Response to: Problem 11479: Average Angular ...

Response to: Problem 11479: Average Angular Swing in a Poncelet Trajectory The American Mathematical Monthly Volume 117, Number 1, January 2010, p. 87 Posed by: Vitaly Stakhovsky National Center for Biotechnological Information Bethesda, MD

From: J. A. Grzesik Allwave Corporation 3860 Del Amo Blvd., #404 Torrance, CA 90503 (310) 793-9620 ext 104 [email protected] [email protected] May 11, 2010

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J. A. Grzesik

1

amm #11479: average angular swing in a poncelet trajectory

2

Solution Pr´ ecis

We establish first the geometric relationships surrounding one single tangent throw, with due attention paid to the question of whether the accompanying angular increment ωj does or does not exceed π . That question is answered simply through momentary recourse to a complex representation, the answer being affirmative whenever this tangent actually feels the need to sweep across the master origin O , a situation characterized by a well-defined sense for the perpendicular linking tangent to origin. And, while they may be gotten analytically, all of these formulae, relying as they do on various apropos angles, are open to immediate geometric interpretation. In essence the single-throw apparatus solves the entire problem in nuce. This is so because, on the one hand, we can build around it an outright numerical simulation based upon tangent throws iterated in computer code, while, on the other, we can invoke the Strong Law of Large Numbers (SLoLN) so as to replace the limiting angular average by its expectation against a continuous distribution, suitably weighted, of Poncelet trajectory (PT) points around the outer rim. The SLoLN formula thus derived, albeit fully explicit, entails nevertheless a cumbersome definite integral which we leave in its native form. Its credibility, however, is reaffirmed through numerical quadrature whose outcomes agree, by way of corroboration, with their counterparts gotten through repeated tangent throw simulation. In general, any given PT is unlikely to close. At the same time, privileged geometric circumstances do exist for any one of which all of its PT cycles close and, by virtue of Poncelet’s great closure porism, do so in the same number of steps N , regardless of start-up angle on the outer rim. Such closure opportunities have an immense bearing on the outcome of this problem because, on the one hand, it is clear, after a moment’s thought, that the SLoLN formula remains unimpaired, while, on the other, it likewise becomes clear that the mandated average is obliged to ring in at the simple value Ω(C, c, P ) = 1 / N .

(1)

And, indeed, we are able to verify this check, in its numerical embodiment, for preferred classes of triangles {N = 3} and quadrilaterals {N = 4} (the so-called Euler-Fuss problem).1 More about that anon.

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Analytic Result

We state an analytic recipe for Ω(C, c, P ) as a sequence of formulae involving angles ϕj−1 , ϕj , α , and β required in the specification of any given tangent throw as made evident in Figure 1 below. Angle ϕj−1 , in particular, gauges the initial azimuth wrt O along outer rim C at the pedestal Pj−1 of any given tangent throw j ≥ 1 . We thus write: q R 2 + d 2 − 2 R d cos ϕj−1 (2) D = (distance between o and Pj−1 ), 1

The values thus obtained from Eq. (1) are consistent with an absolute upper bound Ω(C, c, P ) ≤ 1/2 as set by Eq. (11) below.

J. A. Grzesik

amm #11479: average angular swing in a poncelet trajectory

q

T =

D2 − r2

3

(3)

(distance between Pj−1 and the point of contact between the tangent and inner circle c ), 

β = arcsin

r D



(4)

(tangent to tangent half-angle), and, by way of preparation, compute an SLoLN distribution normalizer W in the form2 Z π dϕ W = . (5) T 0 With this toolkit in hand, we then get Ω(C, c, P ) =

Z π β

1 1 − 2 πW

0

T

dϕ .

(6)

Formulae (2)-(6) respond to both problem parts (a) and (b) in one fell swoop, in the sense of exhibiting an explicit limit for Ω(C, c, P ) , wholly independent of PT starting point P = P 0 . While there seems little hope that we could ever bring (6) into anything resembling a closed form,3 normalizer W from (5) does admit a restatement as W = q

Z π/2

2 ( R − | d | )2 − r2

0

dϑ r

1+

4R | d | ( R−| d | )2 − r2

(7) sin2 ϑ

and thus a natural relabeling W = q

2

F ( π/2 , k )

(8)

( R − | d | )2 − r2

in terms of a complete elliptic integral of the first kind F with a purely imaginary modulus s

k = ±2i

R|d| ( R − | d | )2 − r2

.

