Mathematial Problem SolvingWorkbook

Paul Yiu

Department of Mathematics

Florida Atlantic University

Last Update: April 20, 2010

Student:

Spring 2010

Mathematical Problem Solving (Yiu) Spring 2010Workbook Checklist

Name:

A1 E2 I3 L5A2 E3 I4 L6A3 F1 I5 M1A4 F2 J1 M2A5 F3 J2 M3A6 F4 J3 M4A7 F5 J4 N1A8 F6 J5 N2B1 F7 J6 M3B2 F8 J7 N4B3 F9 J8 N5B4 F10 J9 N6B5 G1 J10 N7B6 G2 K1 N8C1 G3 K2 N9C2 G4 K3 N10C3 G5 K4 O1C4 G6 K5 O2C5 G7 K6 O3C6 G8 K7 O4D1 H1 K8 O5D2 H2 K9 O6D3 H3 K10 O7D4 H4 L1 O8D5 H5 L2 O9D6 I1 L3 O10E1 I2 L4 Total

• Please hand in only those problems with solutions.

• Check the problems with solutions and enter the total number in thelower right hand corner.

• Keep a copy of your workbook.

1

Name:

Problem A1. How many zeros are there at the end of 1000!?

2

Name:

Problem A2. Find all integers n for which �√n� divides n.

3

Name:

Problem A3. Show that 12

+ 13

+ · · · + 1n

is never an integer forn ≥ 2.

4

Name:

Problem A4. Find a closed form expression for the sum of the firstn terms of the series

1+2+4+4+8+8+8+8+16+16+16+16+16+16+16+16+ · · ·where 1 occurs once, and for k ≥ 1, 2k occurs 2k−1 times.

5

Name:

Problem A5. Find the sum

1(1!) + 2(2!) + 3(3!) + · · · + n(n!).

6

Name:

Problem A6. Show that the numbers

729, 71289, 7112889, 711128889, . . . ,

are perfect squares.

7

Name:

Problem A7. A pentagonal number is one of the form 12n(3n− 1).

(a) Find a pentagonal number (apart from 1) that is also a square.(b) Find a number > 1 which is both a triangular number and a pen-

tagonal number.

8

Name:

Problem A8. (a) Starting with the formula for the sum of the firstn squares: 12 + 22 + · · · + n2 = 1

6n(n + 1)(2n + 1), find an expression

for the sum of 2k + 1 consecutive squares centering at n, namely,

(n − k)2 + (n − k + 1)2 + · · · + n2 + · · · + (n + k)2.

(b) There is only one sequence of 49 consecutive numbers the sum ofwhose squares is a square. Identify this sequence.

9

Name:

Problem B1. Find the sum

17 + 187 + 1887 + · · · + 18n7

in closed form, where the last term has exactly n digits equal to 8.

10

Name:

Problem B2. Find all pairs of positive integers a and b satisfying

abba + ab + ba = 89.

11

Name:

Problem B3. Find all integer solutions of

m2 − 3m + 1 = n2 + n − 1.

12

Name:

Problem B4. Find a three-digit number N = abc satisfying thefollowing conditions:(i) the digits a, b, c are distinct;(ii) N is equal to the sum of all two-digit numbers formed from the threedigits a, b, c.

13

Name:

Problem B5. A sequence of numbers (un) satisfies

un+1 =un

1 + aun

for every integer n ≥ 1 and a constant a. Find a if u1 = 1 and u2010 =150

.

14

Name:

Problem B6. Let A = {1, 2, . . . , n}. For each nonempty subsetB ⊂ A, we form the product of its elements. Find the sum of all thepossible products.

15

Name:

Problem C1. Prove or disprove that the equation ax2 + bx + c = 0has no rational root if a, b, c are all odd integers.

16

Name:

Problem C2. Factor x5 + x + 1.

17

Name:

Problem C3. Calculate√7 −

√48 +

√5 −

√24 +

√3 −

√8.

18

Name:

Problem C4. The interior of a rectangular container is 1 meter wideand 2 meters long, and is filled with water to a depth of 1

2meter. A cube

of gold is placed flat in the tub, and water rises to exactly the top of thecube without overflowing.

Find the length of the side of the cube.

19

Name:

Problem C5. Solve the equations

x + y + z = 3,x2 + y2 + z2 = 7

2,

x3 + y3 + z3 = 92.

