Difference between revisions of "2006 UNCO Math Contest II Problems"
m |
m |
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Line 106: | Line 106: | ||
(i) <math>a<b<c</math> | (i) <math>a<b<c</math> | ||
− | (ii) Each of <math>a+b,a+c</math> and <math> | + | (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.
Contents
Problem 1
If a dart is thrown at the target, what is the probability that it will hit the shaded area?
Problem 2
If and are positive integers, how many integers are strictly between the product and ? For example, there are 35 integers strictly between and
Problem 3
The first 14 integers are written in order around a circle.
Starting with 1, every fifth integer is underlined. (That is ). What is the number underlined?
Problem 4
Determine all positive integers such that divides evenly (without remainder) into ?
Problem 5
In the figure is parallel to and also is parallel to . The area of the larger triangle is . The area of the trapezoid is . Determine the area of triangle .
Problem 6
The sum of all of the positive integer divisors of is
(a) Determine a nice closed formula (i.e. without dots or the summation symbol) for the sum of all positive divisors of .
(b) Repeat for .
(c) Generalize.
Problem 7
The five digits and of are such that and ; in addition, . Find another integer such that is also a five digit number that satisfies and .
Problem 8
Find all positive integers such that is a prime number. For each of your values of compute this cubic polynomial showing that it is, in fact, a prime.
Problem 9
Determine three positive integers and that simultaneously satisfy the following three conditions:
(i)
(ii) Each of and is the square of an integer, and
(iii) is as small as is possible.
Problem 10
How many triples of positive integers and are there with and .
Problem 11
Call the figure below a "-tableau" shape. Determine the number of rectangles of all sizes contained within this shape. Note that a square is considered a rectangle, and a rectangle is considered different from a . Express your answer as a binomial coefficient and explain the significance of your expression. Generalize, with proof, to an "-tableau" shape.