16th
Annual J. W. Andrushkiw
Mathematics
Competition
This exam is provided without any claims of accuracy
1. Tom has 2 more pens than pencils, 4 more pencils than markers, and 3 times as many pencils as markers. How many pencils has Tom?
2.
Find the positive real number x for which 
3. How many quarts of water must be added to 12 quarts of 24% acid solution in order to obtain a 19% acid solution?
4. It takes Dan 1 hour to drive 40 miles from A to B on the first part of a trip; his average rate for the next 50 miles, from B to C, is only 25 miles per hour, nd his average rate for the final 30 miles of the trip, from C to D, is double his average rate from B to C. Find his average rate for the entire trip (from A to D) in miles per hour.
5.
Two opposite vertices of a cube, P and Q, are opposite
verices of a rhombus; the other two vertices of the rhombus, T and U, are
midpoints of two opposite edges of the cube (edges which contain neither P nor
Q). If the perimeter of the rhombus PTQU is 
years, find the volume
of the cube.
6.
Let a, b, and c be positive real numbers. There are two real values of x for which 
. Find the product of these two values of x (in terms of a, b,
and c).
7.
Find all real values of x for which 
.
8. In a triangle ABC, median AM meets side BC in M and altitude AH meets side BC in H, with H between B and M. If line segment BH is 12 inches longer than line segment HM and 5 inches shorter than line segment AH, and if side AC is 39 inches long, find the perimeter of triangle ABC.
9. The sum of the cubes of three consecutive positive integers exceeds 300 times the sum of the three consecutive integers by M. And M exceeds the sum of the squares of the three consecutive integers by N. And N exceeds the product of the larger two consecutive integers by 88. Find the three consecutive integers.
10.
The
positive real numbers x which are less than 
and for which the
inequality 
holds, lie in a closed
interval 
. Find a and b.
11. Twelve distinguishable dice are tossed. Find the probability that eleven 1’s and one even number come up, given that the sum on the twelve dice is at most 15.
12.
For
n a positive integer greater than 1, let T(n) equal the total number of
(positive) prime divisors of n, and let t(n) equal the number of distinct
(positive) prime divisors of n. For example, if 
, then 
and 
; and 
and 
Find the number of
3-digit integers n for which 
.
13.
Coplanar
circles with centers A, B, C have areas 
square feet, 
square feet, and 
square feet,
respectively, and each of these three circles is externally tangent to each of
the other two circles. Find the area of the circle inscribed in triangle ABC.
14. Tina bought 50 books at a fair; she bought some hardcover novels (all priced the same), paperback novels (all priced the same), cookbooks (all priced the same), and travel books (all priced the same). If she had paint 15 cents less total for the 50 books, the average price per book would have been 60 cents. Two paperback novels cost only 5 cents more than one hardcover novel; the cost of one paperback novel and one travel book was double the cost of a cookbook; and the cost of one hardcover novel and one paperback novel exceeded the cost of one cookbook and one travel book by 15 cents. The number of paperback novels she bought exceeded the total number of the other three types by 4. The cost of the paperback novels she bought was $3.15 less than the cost of the others, and three times the cost of the hardcover novels exceeded the total cost of the paperback novels by 75 cents. Find the number of cookbooks Tina bought.
15. Ron can paint a certain type of house (alone) in 8 hours, Sam can paint his type of house (alone) in 9 hours, and Tom can paint this type of house (alone) in 6 hours. Ron starts painting a house of this tpe (alone) until it is 3/8 finished; then Sam and Tom help Ron for a while, but Tom leaves before they are finished. Ron and Sam together finish painting the house in one hour after Tom leaves. How much longer did it take to paint the house than if all three had worked together the entire time?
16.
There
are four points on a coordinate plane such that the sum of their distances from
the points A = (4,0) and B = (-4, 0) is
10 and the sum of the squares of their distances from the points A and B is 
. These four points lie on a hyperbola with vertices C =
(2,0) and D = (-2, 0). If the equation of this hyperbola can be written in the
form 
, find the positive real numbers r and s.
Last Modified: Aug 2012
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