Vol. 102(6), October 2002
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Vol. 102(6), October 2002 |
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Volume 102(6) |
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4737: Proposed by Stanley
Rabinowitz, West ford, MA
The lengths of
the side of a triang1e are sinx. sniy and
sinz where x+y+z =p.. Find
the radius of the circumcircle of the triangle.
Show that
every integer of 5n + 1 digits beginning and ending with lâs and having
2âs in between is divisible by the sum of two squares.
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4739: Proposed by Michael Brozinsky,
Central Islip, NY
Given the points O(0,0),A(a,0) and B(b,0) where 0 < a < b and b Õ 2a. Describe the locus of a point P such that for any line through A, P is the point on that line for which the absolute value of the difference of the distances OQ arid BQ for points Q on this line is maximized (uniquely) at P.
ð 4740: Let f(x) = x n + a 1 x n-1 + a 2 x n-2 + a 3 x n-3 + · + a n-1 x + a n be a polynomial with integer coefficient. Suppose there are four distinct integers a, b, c. and d such that f(a) = f(b) = f(c)= f(d)= 8. Can there be an integer k such that f(k) = 3 ?
ð 4741: There are six points in space, no three on the same line. Each pair of points is connected by a line segment. Each segment is colored black or white. Show that among the triangles formed by these segments there is always at least one with all of its sides the same color.
ð 4742: Let A, B and C be three towns, each pair connected by a network of roads. There are 82 routes from A to B, including those via A to C, including those via C and 62 routes from B to C including those via A. Also there are less than 300 routes from A to C, including those via B. How many routes are there?
Solutions
ð 4706: Prove:
Solving this system of equations gives A = -1/8 , B = ¹ , C = 1/8 , and D = ¹.
Now
...............(1)
On the other hand, the last integral in (1) can be evaluated using the residue technique, as it is well known [1]. In fact, the polynomial x4 + 4 has no real zeros and the difference between the degrees of the denominator and the numerator is 4. Then, according to a result on contour integrals (Complex Variables with Applications by A. David Wunsch, 1994, Addison-Wesley, p.344), we have:
where Rjj = 1,2 are the residues of f(z) = 1/(x 4+4) at their poles 1 +i and -1 + i in the upper half plane Im(z) > 0, i.e.,
