Sets
Functions
Trigonometry
Vector
Sequence and series
Inequalities
Parametric Geometry
3D Geometry
Combinatorics
Complex Numbers
Planar Geometry
Polynomials
Induction
Number Theory
Derivatives and limits
Others
1.1 fundamentals
1.1.1 factorials
N!, Pnm
E.g. 3A 2B 2C, not more if already more than others, 3*2*1*3*2*1*3!*2!*2! = 864
1.1.3 word rearrangements
Thm1.1.6 # order a word n!/d!d!d!
E.g. col a b c: 3,2,3, rule:chooe 1 col, break lowest, in how many different orders?
Accbaacb? Word rearrangement, 3a, 2b, 3c rearrange, 8!/(3!2!3!)=560
E.g. amc 10, amosu, 5 leter words, if in alphabetical order, usamo occupies position
1 - A: 4!, M: 4!, S: 4!, O:4! 4*4! = 96
2 - ua: 3!, um: 3!, uo: 3!, 3*3! = 18
3 - usa: mo
96 + 18 + 1 = 115
1.1.4 combination
N choose k = n! / (n-k)!*k! = n choose n-k
1.1.5 probability
E.g. aime, 4 1s, 4 2s,...,4 10s, matching pair(2 cards with the same number) removed from the dock, given 2 rand selected cards also form a pair, m/n prob 2 rand selected cards form a pair, m and n rel prime pos int, find m+n
1, 1: 2 1s left, others 4 left, 1+ 4c2 * 9 = 55, all 38 chooe 2 = 703, 55/703
E.g. amc 10, 20 ball rand 5 bins, p prob some 3 ,another 5 and other 3 bins with 4 each, q the prob every bin 4, p/q=?
Q: 20c4, 16c4, 12c4, 8c4, 1 = , total: 5^20, q =
P: 20c3, 17c5, 12c4,8c4,1 = , *5c2*2 or 5*4 p =
p/q = 5*4*20c3*17c5 / 20c4*16c4 = 16
8c3/8c4 * 20 = 16
1.1.6 susbsets
2^n
E.g. amc 10, 2 subsets of s=abcde, intersection subsets = 2 el, how many ways order no matter
5c2 = 10, 2^3 = 8 , 8 *10/2 = 40, order no matter
E.g. amc12, b&b 5 room, each distinct, 5 pp come, no more 2 per room, how many ways?
1 1 1 1 1, =5!=120
2 1 1 1 0, =5*4*5c2*3! = 1200
2 2 1 0 0, = 5c2*3*5c2*3c2 = 900
E.g. farmer rec field 2by2 grid of 4 subfield, potatoes corn wheat soybean, no corn wheat adjacent , no soybean potatoes adjacent, how many ways?
Case1: ccc(spc) 3, case2:css(scw), cpp(pcw), case3:cc(sc|cs|sp|cp(cp)) = 2*2+2*2=8, case4:cps(cw), csp(cw) 2*2 = 4, 3+8+^+4 * 4
E.g. amc12, azar x in one of 3*3 box, carl O in remain, horizontal vert or diag win, how many winning board
3o in a line, 6c3 - 2) * 6 = 108, 30 diag, 2*(6c3)=40, 148
Complementary counting
E.g. aime Pos int asc if in deci rep at leat 2 digits each less than any to its right, how many ?
2^9 total
Subtract: 1digits:9, zero set 1, 2^9-10
E.g. amc10, 3*3grid,edge wrap around, up down left right, start center, at most 4 hops, prob end on 4 corner?
4 hops without reqaching corner, 4 choices,
4 * 1st
Upwrap 1, back 1 2nd
4 1back 1wrap 3rd
2 4 2 4th
4*(8+6)=56 dont work
4^4-56=200
200/256=25/32
AuB=A+B-AnB
E.g.aime seq 3 letter + 3 digits, eq likely prob plate at least one palindrome m/n, find m+n
26*26*1/26*26*26=1/26 letters
1/10 digits
Sub both true, 1/260
1/10 + 1/26 - 1/260 = 35/260 = 7/52, m+n=59
E.g. s prod free if no abc in s not nec distinct ab=c, for eg. 16,20 is, 4,16 and 2,8 16 not, find number of subsets of 1,2,3,...,9?
1 is never, so rem 8
2*2=4, 2,2,,4 => 2,4
2,3,6
2,4,8=>2,4
3,9
24 236 39
2^6 2^5 2^6 =160
24 236 24 39 236 39
2^4 2^4 2^4 = 48
24 236 39
2^3 = 8
160-48+8 = 120
2^8 - 120 = 136
N identical in k distinct bins = n+k - 1 c k - 1
E.g. amc 10, N, when a+b+c+d+1 ^ N expanded, res exp exactly 1001 terms include all 4 abcd, each to some pos power, what is n?
N distributed among A b c d, 1,
N balls, in a b c d bins, 3 bars + 1
(N-4)+4 c 4 = 1001, n=14
E.g. 19 iden acors 3 squirrels, 1st pos odd, 2nd with any nonneg even, 3rd even >=4
2a+1, 2b, 2c+4 = 19
a+b+c=7
3 distribute among 7
9c2
7.0.1 binomial identity: Nc0 + nc1 +...+ncn = 2^2
7.0.2 vandermondes i\dentity: nc0*mcm + nc1*mcm-1 +...+ncm*m*0 = M+n c n
7.0.3 special case of vandermonde identity: sum i from 0 to k (k c i) ^2 = 2k c k
7.0.4 pascals triangle: binomial coef,
7.0.5 pascals identity: nck + ncK+1 = N+1 c K+1
5c2+5c3=6c3
7.0.6 hockey stick identity
Kck + k+1ck +...+ nck = n+1 c k+1
3c3 + 4c3 +...+7c3 = 8c4
7.0.7 generalization
Jck + J=1ck +...+ nck = n+1 c k+1 - j c K+1
7.0.8 choosing odd even identity
Sum k = 0 to m -1)^k nck = -1)^m n-1 c m, choose even = choose odd
E.g. 11 men 12 women choose committee st women one more than men, 1-23 members, N be number of committees, N=cb, find a+b
1w0m
2w1m
3w2m
…
12w11m
11c012c1 + 11c112c2 + … + 12c1211c11 = 11c1112c1+11c1012c2 + …= 23c12 35
E.g. fin remainder when 3c2 c 2 + 4c2 c 2 + … + 40c2 c 2 div 1000
Nc2 c 2 = nc2 nc2-1 / 2 = n(n-1)/2 n(n-1)/2 - 1 / 2 = n(n-1) (n(n-1) - 2 ) / 8 = n n-1 n-2 n+1 / 8 = n+1 ! / n -3 ! * ⅛ = 3 n+1 c 4
N = 3—-40
3 * 4c4 + 5c4 + 6c4 +...+41c4 = 42 c 5 * 3 = 42*41*39*38 = 1599*1596 div 1000 = 600 - 1 * 600-4 = 360000 - 3000 + 4 div 1000 = 4
8.0.1 num rec of all size in grid m x n is m+1 c 2 * n+1 c 2
E.g. amc10 5x5 grid collection of squares from 1x1 to 5x5, how many contain back center?
