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2006 UNCO Math Contest II Problems: Difference between revisions

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(i) <math>a<b<c</math>
(i) <math>a<b<c</math>


(ii) Each of <math>a+b,a+c</math> and <math>ab+c</math> is the square of an integer, and
(ii) Each of <math>a+b,a+c</math> and <math>b+c</math> is the square of an integer, and


(iii) <math>c</math> is as small as is possible.
(iii) <math>c</math> is as small as is possible.
Line 114: Line 114:
==Problem 10==
==Problem 10==


How many triples of positive integers <math>a,b</math> and <math>c</math> are there with <math>a<b<c</math> and <math>a+b+c=401</math>.


[[2006 UNCO Math Contest II Problems/Problem 10|Solution]]
[[2006 UNCO Math Contest II Problems/Problem 10|Solution]]
Line 120: Line 121:
==Problem 11==
==Problem 11==


Call the figure below a "<math>4</math>-tableau" shape. Determine the number of rectangles of all sizes contained within this shape.
Note that a square is considered a rectangle, and a <math>2\times 1</math> rectangle is considered different from a <math>1\times 2</math>.
Express your answer as a binomial coefficient and explain the significance of your expression. Generalize, with proof, to an "<math>n</math>-tableau" shape.
<asy>
for(int j=0;j<5;++j){
draw((0,j)--(min(j+1,4),j),black);
draw((j,max(0,j-1))--(j,4),black);
}
filldraw((2,2)--(2,3)--(1,3)--(1,2)--cycle,blue);
filldraw((2,2)--(3,2)--(3,3)--(2,3)--cycle,blue);
</asy>


[[2006 UNCO Math Contest II Problems/Problem 11|Solution]]
[[2006 UNCO Math Contest II Problems/Problem 11|Solution]]

Revision as of 21:14, 20 October 2014

UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST FINAL ROUND January 28,2006.

For Colorado Students Grades 7-12.


Problem 1

If a dart is thrown at the $6\times 6$ target, what is the probability that it will hit the shaded area?

[asy] filldraw((2,2)--(4,6)--(6,6)--(6,4)--cycle,blue); filldraw((2,2)--(6,2)--(6,1)--cycle,blue); filldraw((2,2)--(0,0)--(0,1)--cycle,blue); filldraw((2,2)--(0,4)--(0,6)--cycle,blue); for(int i=0;i<7;++i){ draw((0,i)--(6,i),black); draw((i,0)--(i,6),black); } dot((2,2));dot((0,4));dot((0,6));dot((4,6));dot((6,6)); dot((6,4));dot((6,2));dot((6,1));dot((0,0));dot((0,1)); [/asy]


Solution

Problem 2

If $a,b$ and $c$ are positive integers, how many integers are strictly between the product $abc$ and $(a+1)(b+1)(c+1)$ ? For example, there are 35 integers strictly between $24=2*3*4$ and $60=3*4*5.$

Solution

Problem 3

The first 14 integers are written in order around a circle.

Starting with 1, every fifth integer is underlined. (That is $1,6,11,2,7,\ldots$). What is the $2006^{th}$ number underlined?

[asy] draw(unitcircle,black); pair A; for(int j=1;j<15;++j){ A=dir(90-(j-1)*(360/14)); MP(string(j),A,A); } [/asy]

Solution

Problem 4

Determine all positive integers $n$ such that $n^2+3$ divides evenly (without remainder) into $n^4-3n^2+10$ ?

Solution

Problem 5

In the figure $BD$ is parallel to $AE$ and also $BF$ is parallel to $DE$. The area of the larger triangle $ACE$ is $128$. The area of the trapezoid $BDEA$ is $78$. Determine the area of triangle $ABF$.

[asy] draw((0,0)--(1,2)--(4,0)--cycle,black); draw((1/2,1)--(2.5,1)--(2,0),black); MP("A",(4,0),SE);MP("C",(1,2),N);MP("E",(0,0),SW); MP("D",(.5,1),W);MP("B",(2.5,1),NE);MP("F",(2,0),S); [/asy]

Solution


Problem 6

The sum of all of the positive integer divisors of $6^2=36$ is $1+2+3+4+6+9+12+18+36=91$

(a) Determine a nice closed formula (i.e. without dots or the summation symbol) for the sum of all positive divisors of $6^n$.

(b) Repeat for $12^n$.

(c) Generalize.

Solution


Problem 7

The five digits $a,b,c,d$ and $e$ of $55225$ are such that $a=b=e$ and $c=d$; in addition, $55225=235^2=(235)(235)$. Find another integer $m$ such that $m^2$ is also a five digit number $abcde$ that satisfies $a=b=e$ and $c=d$.

Solution

Problem 8

Find all positive integers $n$ such that $n^3-12n^2+40n-29$ is a prime number. For each of your values of $n$ compute this cubic polynomial showing that it is, in fact, a prime.

Solution

Problem 9

Determine three positive integers $a,b$ and $c$ that simultaneously satisfy the following three conditions:

(i) $a<b<c$

(ii) Each of $a+b,a+c$ and $b+c$ is the square of an integer, and

(iii) $c$ is as small as is possible.

Solution

Problem 10

How many triples of positive integers $a,b$ and $c$ are there with $a<b<c$ and $a+b+c=401$.

Solution


Problem 11

Call the figure below a "$4$-tableau" shape. Determine the number of rectangles of all sizes contained within this shape. Note that a square is considered a rectangle, and a $2\times 1$ rectangle is considered different from a $1\times 2$. Express your answer as a binomial coefficient and explain the significance of your expression. Generalize, with proof, to an "$n$-tableau" shape.

[asy] for(int j=0;j<5;++j){ draw((0,j)--(min(j+1,4),j),black); draw((j,max(0,j-1))--(j,4),black); } filldraw((2,2)--(2,3)--(1,3)--(1,2)--cycle,blue); filldraw((2,2)--(3,2)--(3,3)--(2,3)--cycle,blue); [/asy]

Solution

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