(9)

Furthermore, despite the lack of hope for a closed-form reduction, we can likewise bring the integral on the right in (6) into a form similar to (7), viz., Z π β 0

2

T

dϕ = q

2 ( R − | d | )2 − r2

  q Z π/2 arcsin r ÷ ( R − | d | )2 + 4R | d | sin2 ϑ r dϑ , 0

1+

4R | d | ( R−| d | )2 − r2

(10)

sin2 ϑ

Naturally we dispense at this point with any reference to Poncelet throw index j − 1 . Despite repeated entreaties, we were unable to coax from MATHEMATICA any sort of useful contribution in this regard. 3

J. A. Grzesik

amm #11479: average angular swing in a poncelet trajectory

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awkward as this may seem. And finally, on collating Eqs. (6) through (10), we arrive at 

Ω(C, c, P ) =

3

1 1 − 2 πF

Z π/2 arcsin

r÷ r

0

q

( R − | d | )2 + 4R | d | sin2 ϑ 4R | d | ( R−| d | )2 − r2

1+



dϑ .

(11)

sin2 ϑ

Generic Tangent Throw Geometry

Equations (2) through (4) can simply be read off from Figure 1,4 which declares all relevant geometric attributes of a typical tangent throw.

y ζ

Pj -1 C

ϕ j -1

T β D

R

α

2arccos(η/R)

r O

η

o d

x

c

ϕj Pj Figure 1. Primitive Tangent Throw from Pj -1 to Pj 4

Evidently we require that d + r < R and d − r > − R or, more concisely, R > ± d + r (shown in the diagram is a configuration with its incircle offset d > 0 reckoned as positive wrt the background, ζ = x + i y co¨ ordinate system). We abide by the proposer’s preference by keeping all dimensions explicit even though, clearly, some notational advantage would accrue from a normalization across the board by outer radius R , a gesture conveyed by the formal replacement R → 1 .

J. A. Grzesik

amm #11479: average angular swing in a poncelet trajectory

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Also required is angle α separating the radii, respectively of lengths R and D , which emanate from centers O and o and converge upon throw pedestal Pj−1 . The law of sines informs us that 

α = arcsin

d sin ϕj−1 D



(12)

about which we note, for imminent use, that angle α is antisymmetric in its dependence upon ϕj−1 (cf. also Eq. (2)). Angular increment ωj clearly exceeds π in the event that tangent throw Pj−1 Pj is obliged to sweep past the global origin O , a particular situation captured in Figure 1 and gauged by the attribution there of a negative sign to perpendicular offset η . With the sign convention thus set for η , there is placed at our disposal but a single formula5   η (13) ωj = 2 arccos R for angular advance ωj propelling us into the next rim angle ϕj = ϕj−1 + ωj .

(14)

From Figure 1 the required offset parameter η readily follows as6 η = − R sin ( α − β ) ,

(15)

ωj = π − 2 β + 2 α .

(16)

whereupon

Now, in principle, one could evolve an entire Poncelet trajectory and, as a byproduct, evaluate the accompanying angular average Ω(C, c, P ) , merely by iterating Eqs. (13)-(16). But this approach, simpleminded at its core, founders on jagged algebraic rocks even on its first step. And so, in order to establish a formula such as (11), we must accept a shift in viewpoint. At the same time, with vast computing power

5 6

The inverse cosine is taken here to lie on its principal branch, i.e., − π < arccos( η/R ) ≤ π . These purely geometric results can be recovered by following a somewhat more analytic path. This latter begins by setting ζ j−1 = R e i ϕj−1 ζ j = ζ j−1 + L e i {ϕj−1 + π + α − β }

and then insisting that | ζ j | = R . Out of this there easily emerges, besides the trivial solution L = 0 , the required value L = 2 R cos α − β




,

entirely consistent with the length of Pj−1 Pj as that is depicted on Figure 1. Moreover, we can similarly write for η , with all due attention paid as to its pivotal sign,   iL ζ j + ζ j−1 η = . 2 ζ j − ζ j−1 A step or two of algebra suffices then to recover (15).