1+2^2+3^3 + 2^2 + 1 = 19
E.g. how many rec instead of squares
3c1 3c1 + 3c1 3c1 = 18
E.g. polygon 12 rec reg pentagonal fces, with 2 horizontal faces, 5 dj to the top, 5 adj to the bottom, find how many ways from top to bottom by adj faces, no vist each face more than once, move not permitted from bottom to top ring, only going down, can stay
Top -> 5 -> 1 + 2*4 =9 -> 2->9
5*9*2*9=810
9.0.1 try a few examples , mrk extreme cases, use geometry find area of regions
E.g. mc10 ,a: 0,2017 b: 0,4034, pro b>a
4034-2017 * 2017 + ½ * 2017 * 4034-2017 / 4034 * 2017 = .75
E.g. x y ind unif from 0,1, close to prob x,y,1 length of obtuse triangle
Xx + yy < 1, yy< 1-xx, x+y > 1, pi/4 - ½ .29
Sum xP(x)
10.0.4 linearity of expectation
E(x1+x2+xn) = Ex1 + Ex2 + ..+Exn ind or not
E.g. amc10, 5 balls in in circle, chris choses 2 adjacent balls interchange them, silv same, ind of chris, E of balls occupy original positinoss after these two successive transpositions
⅗ * ⅗ = 9/25
⅖*⅖*1/2 = 2/25
11/25*5
E.g. amc12, avg number of pirs of consec int rand sel subset of 5 distinct int from 1,2,3,..,30?, 1 17 18 19 30 has 2 pairs of consec int
12 23 34…29 30
28c3 / 30c5 for 12 = 2/87, 2/87 * 29 = 2/3
Base + rec + iter
E.g. amc 10/12, how many seq of 0 and 1 of length 19 begin with 0, end with 0, contain no two consec 0, no three consec 1
0 1 1|0 1 0
Length 1: 0 or 1 = 1
Length 2: 01, 10, = 0
Length 3: 010 = 1
Fn = Fn-3 + Fn-2
E.g. 8 cubes with edge k 1-8 tower built using all 8: any may be the bottom,at most k+2 immediately on top of k, let t be number of diff towers can be constructed, remainder t div 1000?
F1 = 1
F2 = 2
F3 = 6
Fk = 3*fk-1
E.g. spacy a set of int if contains no more than one out of any three consec int, how many subset of 1,2,3,...,12 including empty set are spicy?
{1}->2
{1,2} -> 2^2 - 1 = 3
f(3) {1,2,3} -> empty + 3 = 4
Fk = fk-1 + fk-3
Assign var to prob win, write equation for prob winning
E.g. amc12, t d h playing game, each flip a fair coin repeated until he gets first head, then stops, wht is prob all three flip coins the same number of times
P = ⅛ + 6/8 *0 + ⅛ p, p = 1/7
E.g. amc10, frieda frog 3x3 grid up down left right wrap, prob reach on corner on one of 4 hops
En = Cen-1+1/4En-1
Cen = 1/4En-1
En + Cen + Con = 1
Co4 = E3 = Ce2 + 1/4E2
Table
Center edge corner
0 1
1 1
2 ¼ ¼ 1/2
3 1/16 5/16 5/8
4 5/64 9/64 25/32
E.g. r s t game each with 1, bell ring every 15sec who have money sim give 1 to other rand ind player, prob bell rung 2019 times, each will have 1
R 1 s 1 t 1
1 1 1
2 1 0
P111n = ⅛*2 p111n-1 + 1/4p(210)n-1 = 1/4
P210n = 1- P111n
Binomial expansion
x+y)^2 = x-y)^2 + 4xy
x-y)^2 = x+y^2 - 4xy
x+y)^2 + x-y)^2 = 2(x^2 + y^2)
x+y)^2 - x-y)^2 = 4xy
x+y+z)^2 = xx+yy+zz + 2(xy+yz+xz)
Binomial cube expansions:
x+y)^3 = xxx + yyy + 3xy(x+y)
x-y)^3 = xxx-yyy-3xy(x-y)
x+y+z)^3 - 3xyz = x+y+z)(xx+yy+zz-xy-xz-yz)
If x+1/x = a, then
xx+1/xx = aa-2
xxx+1/xxx = a3 - 3a
xxxx+1/xxxx = aa-2)^2 - 2
E.g. amc10, a : 4=a+1/a, what is aaaa+1/aaaa?
aa+1/aa = 16-2 = 14
Aaaa + 1/aaaa = 14*14 - 2 = 194
E.g. x-1/x = 3, find xxx-1/xxx
27 = Xxx -1/xxx - 3(x-1/x)
27+9 = 36
13.2 quadratic factorization
13.3 cubic factorization
Xxx-yyy = x-y) (xx+xy+yy)
Xxx+yyy = x+y(xx-xy+yy)
13.4 higher power factorization
x^n - y^n = x-y)(x^n-1 + x^n-2y + x^n-3y^2 +...+xy^n-2 + y^n-1)
X5-y5 = x-y)(x4+x3y+x2y2+xy3+y4)
X5+y5 = x+y)(x4-x3y+x2y2-xy3+y4)
E.g. x>y, x+y=8 xy=4, x4-y4=a sqrt(b), find a+b
X4-y4 = x-y)(x3+x2y+xy2+y3) = x-y)(x+y)^3 - 2xy(x+y)) = 512-2*8*4= x-y)448
x+y)^3 = x3+3x2y+3xy2+y3
x+y)^2 -4xy = x-y)^2, 8^2-4*4 = 48 = x-y)^2
sqrt(48) 448 = 1792sqrt(3)= a sqrt(b), a+b = 1795
13.5 algebraic equations
E.g. solve for a, a2b=4, b2c = 3, c2a = 18
Substitution: b=4/a2, 16/a4 * c = 3, 3a4/16)^2 *a = 18, 9a9/256=18, a9=512,a=2
Elimination: a3b3c3 = 4*3*18 = 216 = 3*3*3*2*2*2, abc=2*3, a/b=2, a2b=4, a^3=8, a=2
E.g. amc12 x,y,z pos int, x+1/y = 4, y+1/z = 1 z+1/x=7/3, what is xyz?
