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Putnam Mathematical Competition...

Maple can solve the easiest two problems of the Putnam Mathematical Competition 2018. link

Problem A1

Find all ordered pairs (a,b) of positive integers for which 1/a + 1/b = 3/2018

>	eq:= 1/a + 1/b = 3/2018;

(1)

>	isolve(%);

(2)

>	# Unfortunalely Maple fails to find all the solutions; eq must be simplified first!

>	(lhs-rhs)(eq);

(3)

numer(%);

(4)

>	s:=isolve(%);

(5)

>	remove(u ->(eval(a,u)<=0 or eval(b,u)<=0),[s]);

(6)

>	# Now it's OK.

Problem B1

Consider the set of vectors P = { < a, b> : 0 ≤ a ≤ 2, 0 ≤ b ≤ 100, a, b in Z}.
Find all v in P such that the set P \ {v} can be partitioned into two sets of equal size and equal sum.

>	n:=100: P:= [seq(seq([a,b],a=0..2), b=0..n)]:

>	k:=nops(P): s:=add(P): numsols:=0:

for i to k do
  v:=P[i]; sv:=s-v;
  if irem(sv[1],2)=1 or irem(sv[2],2)=1 then next fi;
  cond:=simplify(add( x[j]*~P[j],j=1..k))-sv/2;
  try
    sol:=[];
    sol:=Optimization:-Minimize
         (0, {x[i]=0, (cond=~0)[], add(x[i],i=1..k)=(k-1)/2 }, assume=binary);
    catch:
  end try:
  if sol<>[] then numsols:=numsols+1;
     print(v='P'[i], select(j -> (eval(x[j],sol[2])=1), {seq(1..k)})) fi;
od:
'numsols'=numsols;

(7)

Download putnam2018.mw

Edit.
Maple can be also very useful in solving the difficult problem B6; it can be reduced to compute the (huge) coefficient of x^1842 in the polynomial

g := (1 + x + x^2 + x^3 + x^4 + x^5 + x^9)^2018;

The computation is very fast:

coeff(g, x, 1842): evalf(%);
0.8048091229e1147

December 08

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Rouben Rostamian

2416

Extrapolating from the...

The attached worksheet develops a procedure for extrapolating boundary data from a square to its interior. Specifically, let's consider the square [−1,1]×[−1,1] and the continuous functions u₁, u₂, u₃, u₄ defined on its edges. The procedure constructs a continuous function u(x,y) in the interior of the square which matches the boundary data.

The function u(x,y) is necessarily discontinuous at a corner if the boundary data of the edges meeting at that corner are inconsistent. However, if the boundary data are consistent at all corners, then u(x,y) is continuous everywhere including the boundary.

Here is what the extrapolating function looks like: