Vol. 11 No 1 - Pi Mu Epsilon

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48 CLAYTON W. DODGE
971. Proposed by Richard I. Hess, Rancho Palos Verdes, California.
Find an integer-sided obtuse triangle with acute angles in the ratio 7/5.
PROBLEM DEPARTMENT 49
In the array below place sixteen digits to form eight not necessarily distinct
squares without using the digit zero. The answer is unique.
972. Proposed by PaulS. Bruckman, Berkeley, California.
Given three non-collinear points A, B, and C in the complex plane, determine I,
the incenter of triangle ABC as a "weighted average" of these points.
973. Proposed by Ayoub B. Ayoub, Pennsylvania State University, Abington,
Pennsylvania.
Prove that an+l = 2an + an-b given that ao = 0 and
974. Proposed by Kenichiro Kashihara, Sagamihara, Kanagawa, Japan.
Given any positive integer n, the Pseudo-Smarandache function Z(n) is the smallest
integer m such that n divides
m
a) Solve the Diophantine equation Z(x) = 8.
b) Show that for any positive integer p the equation Z(x) = p has solutions.
*c) Show that the equation Z(x) = Z(x + 1) has no solutions.
*d) Show that for any given positive integer r there exists an integers such that
the absolute value of Z(s) - Z(s + 1) is greater than r.
975. Proposed by Doru Popescu Anastasiu, Liceul Radu Greceanu, Slatina, Romania.
For any given fixed positive integer n, determine the positive integers x1, x 2
, •• • ,
Xn such that
X1 + 2 ( X1 + X2 ) + 3 ( X1 + X2 + X3) + ... + n(xl + X2 + ... + Xn) =
2n 3 + 3n 2 + 7n
976. Proposed by Rajindar S. Luthar, University of Wisconsin Center, Janesville,
Wisconsin.
If x + y + z + t = 1r, prove that
tan(x+y)tan(z+t) > 27cotxcotycotzcott.
977. Proposed by Rajindar S. Luthar, University of Wisconsin Center, Janesville,
Wisconsin.
If A, B, and C are the angles of a triangle, then prove that
cot(A/2) + cot(B /2) +cot( C /2) > cot(A) + cot(B) +cot( C).
978. Proposed by Richard I. Hess, Rancho Palos Verdes, California.
6
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*979. Proposed by Murray S. Klamkin, University of Alberta, Edmonton, Alberta,
Canada.
Dedicated to Professor M. V. Subbarao on the occasion of his 78th birthday.
Do there exist an infinite number of triples of consecutive positive integers such
that one of them is prime, another is a product of two primes, and the third is a
product of three primes. Two such examples are 6, 7, 8 and 77, 78, 79.
Correction. Frank Battles of Massachusetts Maritime Academy, Buzzards Bay,
Massachusetts, reported that the web address given in the solution to Problem 914
[Fall 1998, page 744] should have a dot "." instead of a slash "/" between "index"
and "html." He further stated that the alphametic solver at that website could solve
neither Problem 745 (ENID+ DID = DIN E) nor 940. The correct address is
http:/ jwww.ceng.metu.edu.tr/"'selcuk/alphametic/index.html.
Solutions.
924. [Fall1997, Fall1998] Proposed by George Tsapakidis, Agrino, Greece.
Find an interior point of a triangle so that its projections on the sides of the
triangle are the vertices of an equilateral triangle.
III. Comment and solution by Paul Yiu, Florida Atlantic University, Boca Raton,
Florida. Solution I, given by W. H. Peirce, computes the barycentric coordinates
of the desired point P as
(
a 2 sin(A ± 60°) b 2 sin(B ± 60°) c 2 sin( C ± 60°))
D± sin(A) ' D± sin( B) ' D± sin( C) '
where D± = a 2 + b 2 + c 2 ± 4-/3H, H being the area of the triangle. If we homogenize
these coordinates, we find surprisingly simple descriptions of the point P. There
are two such points P ±, with homogeneous barycentric coordinates a sin( A ± 60°) :
bsin(B ± 60°) : csin(C ± 60°). Then one recognizes P± as the isogonal conjugates of
the points
a b c
F± = . ·-----
sin(A ± 60°) · sin(B ± 60°) · sin(C ± 60°)'
which are the isogonal centers of the triangle. [If one erects equilateral triangles
outwardly (respectively inwardly) on the sides of triangle ABC, the three lines from
each vertex of triangle ABC to the third vertex of the equilateral triangle erected on
the opposite side are concurrent at the isogonal center F + (respectively F _)].