x+1/y)(y+1/z)(z+1/x) = xyz+1/xyz + y + x + z + 1/x + 1/y + 1/z = 4+1+7/3 +xyz+1/xyz = 28/3
a+1/a=2, xyz=1
E.g. pos solution to 1/x2-10x-29 + 1/x2-10x-45 - 2/x2-10x-69 = 0
y= x2-10x-29
1/y+1/y-16 - 2/y-40 = 0
y-16)y-40) + y)y-40) - 2y)y-16) = 0
-64y+640 = 0, y = 10,
E.g. abc be solution of x3-xyz = 2, y3-xyz=6, x3-xyz=20, greatest possible value of a3+b3+c3 written in form m/n where m n relatively prime pos int, find m+n
a=xyz, x3=a+2, y3=a+6, z3=a+20
xyz)^3 = a^3 = a+2)(a+6)(a+20) = a3+28a2+176a+240
7a2+43a+60=0, a= -4 | -15/7, -3a+28 = 45/7+28
14.1.3 for quadratics ax3+bx+c=0, sum of root -b/a, prod of root c/a
E.g. root of 3x2+6x-5 are r,s, find r3+s3
r+s=-b/a=-2, rs = c/a = -5/3
r+s)^3
14.1.4 higher polynomials
E.g. x3-6x-5=0, rst root find r3+s3+t3
r+s+t=-0/1=0, rs+rt+st=-6/1=-6, rst=-(-5=5
r3+s3+t3-3rst=r+s+t)(r2+s2+t2-rs-rt-st)=0, 3rst=15
E.g. Amc12 p = x3+ax2+bx+2, avg 0s = prod 0s = sum coef, what is b?
r+s+t=-a/3, rst=-c = -2, 1+a+b+c = -2, a=6,c=2,b=-11
P = anxn+an-1*xn-1 +...+ax+a0
an(x-r1)(x-r2)...(x-rn)
Remainder theorem
E.g. x51 +51 / x+1 rem?
x+1=x- -1
X51+51 = (-1)^51+51 = 50
E.g. in polynomial x4-18x3+kx2+200x-1984 = 0, prod of 2 of its roots is -32, find k
x2+ax-32->r,s,
rs= -32, tu = -1984/-32 = 62, x2+bx+62->t,u
x-r)x-s)x-u)x-t) = x4-18x3+kx2+200x-1984
a+b=-18, 62a-32b =200, a=-4,b=-14, k=86
E.g. z1,z2,...,z673, the polynomial x-z1)3 (x-z2)3… (x-z673)3, can be x2019+20x2018 + 19xx2017+g(x), where g(x) polynomial complex coef and degree at most 2016, the sum |sum(ZjZk)| = m/n, 1<=j<k<=673, find m+n
Z1+z2+...+z673 = -20/3
9(z1z2+z1z3+...+z672z673) + 3(z12+z22+..+z673^2) = 19
A
(Z1+z2+z3+...+z673 )(z1+z2+z3+...+z673) = z12+z22+...+z673^2 + 2(z1z2+z1z3+z1z673) ->a
20*20/3*3 = 2a + z12+z22+...
9a+3*400/9 - 2a = 19, a=-343/9, m+n=352
15.2 Symmetric polynomials
15.2.1, p = a0xn + an-1 xn-1 +...+ a1x+a0
15.2.2 even degree
E.g. x+1/x = sqrt(5), what is x11-7x7 + x3?
x3(x8-7x4+1) -> symmetric
X8- 7x4 + 1 / x4 = x4 - 7 + 1/x4
x7(x4+1/x4 - 7) = 0
E.g. 10000x4 -5000x3 + 825x2 - 50x + 1 = 0, solve for both roots
y=10x, y4 - 5y3 + 8.25y2 - 5y + 1 = 0
Y2 - 5y + 8.25 - 5/y + 1/y2 = 0
y2+1/y2 -5 (y+1/y) + 8.25
A = y+1/y,
A2 - 2 - 5a + 8.25, a = 2.5
Y = 2 | 1/2, x= ⅕ | 1/20
15.2.3 polynomial manipulation, reciprocal roots
P = anxn + an-1 xn-1 + + a1x + a0
Q = a0xn + a1xn-1 + + an-1x + an
1/r1, 1/r2, …,1/rn
15.2.4 roots more or less
P = an xn + an-1 xn-1 +...+a1x+a0
\ q = an(x-k)n + an-1 (x-k)n-1 + … + a1(x-k) + a0
Roots r1+k, r2+k, …, rn+k
15.2.6 1/r-3)3 + 1/s-3)3 + 1/t-3)3, flip coeef,
3x3+7x2+6x-4=0, 3(x+3)^3 + 7(x+3)^2 + 6(x+3) - 4 = 0, 376-4 ->-4673
E.g. g be polyn with leading coef 1, whose 3 roots reciprocal of f=x3+ax2+bx+c, where 1<a<b<c, what is g(1) in terms of a,b,c?
g(x) = A(x)/c = x3+b/cx2+a/cx+1/c
g(1) = a+b+c+1 / c
Common diff, an=a1+(n-1)d
E.g. each pos int k, Sk denote increasing arithmetic seq of int first term is 1, common diff is k, how many value of k does Sk contain term 2005?
1+d(k-1) = 2005, d(k-1) = 2004 = 2*3*2*167, number of possible d -> 3*2*2=12
E.g. int k added to each of 36, 300, 596, one obtains seq of 3 conec of an arithmetic series, find k.
K+36 = a-d)^2, k+300=a^2, k+596=a+d)^2
Aa+dd = k + 316, d = 4, a=35, k=925
Infinit S = g1 / 1-r
E.g. a1,a2,a3 and b1 b2 b3, common ratio, a1=27, b1=99, a15 = b11, find a9
27r14 = 99r10, r4=11/3, a9=27r8 = 27 * 121/9 = 363
E.g. -1<r<1, Sr sum of geom series 12+12r+12r2+12r3+..., let a between -1 and 1 sat S(a)S(-a)=2016, S(a)+S(-1)=?