20

Name:

Problem C6. Completely solve the following system of equations:

x + y + z + w = 10,x2 + y2 + z2 + w2 = 30,x3 + y3 + z3 + w3 = 300,

xyzw = 24.

21

Name:

Problem D1. How many parallelograms are there in the followingconfiguration with with n units along each side of the outermost triangle.

22

Name:

Problem D2. A man has a square field, 60 feet by 60 feet, withother property adjoining the highway. He put up a straight fence in theline of 3 trees, at A, P , Q. If the distance between P and Q is 91 feet,and that from P to C is an exact number of feet, what is this distance?

60

60

91?

A B

CD

P

Q

23

Name:

Problem D3. (a) Give an example of a primitive Pythagorean tri-angle in which the hypotenuse is a square.

(b) Give an example of a primitive Pythagorean triangle in which theeven leg is a square.

(c) Give an example of a primitive Pythagorean triangle in which theodd leg is a square.

24

Name:

Problem D4. Find the smallest Pythagorean triangle whose perime-ter is a square (number).

25

Name:

Problem D5. Find all Pythagorean triangles with area numericallyequal to perimeter?

26

Name:

Problem D6. Find the least number of matches of equal lengths tomake up the following configuration.

27

Name:

Problem E1. Find m so that the equation in x

x4 − (3m + 2)x2 + m2 = 0

has four roots in arithmetic progression.

28

Name:

Problem E2. The perimeter of a right triangle is 60 and the altitudeon the hypotenuse is 12. Find the sides of the triangle.

29

Name:

Problem E3. There are n unknown numbers. For each k = 1, 2, . . . , n,the sum of all the numbers except the k-th one is ak. What are these nnumbers.

30

Name:

Problem F1 . If n is not a prime, then 2n − 1 is not a prime.

31

Name:

Problem F2 . Among five integers, there are always three with sumdivisible by 3.

32

Name:

Problem F3 . Find a 4-digit number of the form aabb which is asquare.

33

Name:

Problem F4 . The sum of squares of five consecutive positive inte-gers is not a square.

34

Name:

Problem F5 . Find all integers x and y satisfying x2 + y2 = x2y2.

35

Name:

Problem F6 . Find all integers solutions of x2 + y2 + z2 = x2y2.

36

Name:

Problem F7 . If 9|a2 + ab + b2, then 3|a and 3|b.

37

Name:

Problem F8 . Are there integers m and n so that m2 + (m + 1)2 =n4 + (n + 1)4?

38

Name:

Problem F9 . Prove that for n a positive integer n4 + 2n3 + 2n2 +2n + 1 is never a perfect square.

39

Name:

Problem F10 . Show that the product of four consecutive positiveintegers is not a square.

40

Name:

Problem G1. (a) How many persons do you need to be sure that 2persons have the same birthday?(b) 3 persons?(c) n persons ?

41

Name:

Problem G2. Of 12 distinct two-digit numbers, there are two witha 2-digit difference of the form aa.

42

Name:

Problem G3. Among n + 1 integers from 1, 2, . . . , 2n, there aretwo which are relatively prime.

43

Name:

Problem G4. Suppose we distribute 5 points in the interior of asquare of side length 2. Prove that some pair of these points must havedistance less than

√2.

44

Name:

Problem G5. Two of six points placed in a 3 × 4 rectangle havedistance ≤ √

5.Hint: What is the largest possible distance between two points in the

shaded region in the diagram below?

45

Name:

Problem G6. Given a sequence of 101 positive integers, it is pos-sible to strike out 90 of them so that the remaining 11 terms form anincreasing or decreasing sequence.

46

Name:

Problem G7. There are nine sticks of different integer lengths,each shorter than 55 units. Prove that it is possible to form a trianglewith three of them.

47

Name:

Problem G8. Given 20 distinct positive integers, all less than 70,show that among their pairwise differences, there are four equal num-bers.

48

Name:

Problem H1. The n−th triangular number is given by Tn = 12n(n+

1). Find all integers n for which the sum of the (n − 1)−st, the n−th,and the (n + 1)−st triangular numbers is a square.

