Add project-euler.190

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+! Copyright (c) 2008 Eric Mertens
+! See http://factorcode.org/license.txt for BSD license.
+USING: kernel sequences sequences.lib math math.functions math.ranges locals ;
+IN: project-euler.190
+
+! PROBLEM
+! -------
+
+! http://projecteuler.net/index.php?section=problems&id=190
+
+! Let Sm = (x1, x2, ... , xm) be the m-tuple of positive real numbers
+! with x1 + x2 + ... + xm = m for which Pm = x1 * x22 * ... * xmm is
+! maximised.
+
+! For example, it can be verified that [P10] = 4112 ([ ] is the integer
+! part function).
+
+! Find Σ[Pm] for 2 ≤ m ≤ 15.
+
+! SOLUTION
+! --------
+
+! Pm = x1 * x2^2 * x3^3 * ... * xm^m
+! fm = x1 + x2 + x3 + ... + xm - m = 0
+! Gm === Pm - L * fm
+! dG/dx_i = 0 = i * Pm / xi - L
+! xi = i * Pm / L
+
+! Sum(i=1 to m) xi = m
+! Sum(i=1 to m) i * Pm / L = m
+! Pm / L * Sum(i=1 to m) i = m
+! Pm / L * m*(m+1)/2 = m
+! Pm / L = 2 / (m+1)
+
+! xi = i * (2 / (m+1)) = 2*i/(m+1)
+
+<PRIVATE
+
+: PI ( seq quot -- n )
+    [ * ] compose 1 swap reduce ; inline
+
+PRIVATE>
+
+:: P_m ( m -- P_m )
+    m [1,b] [| i | 2 i * m 1+ / i ^ ] PI ;
+
+: euler190 ( -- n )
+    2 15 [a,b] [ P_m truncate ] sigma ;