12/1-a * 12/1+a = 2016
12/1-a + 12/1+a = 24 / (1+a)(1-a) = 2016 / 6 = 336
E.g. inf geo series sum 2005, a,ar,ar2,ar3… a2+a2r2+a2r4+...
a/1-r = 2005, a2/1-r2 = 10*2005 = a/1-r * 1/1+r, 10=a/1+r, a=2005-2005r = 10+10r, r = 1995/2005=399/403, sum=802
Sum of odd, 1+3+5+..+n^2, even->n(n+1)
Sum of squares formula: 1^2+2^2+...+n^2 = n(n+1)(2n+1)/6
Sum of cube: 1^3 + 2^3 +...+n^3 = n(n+1)/2 )^2
18.1 telescoping
Expand first last few terms, cancel out
E.g. consider seq ak=1/kk+k, for k>=1, given am+ am+1 + …+ an-1 = 1/29, pos int m and n with m<n, find m+n
Ak = 1/k(k+1) = 1/k - 1/k+1
1/m - 1/n = 1/29 m-29)(n+29)=-841=29*29, m=28,n=812, m+n=840
E.g. an=3^(1+2+3+...+n), a1=3,a2=3^3,a3=3^6, eval ½*⅔*¾*..., write + instead of *, an/an+1
½ ⅔ + ¾ ⅘ ⅚ 6/7 26/27 +27/729 +...
⅓+1/9+1/27+...
sum= ⅓ / 1-⅓ = ½
E.g. x1=211 x2=357 x3=420 x4=523, xn=xn-1 -xn-2 + xn-3 - xn-4 when n>5, find x531+x753+x975
x5=523-420+375-211=267
x6=267-523+420-375=-211
x7=-211-267+523-420=-375
xn+xn+1=xn-xn-4, xn+1=-xn-4
-xa=xa+5
a11=a1, x1+x3+x5 = 211+420+267=898
E.g. finite set s of distinct real number - mean of su{1} = 13 less than mean of S< mean of Su{2001} is 27 more thn mean of S, find mean of S
avg=x, nx+1 /n+1 = x-13, nx+2001/n+1=x+27
40(n+1) = 2000, n = 49, x = 651
sU1 -13
sU2001 27
sUk 0
k=651 = 13/40*2000+1
geo means sqrt(prod a), harmo means = n/sum(reciprocal), quadratic mean sqrt(sum of quadratic / n)
E.g. 4 6 8 17 x, median = mean, x = -5/
E.g. mean median mode of 10 2 5 2 4 2 x, form non constant arithmetic progression, sum of all possible real x? 2 2 2 4 5 10 mode=2, mean = 25+x/7, median=2/4/x, x=3|17, 20
E.g. a b place basketball, a won ⅔, b won ⅝, also b won 7 more lost 7 more than a, how many a played?
2a+7/3a+14 = ⅝, a = 14, 3a=42
20.1 Work and rate
Some can do a work time, some else do b time, together, whey can do in ab/a+b time
E.g. a paint room in 15 hr, b 50% fater, c 2x faster, a begins paint 1.5 hr, then b join until half painted, then c joins until all painted, find the number of mintites after a begins for the three of them to finish
1/10, 90 min, b joins, ⅙ an hour, 4/10 / ⅙ = 144 min, ⅓ / (3/10) = 5/3 hour = 100min
E.g. worker prod widgets whoosits, 1 hr, 100 workers prod 300 wi, 200 wh, 2h 60 workers prod 240 wi 300 wh, in 3 hr, 50 worker prod 150 wi m wh, find m
1 hr 1 worker 3 wi 2 wh
2hr 60 wroker 240 300, 1hr 1 worker 2w 2.5 wh
3hrs 50 woker 150wi m wh, 1hr 1 worker 1wi + m/150 wh
3wi+2wh=2wi+2.5wh, 1wi = 0.5 wh,,m = 450
Wlk ¾ from home to gym, 2km from home, then walk ¾ from where to home, when reached ¾ again from where back to gym, keep so clost to point a km from home b km from home, what is a-b
a+¾(2-a) = ¼ a + 3/2 —> B\
1/4a + 3/2 - ¾(1/4A+3/2) = ¼(1/4a+3/2), a = ⅖, b = 8/5
E.g. al walk down to bottom of escalator moving up and counts 150 steps. Bob walk up to the top of escalator 75 steps. If l speed 3x bob speed, how many teps visible on the escalator at a given time
Bob v = x, al speed 3x. 3x-e
X+e = 2(3x-e), 3e=5x, x=3/5e8/5 * 75 = 120
E.g. 100 ft long moving wlkwy 6ft/sec, l step onto start, b-4ft/sec, c-8ft/sec, certaintime, three persons hlfwy between the other two, find distance in feet between strt of walkway and middle person, 8t 10(r+2) 6(r+4)
B in mid, 10t + 20 = 7t _ 12, t = -8, no
A in mid, t=14/3, 52
Number Theory 257
E.g. prod of 3 ages is 128, sum?
128 = 2^7, 8*8*2, 18
E.g. n=2^31 3^19, pos int div of n2 < n not div n?
63*39 - 1 / 2 - 32*20-1=589
E.g. n smallet pos int mult of 75 and has exactly 75 pos int div, including 1 srlf find n/75
75=3*5^2, 2+1 * 4+1 * 4+1 , n = p1^2 p2^4 p3^4, two of prime 3,5 , smallest=2, 2,3,5
5^2 * 2^4 * 3^4 = n, n/75 = 432
Prod factor = n^(f/2)
Emg. pos int fn quotient obt when sum all pos div of n div by n , f14=1+2+7+14)/14 = 12/7
F768-f384?
768=1*3*2^8, 384=2^7*3, f768=1+2+2^2+...+2^8)(3^0+3) / 2^8*3 = 2^9 - 1)(4)/2^8 * 3
f384=2^8-1 (4) / 2^7 * 3, f768-f384= 1/ 2^6*3 =
E.g. smallest pos int div by 4 and 9 base 10 rep only 4 and 9, as least one each, last four digit?
4444444 9 44 ->4944
Legendre’s theorem, vp n! = number of powers of p in n! =floor m/p + floor n/pp +...
E.g. n number of consec 0 at the right end of decimal rep of prod 1!2!3!4!...99!100!, remainder when n div by 1000
5->96, 10->91, 15->86, ….,100->1
97/2*20=970
25->76, 50->51, 75->26, 100->1
76+51+26+1=154
970+154/1000=124
E.g. base 10 rep for 19! is 121,6t5,100,40m,832,h00, t+m+h?
