THE UNIVERSITY OF VERMONT
DEPARTMENT OF MATHEMATICS AND STATISTICS FIFTY-EIGHTH ANNUAL HIGH SCHOOL PRIZE EXAMINATION
MARCH 11, 2015
1) Express
3 4
–
4 3 5 6
–
6 5
as a rational number in lowest terms.
2) Teresa bought a toy marked 1
4 off the original price. If Teresa paid $60, what was the original price of the toy,
in dollars?
3) If 45% of x is 300, what is 75% of x?
4) In the figureA, B, C and Dare points on the circle andEis the point of intersection of linesAC___andDB___. If the degree measures of anglesCABandCEBare 12 and 36 respectively, what is the degree measure of angleDBA?
12
36
A B
C D
E
5) In right triangle ACD, AD___is perpendicular toDC___, AD= 15 andDB___ is perpendicular toAC___. IfAB = 9, findAC.
A
C D
B
6) Express (2 + 5 )4+ (2 – 5 )4 as an integer.
7) Adam can dig a hole in 2 hours. Ben can dig the same hole in 3 hours. How many hours would it take them to dig the hole if they work together ? Express your answer as a rational number in lowest terms.
8) Express (3 5 17 257 + 1)1/4 as an integer.
9) Find the sum of the prime factors of 2015.
10) Find the degree measure of the angle whose complement is 2
7 of its supplement.
12) Three numbers form a geometric progression. Their sum is 19
2 and the sum of their
reciprocals is 19
18. Find these three numbers.
13) Two circles are concentric, as shown. Chord AB
___
of the larger circle is trisected by the smaller circle so thatAC = CD = DB = 1 . The sum of the radii of the larger and smaller circles equals the length
of the chordAB
___
. Find the radius of the larger circle.
A C D B
14) A brother and sister walk home from school every day at the same constant speed. One day, 15 minutes after leaving school, the boy realized that he had forgotten his lunch bag at school and ran back to get it. In the meantime, the girl continued to walk home at half her usual speed. When the boy caught up with her, they resumed walking at their usual speed and arrived home 6 minutes later than usual. How many minutes did the girl walk alone?
15) Let y be the real number such that 2015y=y20153
.
Find the value of log2015(log2015(y)) – log2015(y). Express your answer in simplest form.
16) The radius of the smaller of the two concentric circles is one meter. The line segments joining the circles consist of the portions of radii of the larger circle that lie outside the smaller circle. These line segments and the smaller circle partition the larger circle into nine sections, each of which has the same area. Find the length of one of the line segments.
17) Four circles of radius 2 are pairwise tangent as shown in the figure. A fifth circle of radius 2 is drawn so as to pass through the common points of tangency. Find the total area of the region
shaded in the figure.
18) Solve the equation log
x(5) + logx4(16) = 1
2 for x. Express your answer as a rational number in
lowest terms.
19) Find the sum of all of the real solutions to the equation x|x– 6 | = 7.
20) Find the area of the region in the plane that simultaneously satisfies the inequalities x2+y2+ 6x– 10y 30 and y 5 – |x+ 3 |.
1 1 1 1 1 1 1
1 1 1 2 2
2 2 2
21) A gameboard in the shape of an equilateral triangle is partitioned into 25 congruent equilateral triangular regions; these are numbered 1 to 25, as shown in the figure. A blue checker is placed on one of the regions and a red checker is placed on a different region. How many ways can this be done so that the two checkers are not in adjacent regions ? Two regions are adjacent if they share a common edge.
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25
22) If k is a non-negative integer, define <k> as the sum of the cubes of the digits of k. For example, < 889 > = 83+ 83+ 93= 512 + 512 + 729 = 1753. The sequence {a
n} is defined by a0= 43 333 and an= <an– 1> for n 1. Find the value of a2015.
23) How many different paths are there from the point labeled START at the bottom of the figure to the point labeled END at the top of the figure, travelling diagonally upward to the left, diagonally upward to the right or straight up along lines in the figure ?
START END
24) Every day a crossword puzzle is placed on Jenny s desk. While Jenny is on vacation, the puzzles accumulate on her desk. When she returns from vacation, Jenny begins to solve the accumulated puzzles as well as the new puzzles that appear each day. Jenny determines that if she solves exactly four puzzles a day, she will completely catch up with her puzzle solving ten days sooner than she would if she solves exactly three puzzles each day. How long was Jenny on vacation?
25) Let g be a function such that g(1) = 1 and g(3n) =n g(n) for any positive integer n. What is the value of log3(g(3200))?
26) Let Sn=
{
1 1 2,
1 2 3 ,
1
3 4 , , 1
n (n+1)
}
. Find the average value of the elements of S2015 .27) Find the positive integer b such that (20b) (15b) = 320b .
28) A circle of radius 25 passes through two adjacent vertices of a square and is tangent to the side opposite the side joining the adjacent vertices. What is the length of one side of the square ?
29) Given that x+ 1
x= 3 , find the integer value of x 6+ 1
x6 .
30) Find the smallest positive integer n such that n! is divisible by 101000.
31) Find the length of either tangent line from the origin to the circle x2+y2– 6x– 8y+ 21 = 0 .
1 2 3 4 5
1 2 3 4 5 6
32) The string of digits 01 001 000 100 001 consists of blocks of zeros followed by a single one. Each successive block of zeros contains one more zero than the previous block. Ones
appear in positions 2, 5, 9, . Find the position of the 100th one.
33) Suppose that x, yandz are positive real numbers such that x(y+z) = 85, y(z+x) = 120 and z(x+y) = 105. What is the value of the product x y z?
34) If k= 0
cos2k(Φ) = 7, determine the value of cos(2Φ). Express your answer as a rational number in lowest terms.
35) Circles with centersCandDhave respective radii 3 and 10. A common tangent intersects the circles atEandF,
respectively. The line segmentsCD___ and EF___intersect atP. Given thatC P= 5, findE F. Express your answer as a
rational number in lowest terms.
C P D
E
F
36) Find the smallest positive integer n such that n – 17
6n+ 11 is a positive rational number that is not in lowest terms.
37) Given that 231– 1 is prime, find the sum of the reciprocals of all of the positive integer divisors
of 230(231– 1).
38) Find the real number k such that log((k– 2015) !) + log((k– 2014) !) + 2 = 2 log((k– 2013) !) .
39) Let S be the set of all 8 -digit positive integers obtained by rearranging the digits of 12345678. For example, 13578642 and 78651234 are elements of S. How many elements of S are divisible by 11 ?
40) Find the smallest positive real number x such that x2 – x x = 6, where t is the largest
integer less than or equal to t. For example, 3.25 = 3 and 13 = 13.
