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Putnam done with Maple

Markiyan Hirnyk 7714 Maple
December 26 2014
2

The William Lowell Putnam Mathematical Competition, often abbreviated to the Putnam Competition, is an annual mathematics competition for undergraduate college students enrolled at institutions of higher learning in the world (regardless of the students' nationalities). One can see some problems and answers here. I find it remarkable that a lot of these problems can be done with Maple. Here is a sample (The DirectSearch package should be downloaded from http://www.maplesoft.com/applications/view.aspx?SID=101333 and installed in your Maple.).

 

rsolve({a(k)=a(k-1)^2-2,a(0)=5/2},a)#2014,A-3

NULL

rs := unapply(rsolve({a(0) = 5/2, a(k) = a(k-1)^2-2}, a), k)

proc (k) options operator, arrow; 2*cosh(arccosh(5/4)*2^k) end proc

 (1)

(2)

evalf(product(1-1/rs(k), k = 0 .. infinity))

.4285714286

 (3)

identify(%)

3/7

 (4)

sol := solve({1/x-1/(2*y) = 2*(-x^4+y^4), 1/x+1/(2*y) = (x^2+3*y^2)*(3*x^2+y^2)}, explicit)

sol[1]; evalf(sol)

{x = 1.122865470, y = .1228654698}, {x = -0.39087502e-2+.3661111372*I, y = -1.003908750+.3661111372*I}, {x = .6924760152-.5923802638*I, y = -.3075239848-.5923802638*I}, {x = .6924760152+.5923802638*I, y = -.3075239848+.5923802638*I}, {x = -0.39087502e-2-.3661111372*I, y = -1.003908750-.3661111372*I}, {x = .3469845126+.1168520057*I, y = 0.3796751170e-1+1.067908527*I}, {x = .7773739670+.4755282581*I, y = .4683569607-.4755282736*I}, {x = .2183569726+.2938926261*I, y = 1.027373941-.2938926802*I}, {x = .3469845126+1.067908522*I, y = 0.3796751830e-1+.1168520056*I}, {x = -.2120324818+.8862728900*I, y = .5969845187+.2984876419*I}, {x = -.3494002531+.8416393955*I, y = -.6584172547-.1094171332*I}, {x = -.2120324818+.2984876377*I, y = .5969845144+.8862728969*I}, {x = -.9084172475+.6600037635*I, y = -0.9940025307e-1+0.7221851120e-1*I}, {x = -.3494002531+.1094171208*I, y = -.6584172336-.8416394020*I}, {x = -.9084172475+0.7221851117e-1*I, y = -0.9940025050e-1+.6600037719*I}, {x = -.9084172475-0.7221851117e-1*I, y = -0.9940025050e-1-.6600037719*I}, {x = -.3494002531-.1094171208*I, y = -.6584172336+.8416394020*I}, {x = -.9084172475-.6600037635*I, y = -0.9940025307e-1-0.7221851120e-1*I}, {x = -.2120324818-.2984876377*I, y = .5969845144-.8862728969*I}, {x = -.3494002531-.8416393955*I, y = -.6584172547+.1094171332*I}, {x = -.2120324818-.8862728900*I, y = .5969845187-.2984876419*I}, {x = .3469845126-1.067908522*I, y = 0.3796751830e-1-.1168520056*I}, {x = .2183569726-.2938926261*I, y = 1.027373941+.2938926802*I}, {x = .7773739670-.4755282581*I, y = .4683569607+.4755282736*I}, {x = .3469845126-.1168520057*I, y = 0.3796751170e-1-1.067908527*I}

 (5)

plots:-implicitplot([1/x+1/(2*y) = (x^2+3*y^2)*(3*x^2+y^2), 1/x-1/(2*y) = 2*(-x^4+y^4)], x = 0 .. 2, y = 0 .. 1, color = [red, blue], gridrefine = 4)

  

"http://kskedlaya.org/putnam-archive/ and https://en.wikipedia.org/wiki/William_Lowell_Putnam_Mathematical_Competition"

Re(convert(int(ln(x+1)/(x^2+1), x = 0 .. 1), polylog))

(1/8)*Pi*ln(2)

 (6)

Im(convert(int(ln(x+1)/(x^2+1), x = 0 .. 1), polylog))

0

 (7)