2:9+4+2+1=16
3:6+2=8 3^8->9
5:3
11:1 ->11
3 factor of 10: H=0, 1+2+1 + 11+t + 1 + 4+m + 13 =33 + m + T = sum of digits hsa to be a mult of 9:
36 45 54, m+n = 2, 12, 21(>18,no), then 3 or 12, odd digit = 13+T, even digit=20+m, either = or 11 more than another, m=t+4, m+7=t, —>t=4,m=8
12
gcd(m,n)lcm(m,n) = mn, gcd(ac,bc)=c gcd(a,b)
E.g. lcm(gcd(24,a), gcd(a,b)) = lcm(gcd(24,b), gcd(a,b))
If a=2b, smallest a+b?
gcd(24,a)=gcd(24,b)=gcd(24,2b), b=2^3, a=16, 24
Euclidean algorithm, gcd(x,y) = gcd(x-ky,y), x>y, k pos int
E.g. an=100+nn, n=1,2,3…, for each n let dn gcd of an and an+1, find max dn as n ranges through pos int
dn=gcd(an,an+1) = gcd(100+nn, 100+n+1)^2)
=gcd(2n+1, 100+nn) = gcd(4nn+400, 2n+1) = gcd(4nn+400-(4nn+2n), 2n+1) = gcd(400-2n, 2n+1) =
= gcd(400-2n+2n+1,2n+1) = gcd(401, 2n+1) = 401
theorem 25.0.2 n=a mod b , d=x mod n, b= y mod n, then ab=xy mod n
111*23 = 1 mod 11
E.g. Rem when 9 x 99 x 999 x…x 99…9 -999 9s div by 1000:
9 = 9 mod 1000
99 = 99 mod 1000
999 = 999 = -1 mod 1000
…
99*9=-891=109 mod 1000
a=b mod n, a^m = b^m mod n, 37^9 mod 3 = 1 mod 9
E.g. s be subset of 1,2,3,...,30, with property no pair of distinct el in s sum div by 5, what is the largest pos size of s?
1+4=5, 2+3 = 5, 0+0,
6 mul of 5, 1 works, 6(1m5)+6(2m5)=13
25.0.4 digit cycles
E.g. k = 2008^2+2^2008, unit digit of kk+2^k?
k*km10, kk-square of k unit digits, 2008^2=4mod 10, 2^2008 mod 10 = 6 mod 10, res 10 mode 10 = 0 mod 10
2^k: k->0 mod 4, 2^k = 6 mod 10, sum 0+6 mod 10
Theorem 25.0.5 euler’s totient function
fi(n) = n * 1-1/pi, mult all
fi(40) = 40 * (1-1/2) (1-1/5) = 16
Fermat’s little theorem, a^p == a mod p, if only if p prime, gcd(a,p) = 1
2^4 = 1 mod 5, 2^8 = 1 mod 5
Eulers totient theorem, a^fi(n) = 1 mod n, if only if gcd(a,n) = 1
Theorem 25.0.9 wilson’s theorem, p-1) ! = p-1 = -1 mod p,
P-1 ! = -1 mod p, 6! = -1 mod 7
E.g. an = 6^n + 8^n, determine rem div a83 by 49
6^83 + 8^83 mod 49,
fi(49) = 42
6^42 = 1 mod 49
6^83 = 6^41 = 6^-1 mod 49
8^83 mod 49, 8^42 = 1 mod 49, 8^-1 mod 49
6a = 1 mod 49, a = 8, 6*-8 = 1 mod 49, 41
8b=1 mod 49, b = -6, , sum=-14 + 49 = 35
Binomial theorem
E.g. hundreds digit of 2011^2011
2011^2011 mod 1000
11^2011 mod 1000
10 + 1 ) ^ 2011 mod 10^3
10^3 2011c2008 + 10^2 2011c2009 + 10 2011c2010 + 2011c2011*10^0
= 55 * 100 + 2011*10 + 1 = 611 mod 1000
Chineses remainder theorem
If a pos number x sat x==a1 mod n1, x===a2 mod n2 … x==ak mod nk, where all n rel prime then x has unique solution mod n1n2n3…nk
Solve:
Algebraic method: find 2 congruences,
n==r1 mod m1,
n==r2 mod m2, s.t. M1 m2 rel prime
Rewrite n = km1 + r1, n = km2 + r2
Set equal,
E.g. smallest pos int > 100, rem 5 div 9, 4 div 17
X = 5 mod 9, x = 4 mod 17
x= 9a +5 = 17b + 4
17b + 4 = 5 mod 9, -b = 1 mod 9, b = -1 mod 9, b = 9c-1, x=17(9c-1)+4 = 153c-13 = 140 mod 153
N = r1+m1(r2-r1) * i, where i = m1^-1 mod m2
E.g. pos int n, n^2 both end in same seq of 4 digits abcd when written in base 10, where digit a is not ero, find three digit number abc
N = n^2 mod 10000,
N^2 - N = 0 mod 10000 -> N(N-1) = 0 mod 10000 = 2^4 x 5^4
N = 0 mod 10000, N = 1 mod 10000,
N(N-1) one even one odd, 2^4 on one number, 5^4 the other
Case1: n=0 mod 16, n-1=0 mod 625, n =1 mod 625
625a + 1 = 0 mod 16
640 - 15 = a+1 = 0 mod 16, a=15 mod 16
625 * (16c + 15) + 1 = 9376
Case2, n-1 = 0 mod 16, n = 0 mod 625, n = 1 mod 16
625b = 1 mod 16, b = 1 mod 16, 625(16d+1)=10000d+625, a=0 no
E.g. find the number of ordered triples a,b,c where pos int a is factor of b, a factor of c, a+b+c=100
B = ka, c = la, (1+k+l)a = 100 = 2^2 5^2 -> 9 factors, sum of factors: 1+2+4)(1+5+25 = 217
1 2 4 5 10 20 25 50 100
No no k+l=3 k+l=4
k+l= 3,4,9,19,24,49,99
# sol: 2,3,8,18,23,48,98 = 200
Quadratic factorizations
E.g. how many values of n from 1 to 100 inclusive make 16^n - 2x4^n + 1 a mult of 49?
