Advanced High-School Mathematics

SECTION 1.6 Mass Point Geometry 51 Example 3. Prove that the angle bisectors of 4 ABCare concurrent. Proof. Assume that AB =c, A ...

52 CHAPTER 1 Advanced Euclidean Geometry Note that in the above diagram,QP :PR= (b 2 +c) : (a+b 1 ) because P is the center of m ...

SECTION 1.6 Mass Point Geometry 53 From the above, it’s easy to compute the desired ratios: EF :FD=^95 : 7 = 9 : 35 and BF :FG=^ ...

54 CHAPTER 1 Advanced Euclidean Geometry 5.^11 In triangleABC, pointDis on [BC] withCD= 2 andDB= 5, pointE is on [AC] withCE = 1 ...

Chapter 2 Discrete Mathematics 2.1 Elementary Number Theory While probably an oversimplication, “number theory” can be said to b ...

56 CHAPTER 2 Discrete Mathematics 2.1.1 The division algorithm Very early on, students learn the arithmetic of integers, namely, ...

SECTION 2.1 Elementary Number Theory 57 Now letr be the smallest element of this set; note thatr≥0, and let qbe defined so thatr ...

58 CHAPTER 2 Discrete Mathematics common divisor is typically obtained by factoring the given numbers into prime factors. Howeve ...

SECTION 2.1 Elementary Number Theory 59 given thatd′ divides both a andb, then obviouslyd′ divides the sum sa+tb=d, i.e.,d′|dals ...

60 CHAPTER 2 Discrete Mathematics (ii) Ifl′is a positive multiple of bothaandbthenl≤l′. We denote the least common multiple ofaa ...

SECTION 2.1 Elementary Number Theory 61 Thend′|a′andd′|band so clearly d′|d. But then d′ divides botha′ andd, forcingd′|gcd(d,a′ ...

62 CHAPTER 2 Discrete Mathematics Letpbe a prime and show that for all integersh, 1 ≤h≤p−1, p ∣∣ ∣∣ ∣ Ñ p h é . Conclude that ...

SECTION 2.1 Elementary Number Theory 63 b a =q 1 + 1 q 2 + 1 q 3 + 1 ... ... qm− 1 + 1 qm . For any positive integern, letUnbe ...

64 CHAPTER 2 Discrete Mathematics (d) Show that for any positive integer n, φ(n) ≥ » n/2. (Hint: prove that for every prime powe ...

SECTION 2.1 Elementary Number Theory 65 2.1.2 The linear Diophantine equationax+by=c Suppose thata, b, care integers and suppose ...

66 CHAPTER 2 Discrete Mathematics x = x 0 − b d t, y=y 0 + a d t, t∈Z (2.1) Finally, we see (by substituting into the equation) ...

SECTION 2.1 Elementary Number Theory 67 Find all solutions of 15x+ 16y= 900, withx, y≥0. Suppose that someone bought a certain ...

68 CHAPTER 2 Discrete Mathematics covers 74 kilometers after biking for 2 hours, jogging for 3 hours, and swimming for 4 hours, ...

SECTION 2.1 Elementary Number Theory 69 An old woman goes to market and a horse steps on her basket and crushes the eggs. The ri ...

70 CHAPTER 2 Discrete Mathematics m 3 = 1 3. Therefore, we are really faced with the task of finding the smallest integermsatisf ...