UNIVERSITY OF NORTHERN COLORADO
MATHEMATICS CONTEST
First Round
For all Colorado Students Grades 7-12
November 1, 2008
• The positive integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
…
.• The Pythagorean Theorem says that a2 +b2 =c2 where a, b, and c are side lengths of a right triangle and c is the hypotenuse.
• A scalene triangle has three sides of unequal length.
1. A unit fraction is a proper fraction of the form 1/n where is an integer greater than 1. The
numerator is always 1. Examples: 1/3, 1/29, 1/100 Find two ways to write 4/5 as the sum of three different unit fractions.
2. Insert all 11 integers 1, 2, 3, , 10, 11 into this shape so that the sum of all the vertical squares is 43 and the sum of all the horizontal squares is 28. What number is in the corner?
3. Find a set of three different positive integers given that their product is 72 and that their sum is a multiple of 7.
4. The area of the scalene triangle shown is 84 sq. units. Two side lengths are given as AB=10 and AC=21. Determine the length of the third side BC.
5. Determine the area of the trapezoid.
6. An army of ants is organizing a peace march across a room. If they form columns of 8 ants there are 4 left over. If they form columns of either 3 or 5 ants there are 2 left over. What is the smallest number of ants that could be in this army?
7. Let . How many subsets of consisting of 8 (eight) different
elements are such that the sum of the eight elements is a multiple of 5?
8. Let denote the maximum number of points of intersection strictly between lines and formed by joining the m points on to the points on in all possible ways. as shown in the diagram.
(a) Compute (b) Compute
(c) Give a formula for
9. (a) How many subsets of have the property that contains at least 2 elements and no two elements of differ by 1? As an example, satisfies these two properties but does not.
Brief Solutions First Round 2008
1.
=
+
2. n = 5; Let n be in the corner, x the sum of the non corner horizontal squares, and y the corner vertical square. Then x + n = 28, y + n = 43; x + y + n = 1 + 2 + + 11= 66; solving gives
x = 23, n = 5.
3. Trial and error yields 1, 3, 24.
4. BC = 17; Let h be the altitude from B. Then and h = 8. Since 10 – 8 – 6 is a
Pythagorean triplet, the base AC could be expressed as the sum 21 = 6 + 15 but then BC is the hypotenuse of a right triangle whose side lengths are 8 and 15. BC = 17.
5. 244; Drop altitudes to form a center rectangle and two triangles. The base could be expressed as the sum 41 = 15 + 20 + 6, noting the potential for 17 –15 – 8 and a 10 – 8 – 6 right triangle. The area is then 60 + 160 + 24 = 244.
6. 92; Let n be the number of ants. Then n = 8a + 4 = 3b + 2 = 5c + 2. Then 8a + 2 is both a multiple of 3 and 5, hence of 15. Trying even k for 8a + 2 = 15k gives us k = 6, a = 11.
7. 9; Since 1 + 2 + 3 + + 10 = 55, the sum of all the integers in S is a multiples of 5. Then the
complement of the subset consisting of 8 elements must also be a multiple of 5. These doubletons are easier to count: 1,4 1,9 2,3 2,8 3,7 4,6 5,10 6,9 and 7,8.
8. (a) f (3, 3) = 9
(b) f (3, 4) = 18
(c) f (2, n) = n (n –1) / 2
A point of intersection is determined by the intersection of two lines. Any choice of 2 points on along with 2 points on will produce such an intersection point. Hence for (c) there are
points.
University of Northern Colorado
MATHEMATICS CONTEST FINAL ROUND 2009
For Colorado Students Grades 7-12
1. How many positive 3-digit numbers abc are there such that ? For example, 202 and 178
have this property but 245 and 317 do not.
2. (a) Let . For how many n between 1 and 100 inclusive is a multiple of 5?
(b) For how many n between 1 and 100 inclusive is a multiple of 5?
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?
4. How many perfect squares are divisors of the product ! · ! · ! · ! · ! · ! · ! · ! ? (Here, for
example, ! means · · · .)
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?
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 and .
7. A polynomial has a remainder of 4 when divided by and a remainder of 14 when divided
by . What is the remainder when is divided by ?
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?
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?
10. Let , , , … , . Determine the number of subsets A of such that A contains at least two elements and such that no two elements of A differ by 1 when
(a) (b) (c) generalize for any .
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 100th row? As an example, the 5th row has 11
numbers. Use exponent notation to express your answer.
Brief Solutions Final Round
point of intersection corresponds to the crossing of the two lines.
9. 832; implies a . Then √ √
There is a one-to-one correspondence between the desired type of subsets of S consisting of k elements and the
number of ways of choosing k out of objects. Each desired k-element subset matches with a way of
choosing k out of a sequence of 10 objects so that no two are adjacent. The following picture illustrates this with
: ○●○●○○○○●○ , ,
Delete objects, one from each gap, to form ○●●○○○●○
In reverse ●●●○○○○○ corresponds to , , . Any of the ways of selecting 3 objects from these 8 corresponds to a desired subset.
11. ⁄ ; The center element in row is . The following chart helps establish the pattern:
Row 2 3 4 5 6 # of elements
Note that
So row 100 contains ⁄ numbers, by summing the geometric sum.