4^n - 1) ^2 = 49k, 7sqrt(k) = 4^n - 1 = 7m, 4^n = 1 mod 7,
4^1 = 4, 4^2 = 2 mod 7, 4^3 = 1 mod 7, 4^4=4 mod 7, n = 3L, 100/3=33
26.2 cubic fact
26.3 simon’s favorite factoring trick
26.3.1 - xy+kx+jy+jk = (x+j)(y+k)
E.g. xy+x+y +1 = 38 = (x+1)(y+1)
E.g. rec floor a by b feet, where a,b pos int with b?a, pint rec on floor with sides of rec parallel to side of floor, unpainted part of floor forms a border of width 1 foot around painted occupies half area of entire floor, how many possibilities for the ordered pair a,b?
a-2)(b-2 = ½ a*b
ab-2a-2b+4=½ ab, ab-4a-4b+8=0
a-4)(b-4 = 8, 2*4, or 1*8, (6,10), (5,12)
26.4 higher power fact
26.5 sophie germain’s identity
E.g. right triangle int leg len a and b and hypotenuse of len b+1 where b<100?
aa+bb=b+1)^2 = bb+2b+1
aa=2b+1 < 201, a<=14, all odd, a=3,5,7,9,11,13, 6 possibilities
E.g. uniq pos int x and y sat eq xx+84x+2008 = yy, find x+y
x+42)^2 + 244 = y^2
Y- x - 42)( y +x + 42 =244 = 4*61
1,244,
2,122, y=62, x=18, x+y=80
4,61
E.g. a,b,c,d pos int, a^5 = b^4, c^3 = d^2, c-a=19, determine d-b
c=m^2, a=n^4, m+n^2)( m-n^2) = 19 = 1*19, m-nn=1, m+nn=19, m=10,n=3,c=100,a=81
E.g. base 9 rep of n is 27006000052, ndiv 5=?
9=-1 mod5
2+ 5*9 + 6*9^6 +7*9^9 + 2*9^10 = 2-5+6-7+2 mod 5 = -2 mod 5 = 3 mod5
E.g. 3 digit pos int a b c whose rep in base 9 is bca where abc are digits
B81 + 9c +a
Abc10 = 100a + 10 b +c
71b + 8c = 99a
8c = 28a mod 71
2c = 7a mod 7
c=7,a=2,b=2, 227
E.g. base 10 three digit n rand, which is closest to prob bae 9 rep and base 11 rep of n are both three digit numerals
1*9^2 +...->81 ->100_9
8+8*9+8*9*9 = 80+648=728->1000_9
10^11+10=121->100_11
10*11^2 +10*11 + 10 = 1210+110+10=1330->1000_11
81->728
121->1330
121->728 608 numbers 608/900=152/225=⅔=0.7
Concept 28.0.2: 0.1_3 = ⅓, 0.1_10 = 1/10 , 0.24_5 = ⅖+4/25
E.g. pos int k, repeating base k rep of bae 10 frac 7/51 is 0.23_k = 0.23232323, what is k?
2/k + 3/kk + 2/kkk + 3/kkkk = 7/51
2k + 3 / kk (1+1/kk + …) = 7/51
2k+3 * 1/kk-1 = 7/51
k->16
29.1 Palindromes
E.g. find arithmetic mean of all 3 digit palindromes
1_1, 2_2, … 9_9
1+9/2=5, 0+9/2=2.5, 5
500+45+5 = 550
29.2 chicken mcnugget theorem
Max val tht cannot be expressed as sum of non neg mult of a and b is ab-a-b if a and b rel prime
a-1)(b-1 / 2 pos int cannot be expres in the form ma+nb
E.g. 3 and 5 coins cannot make 7 max: 15-5-3 = 7
E.g. town 3 people for each horse, 4 sheep for each cow, 3 duck for each person, which cannot be total number of people horses sheep cows nd ducks: 41 47 59 61 66?
Group1: 9+3+1=13, group 2: 5
13a+5b=, max val = 13*5-13-5 = 47
E.g. prod 8 * 888…8, k digits, int whose digits sum of 1000, what is k?
8*8=64, 8*88=704, 888*8 = 7104, 8888*8=71104, 711104
7+x + 4 = 1000, x = 989, k=991
E.g. ab=2ac, triangle abc, bae=acd, f intersection of seg ae and cd, cfe is equilateral, what is acb =90
3 60
4 90
5 108
6 120
7 135
8 140
9 144
Inscribed arc theorem: Angle formed by an arc in the center 2x angle formed by arc and on the edge
<APC = ^BD - ^AC / 2, if inside, <APD = ^BC + ^AD / 2
Tangent R intersects the circle at Q, a chord QP is drawn, then <RQP = ½ of arc angle
Angle of opposite angles = 180 in cyclic quadrilateral
Inrad = a + b + c / 2
Heron’s formula S = sqrt(s(s-a)(s-b)(s-c)), s = a+b+c/2, semiperimeter
Inscribed circle inrad for right triangle r = a+b-c / 2, S = inrad * s, A / S = r, S=½ perimeter
Circumcenter circle rad R, S = abc/4R
Shoelace theorem using coordinates
Pick’s theorem
E.g. aime
E.g. amc12 3 congruent isosceles base on sides of equilateral triangle of side length 1, sum of areas of 3 isosceles triangles is the same as area of equilateral triangle, what is length of one of two congruent sids of one of iso tri. sqrt(3)/3
Centroid is intersection of three medians
Eg.amc10 ab=12,bc=24,ac=18, incenter line // bc x ab at M, ac at N, perimeter of amn?