49

Name:

Problem H2. A large calculus class has n students, with n between100 and 200. Each student is identified with one of the numbers 1, 2,. . . , n. The teacher randomly chooses three students to report on theirperformances. When he entered the first student number on the com-puter, his assistant on seeing this on the screen, immediately says thatthe probability that the other two numbers are both smaller than this firstnumber is exactly 50-50.

What is this first number, and how many students are there in theclass?

50

Name:

Problem H3.A developer wants to build a community in which the n (approxi-

mately 100) homes are arranged along a circle, numbered consecutivelyfrom 1, 2, . . .n, and are separated by the club house, which is not num-bered. He wants the house numbers on one side of club house addingup to the same sum as the house numbers on the other sides. Betweenwhich two houses should he build the club house?

How many houses are there altogether?

51

Name:

Problem H4.A developer wants to build a community in which the n (approxi-

mately 50) homes are arranged along a circle, numbered consecutivelyfrom 1, 2, . . .n, and are separated by the club house, which is also num-bered. He wants the house numbers on one side of club house adding upto the same sum as the house numbers on the other sides.

How many homes should he build, and what is the number of the clubhouse?

52

Name:

Problem H5. Find all integers n such that the triangle with sides2n − 1, 2n and 2n + 1 has integer area.

53

Name:

Problem I1. How many 44-cent and and 90-cent stamps can bepurchased with exactly 50 dollars?

54

Name:

Problem I2. There are three piles of chips, 12, 13, 14 pieces. Youand your opponent take turn removing any (positive) number of chipsfrom any ONE pile. Whoever makes the last move wins. Now it is yourturn. How would you move to ensure winning?

55

Name:

Problem I3. Let n be an odd number. Prove that an odd number ofnumbers from (

n

1

),

(n

2

), . . . ,

(n

n−12

)

are odd.

56

Name:

Problem I4. Decide if the binomial coefficient(10068

)is even or odd.

57

Name:

Problem I5. The positive integers are classified into two types, Aand B. A number is A if and only if it is the sum of two A numbers, orthe sum of two B numbers.

(a) Classify the numbers 1, 2, 3 as A and B.

(b) Prove that if n, n + 1, and n + 2 are A numbers, then every integer≥ n is an A number.

(c) Find the smallest n so that every integer ≥ n is an A number.

58

Name:

Problem J1 . Determine the number of three-digit positive integerswhose digits have a product of 36.

59

Name:

Problem J2 . The positive integers that can be expressed as thesum of 21 consecutive (not necessarily positive) integers are listed inincreasing order. Determine the 21-st integer in the list.

60

Name:

Problem J3 . Determine all pairs of positive integers satisfying4x2 − y2 = 480.

61

Name:

Problem J4 . The sum of a 4-digit number and its four digits is2010. What is the number?

62

Name:

Problem J5 . x, y, z are positive numbers such that

x + y + xy = 8,

y + z + yz = 15,

z + x + zx = 35.

Find these numbers.

63

Name:

Problem J6 . In a two-digit number, the tens digit is greater thatthe units digit. The product of these two digits is divisible by their sum.What is this two-digit number.

64

Name:

Problem J7 . Let a be a positive number such that a + 1a

= 5. Findthe value of a3 + 1

a3 .

65

Name:

Problem J8 . Determine all of the positive integer solutions of theequation

1

x− 1

y=

1

12.

66

Name:

Problem J9 . Two students miscopied the quadratic equation x2 +bx+c = 0 that their teacher wrote on the board. Aaron copied b correctlybut c incorrectly; his equation has roots 4 and 5. Betty copied c correctlybut b incorrectly; her equation has roots 2 and 4. What are the roots ofthe original equation?

67

Name:

Problem J10. The teacher wrote a quadratic equation x2−ax+b =0 on the blackboard, where a and b are positive integers, and b has twodigits. A student erroneously copied the equation by transposing the twodigits of b and also transposing the plus and minus signs. The studentsolved the equation. One of her roots, r, satisfies the teacher’s equation.Find all possible values of a, b, and r.

68

Name:

Problem K1 . Write down the numbers 2, 3, . . . , 100, together withtheir products taken two at a time, the products taken three at a time, andso on up to and including the product of all 99 of them. Find the sum ofthe reciprocals of the numbers written down.

69

Name:

Problem K2 . The ratio of the speeds of two trains is equal to theratio of the time they take to pass each other going in the same directionto the time they take to pass each other in the opposite directions. Findthe ratio of the speeds of the two trains.