NULL

DirectSearch:-GlobalOptima(int(sqrt(x^4+(-y^2+y)^2), x = 0 .. y), {y = 0 .. 1}, maximize)

[.333333333333333, [y = HFloat(0.9999999999999992)], 96]

 (8)

rsolve({T(0) = 2, T(1) = 3, T(2) = 6, T(n) = (n+4)*T(n-1)-4*n*T(n-2)+(4*n-8)*T(n-3)}, T)

GAMMA(n+1)+2^n

 (9)

floor(10^20000/(10^100+3))

99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999970000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000008999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999997300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000809999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999757000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000072899999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999978130000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000006560999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999998031700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000590489999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999822853000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000053144099999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999984056770000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004782968999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999998565109300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000430467209999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999870859837000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000038742048899999999999999999999999999999999999999999999999999999999999999999999999999999999999999999988377385330000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003486784400999999999999999999999999999999999999999999999999999999999999999999999999999999999999999998953964679700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000313810596089999999999999999999999999999999999999999999999999999999999999999999999999999999999999999905856821173000000000000000000000000000000000000000000000000000000000000000000000000000000000000000028242953648099999999999999999999999999999999999999999999999999999999999999999999999999999999999999991527113905570000000000000000000000000000000000000000000000000000000000000000000000000000000000000002541865828328999999999999999999999999999999999999999999999999999999999999999999999999999999999999999237440251501300000000000000000000000000000000000000000000000000000000000000000000000000000000000000228767924549609999999999999999999999999999999999999999999999999999999999999999999999999999999999999931369622635117000000000000000000000000000000000000000000000000000000000000000000000000000000000000020589113209464899999999999999999999999999999999999999999999999999999999999999999999999999999999999993823266037160530000000000000000000000000000000000000000000000000000000000000000000000000000000000001853020188851840999999999999999999999999999999999999999999999999999999999999999999999999999999999999444093943344447700000000000000000000000000000000000000000000000000000000000000000000000000000000000166771816996665689999999999999999999999999999999999999999999999999999999999999999999999999999999999949968454901000293000000000000000000000000000000000000000000000000000000000000000000000000000000000015009463529699912099999999999999999999999999999999999999999999999999999999999999999999999999999999995497160941090026370000000000000000000000000000000000000000000000000000000000000000000000000000000001350851717672992088999999999999999999999999999999999999999999999999999999999999999999999999999999999594744484698102373300000000000000000000000000000000000000000000000000000000000000000000000000000000121576654590569288009999999999999999999999999999999999999999999999999999999999999999999999999999999963527003622829213597000000000000000000000000000000000000000000000000000000000000000000000000000000010941898913151235920899999999999999999999999999999999999999999999999999999999999999999999999999999996717430326054629223730000000000000000000000000000000000000000000000000000000000000000000000000000000984770902183611232880999999999999999999999999999999999999999999999999999999999999999999999999999999704568729344916630135700000000000000000000000000000000000000000000000000000000000000000000000000000088629381196525010959289999999999999999999999999999999999999999999999999999999999999999999999999999973411185641042496712213000000000000000000000000000000000000000000000000000000000000000000000000000007976644307687250986336099999999999999999999999999999999999999999999999999999999999999999999999999997607006707693824704099170000000000000000000000000000000000000000000000000000000000000000000000000000717897987691852588770248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 (10)

int(exp(-1985*(t+1/t))/sqrt(t), t = 0 .. infinity)

(1/1985)*Pi^(1/2)*exp(-3970)*1985^(1/2)

 (11)

l := [seq(LinearAlgebra:-Determinant(Matrix(n, proc (i, j) options operator, arrow; 1/min(i, j) end proc)), n = 1 .. 10)]

[1, -1/2, 1/12, -1/144, 1/2880, -1/86400, 1/3628800, -1/203212800, 1/14631321600, -1/1316818944000]

 (12)

with(gfun):

rec := listtorec(l, u(n))

[{u(n+1)+(n^2+5*n+6)*u(n+2), u(0) = 1, u(1) = -1/2}, ogf]

 (13)

rsolve(rec[1], u)

(-1)^n*(n+1)/GAMMA(n+2)^2

 (14)

``

Hope the reader will try to continue the above.

Download Putnam_done_with_Maple.mw

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