Incenter ½ bisects each angle, 12+18 =30
Important pythagorean triangle, 345 5/12/13 9/40/41, 7/24/25, 8/15/17, 9/40/41, 20/21/29
13/14/15 -> 5/12/13 + 9/12/15
Congruent test: AAS, SAS, SSS, HL in right triangle, LL in right triangle, SSA is not
Similarity test: AA, SAS, SSS, right triangle HL, LL, SSA is not
Rignth triangle B=90, ,BD perpendicular AC, AD *CD = BD *BD
Angle bisector theorem, ad bisect a, then ab / bd = ac/cd = pk^2 k tangent
E.g. amc construct similar triangle
Area of rhombus, S=½ d1 d2, Perimeter = 2sqrt(d1^2+d2^2)
Parallelogram with diag d1 and d2 has area ½ d1 d2 sin theta, the center angle
Trapezoid area, diagonal lines, 4 triangle, same twp sides, top bottom ratio line
Area, length
Power of point for 2 tangents
Tangent equal
Power of point inside circle, P
PA*Pb=Pc*Pd = r^2-OP^2, angle = ½ arc , similarity
Power of point outside circle P
Pa *pb = pc *pd = op^2 - r^2
Cyclic quadrilateral
Sum of opposite angle = 180
Ptolemy’s theorem
Ac * bd = ab*cd+ad*bc
Brahmagupta’s formula
A = sqrt(s-a * s-b * s-c * s-d), s = a+b+c+d/2
Sum of interior angle = n-2 * 180
Interior angle of regular polygon = n-2 / n * 180
Exterior angle = 360 / n
Hexagon:
Interior angle = 6 - 2 / 6 * 180 = 120
Exterior angle = 60
Area = 6 * sqrt (3) / 4 * s^2
Length of diagonal = 2 s
Octagon:
Sum interior angle = 8-2 * 180 = 1080
Interior 135, exeter 45, A = 2 (1+sqrt(2))*s^2, side length s
Volume of tetrahedron = ⅓ base height, regular(all sides equal) = sqrt(2)/12s
V of pyramid = ⅓ base height, regular sqrt(2)/6 s
SHoelace formula, for area of polygon
X,y and ax+by+c=0, dist = ax+by+c / (sqrt)(aa+bb
frac(2.7) = 0.7, frc(-3.3) = 0.7
E.g. fl x/2 + x/4 + x/6 + x/8 x real how many in 1000
3/24 6/24 9/24 12/24 15/24 18/24 21/24 24/24
4/24 8/24 12/24 16/24 20/24 24/24
1 1 2
0 20
Hit 12 vals
x=1->20
x=2->40
x=n->20n
x=50->1000
12*50=600
E.g. xx + 10000fl(x) = 10000x
Real x, how many
x==fl(x) + {x}
(fl(x) + {x} )^2= 10000{x} <10000
-100 < fl(x) +{x}= 100{x} <100
-100<= fl(x)<=99
99+{x} )^2 = 10000{x}
99*99 +2*99*{x} + {x}^2 = 10000{x}
No valid {x} except 1, no
Graphing 199
E.g. how many pos int n n+1000 / 70 = fl(sqrt(n))
Inequality
fll(d\qrt(n)) <= < sqrt(n) + 1
n+1000/70 <= sqrt(n) <= n+1070/70
n=50 mod 70
sqrt(n) in 20,50
n=[400, 2500]
n=[22+13/24)^2 , 47+11/24)^2]
n=[506.25, 2256.25]
sqrt(155)= 12 + 11/2*12 approx
400 4700 2500 2430 2360 2290
a+b/2>sqrt(ab) arithmetic mean > geometric mean
Am-gm inequalities , for non neg 1,a2..an, a1+ a2 +... +an / n > Nqrt(a1 a2 a3 … an) max when =
Weighted am-gm c1a1+... > ca+c2..)qrt(a1a2..an)
Cauchy schwarz a1b1+...)^2 < a1^2+a2^2…)(b1^2+b2^2+...), = when ai / bi same for all
a2+b2+c2)(a4+b4+c4 <= a3+b3+c3)^2
A2 + b6 + c8)(a12+b2+1) <= a7+b4+c4)^2
a4+b6+c8)(1+1+1) >= a2+b3+c4)^2
Eg. a+b+c = 2, a2+b2+c2=12 Diff of Max min of possible of c?
a2+b2+c2)(1+1+1) >= a+b+c)^2
A+b = 2-c, a2+b2 = 12-c2
a2+b2)(1+1) >= a+b)^2, 0>=3c2-4c-20
R-s
R+s = 4/3, rs = -20/3
r+s)^2 - 4rs = r-s)2 , r-s = 16/3
E.g. Log_3x_4 = log_2x_8, x pos rel other thn ⅓ ½ written as p/q rel prime pos, p+q?
log4/log3x=log8/log2x
log3/log3+logx = log7/log2+logx, choose base 2
log2_x=y, x=2^y
2/log2_3 + y = 3/1+y, y = 2-3log2_3, 2^2-3log2_3 = 4 / 2^3log2_3 = 4 / 2^log2_27 = 4/27
31
E.g. 3log(sqrt(x)logx) = 56, log_logx_x = 54, find b, base not spec assume b
log(sqrt(x)logx) / logb = 56/3
log(sqrt(x)) + log(log(x)) / log x = 56 /3
Base x, ½ + logx(logb_x) / logx_b = 56 / 3
log(x) / log(log(x)/log(b)) = 54, logx / log(logx) - log(logb) = 54 base x
1 / logx(logb_x) = 54, same term
½ + 1/54 / logx_b = 56 /3
Logx_b = 1/36, logb_x = 36, 1/54 = logx(36), logx = /logb = 36
logx_b/logx_36 = logb / log36 = log36_b = 54/36 = 3/2, b = 216
E.g. logx a = 5, log3b = 4, log4c = 4
A = xyz-log5x = 35, b = xyz-log5y = 84, xyz-log5z = 259
Find log5x + log5y + log5z
3xyz + log5xyz = 378 = 375 + 3, xyz = 125
log5x=35-125, log5y=84-125=-41, log5z=259-125=134, add = 265
E.g. xyz pos real 2logx(2y) = 2log2x(4z) = log2x^4(8yz) != 0
The val of xy^5z exp in ½^(p/q) p q rel prime pos int, p+q=?
2log2(2y)/log2(x) = 2log2(4z)/log2(2x) = log2(8yz) / log2(2x^4)
a=log2(x), b= log2(y), c = log2(z)
2 1+b/a = 2 2+c/1+a = 3+b+c/1+4a, d=1+b, e=2+c, 2d/a=2e/1+a=d+e/1+4a, a=-1/6,b= ,c=
log2(xy^5z) = a+5b+c
E.g. sin a + sin b = rt(5/4 cosa+cosb = 1, what is cos(a-b) = cosabosb + sinasinb = ⅓
E.g. sin^10x+cos^10x=11/36, sin^12x+cos^12x=m/n
sin^2x=a, cos^2x=b, a+b=1, a6+b5=11/36, a6+b6?