70

Name:

Problem K3 . A circle of radius 6 has an isosceles triangle PQRinscribed in it, where PQ = PR. A second circle touches the first circleand the midpoint of the base QR of the triangle as shown. The side PQhas length 4

√5. Find the radius of the smaller circle.

71

Name:

Problem K4 . Find all right-angled triangle with integer sides ifone of the sides is 2001 units long.

72

Name:

Problem K5 . What is the smallest number with the property thatwhen the leftmost digit is moved to the rightmost position, the new num-ber is three times the original?

73

Name:

Problem K6 . Find√

ab + 1 where a = 1n and b = 10n−15.

74

Name:

Problem K7 . (a) Find all quadratic polynomials x2 + ax + b withinteger roots, and with 1, a, b in arithmetic progression.

75

Name:

Problem K8 . Find the sum of the first 2010 terms of the sequence

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, . . . .

76

Name:

Problem K9 . Find positive integers N in base 10 such that N ! inbase 6 has exactly 99 trailing zeros.

77

Name:

Problem K10 . The increasing sequence

1, 5, 6, 25, 26, 30, 31, 125, 126, . . .

consists of positive integers which can be formed by adding distinctpowers of 5. What is the 75-th integer in the sequence?

78

Name:

Problem L1 (Multiple choice questions).

(a) The sum a + b, the product ab and the difference a2 − b2 for twopositive numbers a and b are all equal. What is the larger of the twonumbers a and b?

A 2 B√

5 C√

5+12

D√

5+32

E NOTA

(b) ABCD is a rectangle in which the shorter side AD has length 1.The perpendiculars from B and D to the diagonal AC divide thediagonal into three equal parts. Find the length of AB.

A√

2 B√

3 C√

5 D 3 E NOTA

(c) The tangents to a circle from a point P outside have lengths 10units. The point on the circle nearest to P is 7 units from it. Findthe radius of the circle.

A√

151−72

B√

14 C 5114

D 72

E NOTA

(d) What is the smallest possible value of x2+2xy+3y2+2x+6y+5?

A 1 B 2 C 3 D 4 E NOTA

(e) The base diameter and the height of a right circular cone are bothequal to the diameter of a sphere. The volumes of the cone and thesphere are in the ratio of

A 1 : 3 B 2 : 3 C 1 : 2 D 2 : 9 E NOTA

79

Name:

Problem L2 (Multiple choice questions).

(a) Find the greatest value of 12+sin θ+cos θ

.

A −2+√

22

B 2+√

22

C 2−√2

2D −2−√

22

E NOTA

(b)√

7 + 4√

3 −√

7 − 4√

3 =

A 4 B 2√

3 C√

6 D 2 E NOTA

(c) A rectangle contains three circles as in the diagram, all tangent tothe rectangle and to each other.

If the height of the rectangle is 4, then the width of the rectangle is

A 3 + 2√

2 B 4 + 4√

23

C 5 + 2√

23

D 6 E NOTA

(d) Which of the following conditions does not guarantee that the con-vex quadrilateral ABCD is a parallelogram?

A AB = CD and AD = BC B ∠A = ∠C and ∠B = ∠DC AB = CD and ∠A = ∠C D AB = CD and AB//CDE NOTA

(e) A line AE is divided into four equal parts by the points B, C,D. Semicircles are drawn with segments AC CE, AD and DE asdiameters.

A

B C

D

E

The ratio of the area enclosed above the line AE to the area en-closed below the line is

A 4 : 5 B 5 : 4 C 1 : 1 D 8 : 9 E NOTA

80

Name:

Problem L3 (Multiple choice questions).

(a) When x3 + px + 3 is divided by x − 1, the remainder is the sameas when it divided by x + 1. Find the value of p.

A 0 B 1 C −1 D 2 E −2

(b) Suppose x + y = 1 and x2 + y2 = 2. Find x3 + y3.

A 1 B 112

C 2 D 212

E 3

(c) The integers

1, 3, 4, 9, 10, 12, 13, 27, 28, 30, . . .

are the sums of distinct powers of 3 arranged in increasing order.What is the 100th term of the sequence?

A 100 B 345 C 543 D 981 E 1024

(d) A number is formed by concatenating the first 1000 positive inte-gers in increasing order:

123456789101112131415161718192021222232425 . . .

