THE UNIVERSITY OF VERMONT
DEPARTMENT OF MATHEMATICS AND STATISTICS FIFTY-SEVENTH ANNUAL HIGH SCHOOL PRIZE EXAMINATION
MARCH 19, 2014
1) Express 1+ 1
20
1+ 1 14
as a rational number in lowest terms.
2) Express 84ê3I2-3-9-1ê2M
as a rational number in lowest terms.
3) Express log2H125Llog3H49Llog5H81Llog7H64L as an integer.
4) The formula for limeade calls for 4 ounces of lime juice for every 12 ounces of water. Karla initially uses 18 ounces of water to make her limeade. If her limeade has 40% too much lime juice, how many ounces of water does she need to add to her mixture to have the correct ratio of lime juice to water? Express the answer as a rational number in lowest terms.
5) During the winter, 60% of Vermonters ski and during the summer, 45% of Vermonters hike. If 15% of Vermonters do both activities, what percent of Vermonters do neither?
6L Find the total area of the shaded regions if the area of rectangle
ABCDis 40 square units. A B
C D
7) A candy store sold bags of 40 caramels for $3.20, bags of 40 chocolates for $4.00 and mixed bags of chocolates and caramels for $3.50. If the mixed bags also have 40 pieces of candy, how many caramels are in each mixed bag?
8) The function f satisfies 2 fHxL-6 fJ1
xM=x
2 for all x∫0. Find f(2) and express the answer as a rational number in lowest terms.
9) Find the real number x such that log3I xM+4 logxH9L=4.
10) Suppose that f is a function such that fH3xL= 3
3+x for all real x>0. Determine the value of fH10L.
11) Express 125
10 + 2510
12) In how many ways can 24 cents be paid using any combination of pennies, nickels and dimes?
16) The probability that Sheila hits the bullseye when playing darts is 1
4. If she tosses three darts, what is the probability she will hit the bullseye at least once? Express your answer as a rational number in lowest terms.
17) Find the coordinates of the center of the circle that passes through the points H7, 0L,H2,-1L and H2,-5L. Express
19) Ticket prices for a local community orchestra are $15 for adults, $12 for seniors and $7 for students. At a recent concert, the orchestra sold 120 tickets for a total of $1481. What is the maximum possible number of student tickets that were sold? is located onAB so that AQ=QC=CB. Determine the degree measure of angleA.
A
B C
25) For a real number x, define fHxL= 16x- x2 - 30x- x2-224 . Determine the largest possible
positive value of fHxL.
26) In a list of the base 4 representations of the decimal integers from 0 to 1023, the digit 3 appears a total of k times. Find k.
27LCirclesCPandCQwith centers atPandQare externally tangent and have radii 2 and 1, respectively. Line segment
ACis tangent to circleCPatAand line segmentBCis tangent
toCPandCQatTandS, respectively. Find the length AC.
A B
C
S T
P Q
28L Suppose thatAandBare points on a circle with centerO. If the perimeter of sectorOABis 10 units and the area of sectorOABis 4 square units, find all
possible values of the length of arcíAB. O
A B
29) If the roots of x2+a x+b=0 are the cubes of the roots of x2+x+2=0, find aandb.
30) How many positive integers x have the property that 14 is the remainder when 2014 is divided by x ?
31) Find the smallest positive value of x (in radians) such that tanH2xL= cosHxL-sinHxL cosHxL+sinHxL.
32L How many paths are there fromAtoB, if at each intersection you can only move in the indicated directionHsL?
B
A
33) If x2+6x-6=0 , what is the value of x3+7x2+2014 ?
34) For how many integers n is 20-n
14-n an integer ?
35) The integer M consists of 500 threes and the integer N consists of 500 sixes. What is the sum of the digits in the base 10 representation of the product MÿN ?
37) The geometric mean of a set of k positive real numbers 8x1,x2,x3,ÿ ÿ ÿ,xk< is Hx1ÿx2ÿ ÿ ÿxkL1êk.
Find the positive integer n such that the geometric mean of the set of all positive integer divisors of n is 70.
38L An equilateral triangle with side length one is divided into four congruent triangles and the central triangle is shaded. Let the shaded area be A1. The remaining three triangles are similarly divided and each central triangle is shaded; the area of the three shaded triangles isA2. This process is continued . The shaded areasA1, A2andA3are
shown. Find ⁄
n=1
¶
An
39) How many different rectangles can be formed using edges in the left-hand figure below? Two such rectangles are shown in the right-hand figure.
40LA grassy park in the shape of an equilateral triangle is to be surrounded by a gravel
walkway whose outside edges form an equilateral triangle. If the parallel sides of the walkway are 2 meters apart and the area of the grassy park
is 30,000 3 square meters, what is the area of the gravel walkway?