a+b)(a5+b5 =a6+b6+ab(a4+b4)
a+b)(a4+b4)=a5+b5+ab(a3+b3)
a+b)(a3+b3)=a4+b4+ab(a2+b2)
a+b)(a2+b2)=3+b3+ab(a+b)
a+b)(a+b = a2+b2+2ab
ab=⅙ or ⅚, sinxcosx<½, ab=⅙, 11/36-⅙(7/18)=13/54
E.g. n>1 int, let F(n) num of solu to eq sinx=sin(nx) on interval [0,pi], what is sum(n=2…2007)Fn
n=2, 3
n=3, 4
n=4, 5
n=5, 5 period 2pi
n=6, 7
3+4+,,,+2008, 1mod4 except, 501 of these
Sum 3+4+...+2008 - 501 = 2016532
A triangle = ½ ab sin(C), a/sinA=b/sinB=c/sinC=2R, circumradius outtan circl
E.g. pqr, pr=15,qr=20,pq=25, a,b on pq, c,d on qr, e,f on pr, pa=qb=qc=rd=re=pf=5, area of abcdef
150-25/2-25/2(sinQ+sinP) ->7/5 = 120
z*z’ = 1, z^2 = 1, z real, z=z’, z im, z = -z’
A’ + b’ = a+b)’
A’b’ = ab)’
a’^n = a^n)’
E.g. z^6! - z^5! = its conjugate
z^6! - z^5! = 1/z^6! - 1/z^5!, z^6!=z1440 , 1440 solutions
z^1440-z^840+z^600-1, z^840+1)(z^600-1)=0, z^600=1, z^840=-1, z^240=-1, z^120=1, no dup
E.g. zw+12i+20i-240/wz=-30+46i, what is smallest possible zw^2,
let y=zw, y-240/y=-30+14i, y=36-12i|-6-2i, 40
E.g. a+164i = z, pos int n .t. z/z+n = 4i, find n
a=-656, 164=4a+4n,n=697
Polar form
Than theta = b/a
Z = a+bi = r(cos that + i sin theta) = r cis theta
cis= cos i sin
Cos = b/r, sin = a/r
Eulers formula: e^pi i = -1
Roots of unity:
w^3 = 8, (r cistheta)^3 = 8, r=2, cistheta)^3 = 1, cis(3theta)=1, 3theta= 2kpi
E.g. how many n>=2 st whenever |z1| = |z2| =..= |z_n| = 1, and z1+z2+...+zn=0, zi equally spaced on unit of circle
n=2,3
De moivre:
For z=re^itheta, z^n = re^itheta)^n = r^n (cos ntheta + i sin ntheta) , ^n ==rotate n*theta ccw
Eval: sqrt(3) +i )^8 = 2^8cis(60*8)
E.g. z^13=w, w^11=z, w143=w w^142=1, w = cis(2pi/142), cis(4pi/142),...,cis(284pi/142),
w^11 cis(22pi/142), cis(44pi/142)... sin(11pi/71)
E.g. =1+i / sqrt(2) = cis(45), what is z1+z4+...+z144)(z-1++...+z-144
z^8 = 1, 1^2=1 , 2^2 = 4, 3^2 = 1, 4^2 = 0, 5^2 = 1, 6^2=4,7^2=1,8^2=0 mod8,
3(cis145+cis180+cis145+cis0) 3(cis-145 + cis-180 + cis-45 +cis0) = 36
Complex number to solve geometry problem
Rotate in complex plane by theta, == *cis(theta)
E.g. (0,0), (a,11), (b,37) vertices of an equilateral triangle, find ab
0, A+11i, b+37i, ange is 60,
a+11i) cis(60) = b+37i, cis60 = ½ + qrt(3)/2 i, a=21sqrt(3), b=5sqrt(3), ab = 315
E.g. f=z^2-19z, right triangle 0, z-fz, f(fz)-fz, z-fz cis(90) = f(fz)-fz, fz-z/fz-f(fz) = -(z-19)(z+1) = a+11i-19)(a+11i+1, a+11i)^2 - 18(a+11i) - 19, only im, a^2-18a-140 = 0, a-9)^2=221,
Complex roots in polynomial
E.g. fz = az^2018 + bz^2017 + cz^2016 real coef not exceeding 2019, f(1+rt(3)i/2) = 2015 + 2019rt(3) i, remainder when f(1) is div by 1000
Fz = z^2016(az^2+bz+c), f(cis60) = 2015 + 2019rt(3) i = cis60)^2016 (acis120+bcis(60)i),
f(1) = 1^2016(a+b+c) = a+b+c, cis60=1 2019rt(3) = a rt(3)/2 + b rt(3)/2, a+b = 4038
2015=a(-1/2)+b1/2 +c, a=b=2019,c=2015, sum=6053 div 1000, 53
E.g. x10+13x-1)^10 = 0, 10 complex roots , 1/rr1’ + 1/rr2’+..+1/rr5’, let y=1/x, find s1s1’+s2s2’+..+s5s5’, 1/y^10+13/y-1)^10=0, 13-y)^10 = -1, cis18,cis54,...cis342 = 13-y, y = 13-cis18, 13-cis54, …13-cis342, conjugate: 13-cis18)(13-cis342 = 170 - 13(cis18+cis342), same for 13-cis54)(13-cis308 = 170 - 13(cis54+cis306), … = 1, conjugates sum=0, 175*5
Complex number solve trig problem
SinA = cisA + cis180-A / 2i = e^iA + e^i(180-A) / 2i
cosA = e^iA + e^i(-A) / 2
tanA = sin/cos
E.g P=x^3+ax^2+bx+c root cos2pi/7, cos4pi/7, cos6pi/7, what is abc
-a = cos2pi/7 + cos(4pi/7) + cos6pi/7 = cis2pi/7 + cis(-2pi/7) / 2 + 4pi + 6pi, let cis2pi/7 = W
w^2+1/w^2 / 2 + w4+1/w^4 / 2 + w6+1/w^6 / 2, letW=w^2,
W+W^2+W^3 + 1/W + 1/W^2 + 1/W^3/2 = -a, w^7=1,.
W+W^2+...+W^6 / 2 = -½ = -a, a = 1/2,, same for b and c
1+W+W2+W3+W6=0, sum off all unities =0, W^7-1/W-1 = 0
E.g. sin2sin4…sin90 - prt(5)/2^50, find p
Sinx = cisx+cis(180-x)/2i = cisx-1/cis(x) / 2i, w = cis2, now cis2 - 1/cis2 / 2i = w-1/w = 2i, cis4:w^2, ..
w-1/w)(w2-1/w2)...(w45-1/w45) / (2i)^45, sin2=sin178, sin4 = sin(176),...
Find sin2sin4…sin90sin92…sin178
w^2-1)(w^4-1…w^178-1 / w^89*90/2->w^45->i^(2)^89 i^89-> 2^89 and sqrt at the end
1-w2)(1-w4…1-w178 / 2^89, let 1=x, polynomial 1+x+x2+.../2^89, 90/2^