What is the 1000−th digit from the left?

A 0 B 1 C 2 D 3 E 7

(e) An equilateral triangle and a regular hexagon have equal perime-ters. What is the ratio of their areas?

A 1 : 1 B 2 : 1 C 2 : 3 D 3 : 2 E 1 : 2

81

Name:

Problem L4 (Multiple choice questions).

(a) If you write all integers from 1 to 100, how many even digits willbe written?

A 50 B 71 C 80 D 89 E 91

(b) In a farm there are hens (no hump, two legs), camels (two humps,four legs), and dromedaries (one hump, four legs). If the numberof legs is four times the number of humps, then the number of hensdivided by the number of camels will be

A 12

B 1 C 32

D 2 E Not enough information

(c) A cubic box of side 1 foot is placed on the floor. A second cubicbox of side 2

3foot is placed on top of the first box so that the center

of the second box is directly above the center of the first box. Apainter paints all of the surface area of the two boxes that can bereached without moving the boxes. What is the total area in squarefeet that is painted?

A 499

B 579

C 619

D 709

E NOTA

(d) What is the units digit of 22010?

A 0 B 2 C 4 D 6 E 8

(e) The numbers 1, 2, 3, 4, 5, 6 are to be arranged in a row. In howmany ways can this be done if 2 is always to the left of 4, and 4 isalways to the left of 6? (For example, 2, 5, 3, 4, 6, 1 is an arrange-ment with 2 to the left of 4 and 4 to the left of 6).

A 20 B 36 C 60 D 120 E 240

82

Name:

Problem L5 (Multiple choice questions).

(a) Three digit numbers are formed using only odd digits. The sum ofall such three digit numbers is:

(A) 19375 (B) 34975 (C) 6253 (D) 34975 (E) 69375

(b) Equilateral triangle ABC is inscribed in a circle. Point P is chosenon arc BC and lines AP , BP , and CP are drawn with PB = 5and PC = 20. If AP intersects BC at point D, what is the lengthof AD?

A

B C

P

D

(A) 18 (B) 19 (C) 20 (D) 21 (E) NOTA

(c) Peter takes 4 hours to paint a wall. Paul takes 6 hours to do thesame job. What is their average time to paint the wall?

(A) 5 hours (B) 512

hours (C) 412

hours(D) 4 hours 48 minutes (E) 5 hours 12 minutes.

(d) John and Mary both celebrate their birthday today. In three years,John will be four times as old as Mary was when John was twoyears older than Mary is today. If Mary is a teenager, how old isJohn now?

(A) 17 (B) 21 (C) 25 (D) 29 (E) 33

(e) For x2 + 2x + 5 to be a factor of x4 + px2 + q, the values of p andq must be, respectively

(A)−2, 5 (B) 5, 25 (C) 10, 20 (D) 6, 25 (E) 14, 25

83

Name:

Problem L6 (Multiple choice questions).

(a) One of the factors of x4 + 4 is:

(A) x2 + 2 (B) x + 1 (C) x2 − 2x + 2 (D) x2 − 4(E) NOTA

(b) The points of intersection of xy = 12 and x2 + y2 = 25 are joinedin succession. The resulting figure is

(A) a straight line (B) an equilateral triangle(C) a parallelogram (D) a rectangle(E) a square

(c) If the area of a circle is doubled when its radius is increased by n,the radius is equal to

(A) n(√

2 + 1) (B) n(√

2 − 1) (C) n (D) n(2 −√2)

(E) NOTA

(d) If 9x+2 = 240 + 9x, the x is equal to:

(A) 0.1 (B) 0.2 (C) 0.3 (D) 0.4 (E) 0.5

(e) The sum of two numbers is 10; their product is 20. The sum of theirreciprocals is:

(A) 110

(B) 12

(C) 1 (D) 2 (E) 4

84

Name:

Problem M1.A sequence is generated by listing (from smallest to largest) for each

positive integer n the multiples of n up to and including n2. Thus, thesequence begins

1, 2, 4, 3, 6, 9, 4, 8, 12, 16, 5, 10, 15, 20, 25, 6, 12, . . .

Determine the 2010−th term of the sequence.

