UNIVERSITY OF VERMONT
DEPARTMENT OF MATHEMATICS AND STATISTICS FIFTY-FIFTH ANNUAL HIGH SCHOOL PRIZE EXAMINATION
MARCH 14, 2012 1) What is the smallest positive integer n such that 2n + 1 and 3n + 1 are both primes?
2) Every student in Ms. Math’s class took the AP Calculus Exam and received a score of either a 4 or a 5. If 32 of the students earned a 5 and that was 80% of the class, how many students earned a 4 on the AP Exam?
3) Express 1
2 3
4 5
6
as a rational number in lowest terms.
4 M,A,BandNare collinear withMA AB BN. ArcsMA, AN, NBand MB are all semicircles. IfMN 9, what is the perimeter of the shaded region ?
5) If p and q are the roots of the equation x2– 6x 2 0, find the value of 1
p 1 q .
6) Among the 41 students in a class, 12 have a cat,
5 have a dog, 8 have a robot, 2 have a cat and a dog, 6 have a cat and a robot, 3 have a dog and a robot, and 1 has a cat, a dog and a robot.
How many students have neither a cat nor a dog nor a robot?
7) The total cost of tickets to the school play for one adult and 3 children is $27. If the cost of a ticket for an adult is $1 more than the cost of a ticket for a child, what is the cost of an adult ticket?
8) A line passing through the points (3 , – 6) and (9 , 3) intersects the x-axis at the point (a , 0). What is the value of a ?
9) Let f(n) 3 n
3n . Find f 4 f 5 .
10) For non-zero real numbers a and b, define a b a b – 1 a –
1
11 Two circles, one of radius 3 and the other of radius 5, are externally tangent and are internally tangent to a larger circle. What is the area of the shaded region inside the larger circle that is not contained in either of the two smaller circles?
12) Find all values of k so that the curves y x2– 3k x–1 and y 9 k x – 17 intersect in exactly one point.
13) Find all real numbers x such that 4 –x 4 x 2x .
14) Let x and y be two positive real numbers such that logx y logy x 3. Find the value of logx y 2 logy x 2.
15) Let g be a function such that 3g(x) + 2g(1 – x) 9 + 2x. Find the value of g(2).
16) Find the exact value of cos3(15°) sin(15°) – cos(15°) sin3 15 ° .
17) What integer n satisfies n < 5 2 2 5 – 2 2 n 1?
18) In a random arrangement of the letters of GREENMOUNTAINS, what is the probability that the vowels are in alphabetical order within the arrangement? Express your answer as a rational number in lowest terms.
19) If is an acute angle such that tan( ) 2, find the value of cos(4 ). Express your answer as a rational number in lowest terms.
20) Find the area of the region in the plane consisting of all points (x , y) that satisfy | 4x – 8 | y 12 .
21) Find all positive real numbers x such that log2 x log2 x– 12 6 .
22) Let f be a function such that f x
5 x
2 x 2 . Find the sum of all values of w such that f(5w) 2012.
23) Suppose that 9a 12, 12b 15, 15c 18, 18d 21, 21e 24 and 24f 27. What is the value of a · b · c · d · e · f ?
24) If x, y and z are positive real numbers such that x y 40, x z 60 and y z 96, what is the value of x + y + z ?
25) Let S 5, 52, 53, , 510 . Suppose that a and b are distinct integers chosen from S. For how many ordered pairs a, b is logab an integer?
A B
C D
E
26 SquareABCDhas side length 6. A semicircle whose diameter isABis drawn internally. The line that is tangent to the semicircle atTand passes through vertexC intersects sideADatE. FindET.
A B
C D
E
T
27) Let S be the set of all three-digit positive integers, each of whose digits is 1 , 3 , 5 , 7 or 9. For example, 197 and 331 are two elements in S. How many integers in S are divisible by 3?
28) Consider the binary sequence 01001000100001000001··· , where each block of 0’s contains one more 0 that the preceding block of 0’s. Note that the first 1 appears in position 2. In what position does the 200th 1 appear?
29) Define the sequence an by a1 2012 and an
1
1–an–1 for n 2. Find a2012.
30) Let a1, a2, a3 and a4 be positive real numbers. If a0 20, a5 12 and ak ak–1ak 1 for k 1, 2, 3 and 4, find a4.
31) Let S { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 }. Let T be a subset consisting of three distinct elements chosen randomly from S. What is the probability that the elements of T can be arranged to form an arithmetic progression with positive common difference? Express your answer as a rational number in lowest terms.
32) Determine the integer m so that the equation x4– 3m 2 x2 m2 0 has four real roots in arithmetic progression.
33 Right triangleABChasAB 24,AC 32 andBC 40. PointsDandEare chosen on sidesBCandACrespectively so that triangleADEis similar to triangleABCandAE AD DE. Find the area of triangleADE.
A B
C
D E
34) In how many ways can the squares in a 2 by 5 array be colored red (R), green (G) or blue (B) so that no two squares with a common edge are the same color? One such example is
R
B
G
B
R
35 LetS1be a square with side length 82 . A sequence of smaller squares is
constructed by joining the midpoints of the sides of each previous square to form a new square. That is, the corners of squareS2are the midpoints of the
sides of squareS1; the corners of squareS3are the midpoints of squareS2;
and so on. The first 5 such squares are shown in the figure. LetPibe the
perimeter of squareSi. Find Each side of this square is trisected and the points of trisection are joined as indicated in the sketch to form two overlapping squares. Find the area of the region common to these two squares.
41 Two circles of radius 16 have their centers on the circumference of each other. ABis the diameter of the right–hand circle that passes through the
centers of the two circles. A smaller circle is constructed tangent toABand the two given circles as in the sketch.