J. A. Grzesik

amm #11479: average angular swing in a poncelet trajectory

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now always close to hand, we can, and indeed have, implemented the iteration of (13)-(16) as computer code tanthrow 06 written in FORTRAN77. And this code should not be dismissed too lightly, since it allows us to check the numerical quadratures of (11) which it runs in parallel, as well as the claimed invariance of Ω(C, c, P ) against trajectory startup P 0 .

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SLoLN to the Rescue!

Since direct iteration of (13)-(16) is simply not feasible in analytic terms, we are forced to seek refuge in the Strong Law of Large Numbers, which aims to replace the limiting average whereby Ω(C, c, P ) first presents itself by the mean of some governing continuous distribution. The required distribution is found in Reference 1 in the context of the invariant mapping, automatically compensating for arc length interval stretch/shrink, which the tangent throws induce from the outer circle C upon itself. From pp. 613-614 and Figure 1.7 there, one is led to consider the two tangent legs emanating from a fulcrum at the contact point with inner circle c , and to infer thereby that the weighting function should be taken as 1/T , a feature first revealed in Eq. (5) above. With note taken of (16) we thus write Ω(C, c, P ) =

Z π π − 2β + 2α −π

T

!

dϕ

÷

2π

Z π 1 −π

T

!

dϕ

(17)

and, because both β and T are symmetric in angle ϕ whereas α from (12) is antisymmetric, this latter condenses into ! ! Z π Z π 1 1 β Ω(C, c, P ) = − dϕ ÷ π dϕ , (18) 2 0 T 0 T which is the genesis of (5)-(6).

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Numerical Corroboration

So bold a step as (17)-(18) pleads for some measure of reassurance, which is provided when the outcome of computer iteration of (13)-(16) is compared against numerical, Gauss-Legendre quadrature (GLQ) applied to Eq. (11). This is done in Figure 2, which considers inner-circle radii having normalized values r/R = 0.1, 0.2, 0.3, and 0.4 , while at the same time normalized inner-center offset values d/R range between limits of tightest inner/outer circle approach set at d/R = ± ( 0.9 − r/R ) . The computer-generated iterations of (13)-(16) were undertaken with start-up rim angles arbitrarily taken at ϕ 0 = 30 0 , 60 0 , 90 0 , and 120 0 , involving 250,000 tangent throws for each ( r/R, d/R ) pair, without doubt an almost obscene instance of numerical overkill.7 At any rate, the four traces shown (and scrutiny of the supporting tabular output) confirm a harmonious agreement between (17)-(18) (sparse symbols) when confronted with its raw simulation counterpart (13)-(16) (solid, continuous lines). And besides, that agreement remains indifferent to the start-up value ϕ 0 , the seed, as it were, for the latter. Simply stated, the two calculation flavors mask each other to a point of virtual perfection. 7

Hey, (speaking strictly on behalf of the computer, of course) if you’ve got it, why not flaunt it?

J. A. Grzesik

amm #11479: average angular swing in a poncelet trajectory

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Normalized average angular increment Ω

Figure 2. Normalized Average Angular Increment Ω per Tangent Throw in a Poncelet Trajectory 0.48 0.46 0.44 0.42 0.4 0.38 0.36 0.34 0.32 0.3 -1

-0.8

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 Inner circle center displacement d/R from master center O

r/R=0.1, primitive average r/R=0.1, GLQ vis-a-vis Eq. (11) r/R=0.2, primitive average r/R=0.2, GLQ vis-a-vis Eq. (11)

0.8

1

r/R=0.3, primitive average r/R=0.3, GLQ vis-a-vis Eq. (11) r/R=0.4, primitive average r/R=0.4, GLQ vis-a-vis Eq. (11)

Figure 3. Throw Angle ω versus Tangent Pedestal ϕ 300 r/R=0.1, d/R=0.8 r/R=0.2, d/R=0.7 r/R=0.3, d/R=0.6 r/R=0.4, d/R=0.5

Throw angle ω [degrees]

250

200

150

100

50 0

30

60

90

120 150 180 210 240 Tangent pedestal ϕ [degrees]