85

Name:

Problem M2. A Pythagorean triangle has sides a, b, hypotenuse c,and altitude h on the hypotenuse. Determine all such triangles with

1

a+

1

b+

1

h= 1.

86

Name:

Problem M3.Determine all pairs of positive integers a, b such that a+1

band b+2

aare

positive integers.

87

Name:

Problem M4.Find the fraction p

qwith p, q positive integers and q < 1000 which is

closet, but not equal to 1972

.

88

Name:

Problem N1. Altitude AD of an equilateral triangle ABC is adiameter of circle O. If the circle intersects AB and AC at E and Frespectively, find the ratio EF : BC.

89

Name:

Problem N2. Triangle ABC is inscribed in a circle with diameterAD. A tangent to the circle at D cuts AB extended at E and AC ex-tended at F . If AB = 4, AC = 6, and BE = 8, find(a) CF ,(b) ∠DAF ,(c) BC.

90

Name:

Problem N3. ABCD is a quadrilateral inscribed in a circle. Di-agonal BD bisects AC. If AB = 10, AD = 12, and DC = 11, findBC.

91

Name:

Problem N4. In circle O, perpendicular chords AB and CD inter-sect at E so that AE = 2, EB = 12, and CE = 4. Find (a) the radiusof the circle, (b) the shortest distance from E to the circle.

92

Name:

Problem N5. A circle with radius 3 is inscribed in a square. Findthe radius of the circle that is inscribed between two sides of the squareand the original circle.

93

Name:

Problem N6. An equilateral triangle ABC is inscribed in a circle,and a point P is chosen on the minor arc AC. Show that PB = PA +PC.

94

Name:

Problem N7. E is the midpoint of the side BC of a unit squareABCD, and F is the point on DE such that AF ⊥ DE. Calculate thelength of BF .

A B

CD

E

F

95

Name:

Problem N8. In triangle ABC, in which AB = 12, BC = 18, andAC = 25, a semicircle is drawn so that its diameter lies on AC, and sothat it is tangent to AB and BC. If O is the center of the circle, find OA.

96

Name:

Problem N9. The triangle is isosceles and the three small circleshave equal radii. Suppose the large circle has radius R. Find the radiusof the small circles.

97

Name:

Problem N10. ABC is an isosceles triangle with ∠B = ∠C =80◦. D and E are points on AC and AB respectively such that ∠DBC =60◦ and ∠ECB = 50◦. Calculate ∠BDE.

60◦ 50◦

A

B C

D

E

Hint: Let F be the point on AC such that BF = BC. Show thatDF = EF = BF .

98

Name:

Problem O1. If squares on constructed externally on the sides ofa convex quadrilateral, the centers of the squares form a quadrilateralwhose diagonals are equal and perpendicular to each other.

99

Name:

Problem O2. What is the area of a triangle in terms of its medians?

100

Name:

Problem O3. Prove that if two circles touch externally, their com-mon tangent is a mean proportion between their diameters. [Note: c isthe mean proportion of a and b if c2 = ab].

101

Name:

Problem O4. From the centers of each of two nonintersecting cir-cles tangents are drawn to the other circle. Prove that the chords PQ andRS are equal in length.

A B

P

Q

R

S

102

Name:

Problem O5. In parallelogram ABCD, angle A is acute and AB =5. Point E is on AD with AE = 4 and BE = 3. A line through B,perpendicular to CD, intersects CD at F . If BF = 5, find EF . Ageometric solution (no trigonometry) is desired.

103

Name:

Problem O6. A circle is inscribed in a square ABCD. A secondcircle on diameter BE touches the first circle. Show that AB = 4BE.

A B

CD

O

E

104

Name:

Problem O7. Find the area of the region which is common to fourquadrants that have the vertices of a square as centers and a side of thesquare as a common radius.

105

Name:

Problem O8. ABCD is a square and ECD an isosceles trianglewith base angles 15◦, as shown in the figure. Prove that ∠AEB = 60◦

(and therefore triangle AEB is equilateral), without using trigonometry.

D C

BA

E

106

Name:

Problem O9. In the diagram below, AB = BC and ∠ABC = 60◦.Prove that CD = OA

√3.

60◦

A

O

B

C

D

107

Name:

Problem O10. In the diagram below, OA = BC and ∠ABC =30◦. Prove that CD = AB

√3.

30◦A

O

B

C

D