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MM 1410. [1992:348; 1993:341] Proposed by Seung-Jin Bang, Seoul, Republic of
Korea.
Prove that ë n1/3
+ (n+1)1/3û =
ë (8n+3)1/3û for
every positive integer n.
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MM 1420. [1993:126; 1994:148] Proposed by Cristian Turcu, London, England.
If á, â, ã, ä are real numbers, n is an
odd integer, cos á + cos â + cos ã+ cos ä = 0, sin á + sin â + sin ã + sin ä
= 0, prove that cos ná + cos nâ + cos nã+ cos nä = 0 and sin ná + sin nâ +
sin nã + sin nä =0
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MM 1422. [1993:127; 1994:151] Proposed by David Callan, University of
Wisconsin, Madison, Wisconsin.
Let A and B be r ´
n matrices with r £ n and rank A = r. Suppose that there is a
nonzero constant k such that the determinant of every one of the C(n, r)
r-square submatrices of B is k times the corresponding subdeterminant of A.
Show that A and B have the same row space.
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MM 1425. [1993:192; 1994:227] Proposed by William P. Wardlaw, U.S..
Naval Academy, Annapolis, Maryland.
Let p be a prime number and let A
be a (p-1) ´ (p-1) matrix over the field of rational numbers
such that Ap = I ¹ A . Show that
if f(x) is any nonzero polynomial with rational coefficients and degree less
than p-1, then f(A) is nonsingular.
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MM 1465. [1994:68; 1995:67] Proposed by Steven W. Knox, University of
Illinois, Urbana, Illinois.
It is well known that given any
coloring of the plane by two colors, there exists an equilateral triangle
with monochromatic vertices. As a generalization, show that given any
two-coloring of the plane and any triangle T, there exists a triangle similar
to T with monochromatic vertices.
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MM 1469. Proposed by Roger Izard,
Dallas, Texas.
In triangle EDB, A and C lie on EB and
ED, respectively; CB and DA intersect at F. Also,
DC×
AB =4, and Area(Ä CFA) + Area(Ä DFB) = 14/5.
Prove that DEB is a right angle.
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MM 1471. Proposed by Bill
Correll, Jr., student, Denison University, Granville, Ohio.
For positive integers n, define f(n) to
be the smallest positive integer j such that
where  denotes the floor function. Let
Prove that
(i) {f(n)}n³
1 consists of all integers
except the perfect squares, and
(ii) f(n) + c(n) = 2n.
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MM 1472. Proposed by Erhan Gurel,
Middle East Technical University, Ankara, Turkey.
Let Q denote an arbitrary convex
quadrilateral inscribed in a fixed circle, and let F(Q) be the set of
inscribed convex quadrilaterals whose sides are parallel to those of Q. Prove
that the quadrilateral in F(Q) of maximum area is the one whose diagonals are
perpendicular to one another.
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MM 1473. Proposed by Gerald A.
Heuer, Concordia College, Moorhead, Minnesota.
Let Bn(z) denote the
determinant of the (2n+1)´ (2n+1) matrix whose entries are given by b(1,j)
= 1 for all j, b(j,j) = 2 for j=2,3,…, 2n+1, b(i+1,n+i+1) = b(n+i+1,i) = -z
for i=1,2,…,n, and all other b(i,j)=0. For example
Find all (complex) roots of Bn(z).
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MM 1491. [1996:66; 1997:68] Proposed by Wu Wei Chao,He Nan Normal
University, Xin Xiang City, He Nan Province, China.
Find all functions f: R® R such
that
(i) f(x + f(y) + y×
f(x))= y + f(x) + x× f(y) for all x,y in R;
(ii) {f(x)/x: x Î
R, x ¹ 0} is a finite set.
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MM 1509. Proposed by David
Callan, University of Wisconsin, Madison, Wisconsin.
Let A be a real n´
n matrix satisfying (i) each row sums to 1; (ii) each entry immediately above
the main diagonal is 1/2; (iii) all other entries above the main diagonal are
0. Prove that the permanent of A is 1/2n-1.
(The permanent of a matrix is . Thus, it is
similar in form to the determinant:
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MM 1512. Proposed by Arthur L.
Holshouser, Charlotte, North Carolina, and Benjamin G. Klein, Davidson College,
Davinson, North Carolina.
Let R be a commutative ring such that x3
= x for every x Î R. For x, y Î R, let F(x, y) =
xy-x2y-xy2-x2y2. If F(a,b) = a
and F(a, c) = b, prove that F(a, c) = a.
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MM 1528. [1997:224; 1998:230] Proposed by Florin S. Pirvanescu, Slatina,
Romania.
Let M be a point in the interior of
convex polygon A1A2...An. If dk
is the distance from M to AkAk+1 (An+1 = A1),
show that
(d1
+ d2)(d2 + d3) × × ×
(dn + d1) £ 2n × cosn
(ð/n) × MA1× MA2× × × × × MAn
and determine when equality holds.
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MM 1533. Proposed by Joaquin
Gomez Rey, I.B., " Luis Bunuel", Alcorcon, Madrid, Spain.
Solve the recurrence relation
in terms of á0.
Solution (PDF)
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MM 1557. [1998:316; 1999:329] Proposed by Peter Y.Woo, Biola University, La
Miranda, California.
Let PQ be the diameter of a circle with
A and B two distinct points on the circle on the same side of PQ. Let C be
the intersection of the tangents to the circle at A and B. Let the tangent to
the circle at Q meet PA, PB, and PC at A', B', and C', respectively. Prove
that C' is the midpoint of A'B'.
Solution (PDF)
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MM 1572. [1999:149; 2000:160] Proposed by Western Maryland College Problems
Group, Westminster, Maryland.
Let b0=1 and b1
satisfy 0< b1 < 1. For n ³ 1,
define bn+1 by
 .
Show that (bn)(n³
0) converges, and compute its
limit in terms of b1.
Solution (PDF)
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MM 1573. [1999:149; 2000:161] Proposed by Jiro Fukuta, Proffesor
Emeritus,Gifu University, Gifu-ken,Japan.
Given
Ä ABC, let AD be a cevian to the side BC, and let E be on segment AD. The
circumcircle of Ä ACD intercects the line BE at points M and N, and the
circumcircle of Ä ABD intercects the line CE at points P and Q. Prove that
the points M, N, P, and Q lie on a common circle and its center is on the
line perpendicular to the side BC at the point D.
Solution (PDF)
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for Euler's 
totient function,
evaluate
.