270

300

330

360

J. A. Grzesik

amm #11479: average angular swing in a poncelet trajectory

8

The four traces on Figure 2 are perhaps not dramatic in appearance, but they do respect both an inherent positivity and the upper bound at 1/2 evident from Eq. (11). Moreover, all four dip as the inner/outer circle passage undergoes constriction and the opportunities for large angular advances ω are simply choked off. As already stated, and because of its antisymmetry against rim angle ϕ , any possible aggregate contribution to Ω from angle α in quadrature (17) evaporates during its transition into (18). Such, however, is certainly not the case as regards each throw angle ω on a local level, circumstance brought to vivid life in the four traces from Figure 3, these latter being the histories of ω against ϕ which ensue when inner circle o has been pushed toward its right-hand extreme in the illustration preceding. We see confirmed therein our expectation that the smaller the inner radius the larger the angles ω swept out by the tangents emerging from the R/10 gap in the general vicinity of ϕ = 0 0 . And then, on return to that narrows when finally ϕ → 360 0 − , angular increments ω are compelled to shrink down to an approximately common minimum attained at an essentially common location. Along the way, of course, angular increments ω in excess of 180 0 (i.e., η from (15) negative) are indeed encountered.

5.1

Closed Poncelet Trajectories at Levels N = 3 and N = 4

In general, the vertices of a Poncelet trajectory never quite repeat and, indeed, they populate the outer rim in a dense scatter (assertion (1.11) on p. 615 of Reference 1). Nevertheless there do exist certain privileged relationships between parameters r, d, and R for which such orbits will close and, by virtue of Poncelet’s great closure theorem, will do so in the same number of steps N (for all triplets r, d, and R in a category dedicated to the latter), regardless of inception angle ϕ 0 .8 The requisite relationships for categories N = 3 and N = 4 , attributed, respectively, to none other than Euler and to one of his disciples named Nicolaus Fuss,9 can be found in Reference 2,10 and read:

8

Since any such polygon will then visit just N rim points, it is not unreasonable to entertain a nagging suspicion as to how in the world one could possibly harness a continuous distribution, after the fashion of Eq. (17), and, in particular, expect (17) to validate (1). Such doubts are allayed, perhaps, by invoking the indifference of the N −legged trajectory closure to inception angle, and the patent fact that, when this indifference is exploited, each one of the N rim vertices individually explores the full, 0 < ϕ < 2 π range. Arguing, in some sense, in the opposite direction, one may note that the definition of Ω(C, c, P ) , first suggested as a limiting average, can, for closed Poncelet polygons, be drastically curtailed by allowing iteration index j to advance only up to N. Further advances of j in multiples of N, while technically permissible, are clearly redundant and superfluous. 9 Fuss was not only a disciple of Euler but also his secretary, friend, and son-in-law. This last rˆ ole seems to confirm the old saw that mathematical talent in Germany passed not from father to son, but, rather, from father to son-in-law. Well, OK, both Euler and Fuss were Swiss, German Swiss beyond a doubt, and both flourished in, of all places, the Russia of Catherine the Great (herself a German, one may note). But still...... History has not been overly kind to Fuss, a man of obvious ability eclipsed by the unsurpassed radiance of his master. 10 Heinrich D¨ orrie, at the head of his p. 192 in Reference 2, further asserts that Fuss had obtained formulae analogous to Eqs. (19) and (20) for pentagons, hexagons, heptagons, and octagons {N = 5, 6, 7, and 8}, no mean feat indeed. He, D¨ orrie, further indicates that these results can be traced from Nova Acta Academiae Scientiarum Imperialis Petropolitanae, Volume XIII, 1798. As a grateful beneficiary of Google Books, we have indeed done so and, following download, Fuss’ scholarly article, in Latin, no less, begins on p. 281 in its *.pdf incarnation (p. 166 of Nova Acta). The intrepid reader can then attempt to decipher this truly heroic work, or maybe even essay its reconstruction ab initio. Such efforts, while desirable, would tax our feeble powers and available time way beyond their elastic limit. One should perhaps also point out that D¨ orrie, in his lust for concision, makes a statement on the way to (19) which, while true, does not appear to be obvious a priori, but emerges instead only as part of the unfolding argument. The statement at issue, in D¨ orrie’s notation, reads DM = DA = DB , of which only the part DA = DB seems to be self-evident.

