10,5)--(10,0)--cycle,dot); draw((0,0)--P,black); draw((0,5)--P,black); draw((10,5)--P,black); draw((10,0)--P,black); dot(P); MP("P",(1,1),N); MP("5",(1.5,2.9),N); MP("10",(6.5,2.8),N); MP("11",(6.5,.9),N); MP("A",(0,5),NW); MP("B",(10,5),NE); MP("C",(10,0),SE); MP("D",(0,0),SW); [/asy]

Solution

Problem 10

The scalene triangle $ABC$ has side lengths $51, 52, 53.$ $AD$ is perpendicular to $BC.$ [asy] draw((0,0)--(52,0)--(24,sqrt(3)*26)--cycle); draw((24,0)--(24,sqrt(3)*26)); draw((0,-8)--(52,-8),arrow=Arrow()); draw((52,-8)--(0,-8),arrow=Arrow()); draw((24,3)--(21,3)--(21,0),black); MP("B",(0,0),SW);MP("A",(24,sqrt(3)*26),N);MP("C",(52,0),SE);MP("D",(24,0),S); MP("52",(26,-8),S);MP("53",(38,sqrt(3)*13),NE);MP("51",(12,sqrt(3)*13),NW); [/asy]

(a) Determine the length of $BD.$

(b) Determine the area of triangle $ABC.$


Solution

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