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2012 UNCO Math Contest II Problems

Revision as of 20:29, 29 March 2019 by Eliteye (talk | contribs)

Twentieth Annual UNC Math Contest Final Round January 21, 2012. Three hours; no electronic devices. Show your work for each problem. Your score will be based on your answers and your written work, including derivations of formulas you are asked to provide.

• The positive integers are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,\ldots$

• An ordinary die is a cube whose six faces contain $1, 2, 3, 4, 5$, and $6$ dots.

Problem 1

(a) What is the largest factor of $180$ that is not a multiple of $15$?

(b) If $N$ satisfies $2^5 \times 3^2 \times5^4 \times 7^3 \times 11=99000 \times N$, then what is the largest perfect square that is a factor of $N$?


Solution

Problem 2

Four ordinary, six-sided, fair dice are tossed. What is the probability that the sum of the numbers on top is $5$?

Solution

Problem 3

Mrs. Olson begins a journey at the intersection of Avenue $A$ and First Street in the upper left on the attached map. She ends her journey at one of the Starbucks on Avenue $D$. There is a Starbucks on Avenue $D$ at every intersection from First Street through Sixth Street! If Mrs. Olson walks only East and South, how many different paths to a Starbucks on Avenue $D$ can she take? Note that Mrs. Olson may pass one Starbucks on her way to another Starbucks farther to the East.

[asy] size(300,100); path R=((0,0)--(0,1)--(2,1)--(2,0)--cycle); for(int rw=0;rw<5;++rw){ for (int cl=0;cl<7;++cl){ filldraw(shift(3*cl,2*rw)*R,green);} } MP("Avenue \: D",(12,1+.5),E); MP("Avenue \: C",(12,3+.5),E); MP("Avenue \: B",(0,5+.5),E); MP("Avenue \: A",(0,7+.5),E); MP(90,"1st\: Street",(3-.5,0),N); MP(90,"2cnd\: Street",(6-.5,0),N); MP(90,"3rd\: Street",(9-.5,0),N); MP(90,"4th\: Street",(12-.5,4),N); MP(90,"5th\: Street",(15-.5,4),N); MP(90,"6th\: Street",(18-.5,4),N); [/asy]

Solution

Problem 4

(a) What is the largest integer $n$ for which $n^3+1631$ is divisible by $n+11$ ?

(b) For how many positive integer values of $n$ is $n^3+1631$ divisible by $n+11$?


Solution

Problem 5

What is the remainder when $12^{2011}+11^{2012}$ is divided by seven?

Solution

Problem 6

How many $5$-digit positive integers have the property that the product of their digits is $600$?


Solution

Problem 7

A circle of radius $1$ is externally tangent to a circle of radius $3$ and both circles are tangent to a line. Find the area of the shaded region that lies between the two circles and the line.

[asy] draw((-1,-1)--(7,-1),black); filldraw((0,-1)--(0,0)--(2*sqrt(3),2)--(2*sqrt(3),-1)--cycle,grey); filldraw(circle((0,0),1),white); filldraw(circle((2*sqrt(3),2),3),white); [/asy]

Solution

Problem 8

An ordinary fair die is tossed repeatedly until the face with six dots appears on top. On average, what is the sum of the numbers that appear on top before the six? For example, if the numbers $3, 5, 2, 2, 6$ are the numbers that appear, then the sum of the numbers before the six appears is $3+5+2+2=12$. Do not include the $6$ in the sum.

Solution

Problem 9

Treasure Chest . You have a long row of boxes. The 1st box contains no coin. The next $2$ boxes each contain $1$ coin. The next $4$ boxes each contain $2$ coins. The next $8$ boxes each contain $3$ coins. And so on, so that there are $2^N$ boxes containing exactly $N$ coins.

(a) If you combine the coins from all the boxes that contain $1, 2, 3$, or $4$ coins you get $98$ coins. How many coins do you get when you combine the coins from all the boxes that contain $1, 2, 3,\ldots,$ or $N$ coins? Give a closed formula in terms of $N$. That is, give a formula that does not use ellipsis $(\ldots)$ or summation notation.

(b) Combine the coins from the first $K$ boxes. What is the smallest value of $K$ for which the total number of coins exceeds $20120$? (Remember to count the first box.)

Solution

Problem 10

An integer equiangular hexagon is a six-sided polygon whose side lengths are all integers and whose internal angles all measure $120^{\circ}$.

(a) How many distinct (i.e., non-congruent) integer equiangular hexagons have no side length greater than $6$? Two such hexagons are shown.

[asy] draw((0,0)--(1,0)--(4,3*sqrt(3))--(3,4*sqrt(3))--(-1,4*sqrt(3))--(-1-3*sqrt(3)/2,4*sqrt(3)-1.5)--cycle,black); MP("1",(.5,0),S);MP("6",(2.5,1.5*sqrt(3)),SE);MP("2",(3.5,3.5*sqrt(3)),NE);MP("4",(1,4*sqrt(3)),N);MP("3",(-1-.75*sqrt(3),4*sqrt(3)-.75),NW);MP("5",(-.5-.75*sqrt(3),2*sqrt(3)-.75),W); draw((8,0)--(11,0)--(13,2*sqrt(3))--(11.5,3.5*sqrt(3))--(7.5,3.5*sqrt(3))--(6,2*sqrt(3))--cycle,black); MP("3",(9.5,0),S);MP("4",(12,sqrt(3)),SE);MP("3",(12.25,2.75*sqrt(3)),NE);MP("4",(9.5,3.5*sqrt(3)),N);MP("3",(6.75,2.75*sqrt(3)),NW);MP("4",(7,sqrt(3)),W); [/asy]


(b) How many distinct integer equiangular hexagons have no side greater than $n$? Give a closed formula in terms of $n$.

(A figure and its mirror image are congruent and are not considered distinct. Translations and rotations of one another are also congruent and not distinct.)


Solution

Problem 11

Construct a $4$th degree polynomial $x^4+ax^3+bx^2+cx+d$ that meets as many of the following conditions as you can: the sum of the roots is $1$, the sum of the squares of the roots is $2$, the sum of the cubes of the roots is $3$, and the sum of the $4$th powers of the roots is $4$.

Solution

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