J. A. Grzesik

amm #11479: average angular swing in a poncelet trajectory

9

N=3 r = and

R2 − d2 2R

(19)

N=4 R2 − d2 r = r   . 2 R2 + d2

(20)

With inner radius r subordinated in this way to offset d , we again subject (13)-(16) to computer iteration and compare its output against that inferred from (11) when that latter is placed beneath the sway of Gauss-Legendre quadrature.11 The outcome of such comparison is revealed by Figure 4 to be a stunning success, with Eq. (1) accorded unconditional respect and GLQ values shadowing essentially to perfection their primitive average analogues.12 At its base, of course, Figure 4 offers testimony to the truth of, and pays homage to Poncelet’s great closure porism. Figure 4. Average Angular Increment Ω for Closed Poncelet Trajectories at Levels N = 3 and N = 4 0.34

Normalized average angular increment Ω

0.33 0.32 0.31 0.3 0.29 0.28 0.27 0.26 0.25 0.24 -0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Inner circle center displacement d/R from master center O N = 3, primitive average N = 3, GLQ vis-a-vis Eq. (11) 11

N = 4, primitive average N = 4, GLQ vis-a-vis Eq. (11)

As was the case beforehand, in connection with Figure 2, the hopes for any sort of closed-form quadrature in (11) are hardly improved by having r and d yoked together by either of (19) or (20). 12 In the computations reported on Figure 4, the respective ranges for inner-circle offset d/R are ± 0.645 {N = 3} and ± 0.60456493 . . . {N = 4}. The corresponding incircle radii r/R register then at the values r/R = 0.2919875 {N = 3} and r/R = 0.38394748 . . . {N = 4}. Inner/outer circle clearances neck down dramatically at these extremes, with tangent trajectories being obliged to thread their way across straits no wider than R − | d | − r = 0.0630125 R {N = 3} and R − | d | − r = 0.011487585 . . . R {N = 4}.

J. A. Grzesik

amm #11479: average angular swing in a poncelet trajectory

10

It is of course quite remarkable that, under the aegis of Eqs. (19) and (20) (and all of their higher-order analogues adumbrated beneath Footnote 10), the ratio of integrals on the right-hand side of (11) should remain constant qua function of offset d . A direct analytic proof of this fact would surely be preferred, but, failing that, one can view Poncelet’s porism as an oblique underpinning. One further verifies that both (19) and (20) permit offset d to rise all the way up to R , with incircle r collapsing then down to zero. But, in spite of the extreme gap constriction thus allowed, angular average Ω retains a constant value to the very end, remaining immune, quite incredibly, to the decline on its flanks as previously encountered in Figure 2.

6

References

1. Jonathan L. King, Three Problems in Search of a Measure, The American Mathematical Monthly, Volume 101, Number 7, August-September 1994, pp. 609-628. 2. Heinrich D¨orrie, 100 Great Problems of Elementary Mathematics, Dover Publications, Inc., New York, 1965, Problem 39: Fuss’ Problem of the Chord-Tangent Quadrilateral, pp. 188-193.

PROBLEMS AND SOLUTIONS Edited by Gerald A. Edgar, Doug Hensley, Douglas B. West with the collaboration of Itshak Borosh, Paul Bracken, Ezra A. Brown, Randall Dougherty, Tam´as Erd´elyi, Zachary Franco, Christian Friesen, Ira M. Gessel, L´aszl´o Lipt´ak, Frederick W. Luttmann, Vania Mascioni, Frank B. Miles, Bogdan Petrenko, Richard Pfiefer, Cecil C. Rousseau, Leonard Smiley, Kenneth Stolarsky, Richard Stong, Walter Stromquist, Daniel Ullman, Charles Vanden Eynden, Sam Vandervelde, and Fuzhen Zhang. Proposed problems and solutions should be sent in duplicate to the MONTHLY problems address on the inside front cover. Submitted solutions should arrive at that address before May 31, 2010. Additional information, such as generalizations and references, is welcome. The problem number and the solver’s name and address should appear on each solution. An asterisk (*) after the number of a problem or a part of a problem indicates that no solution is currently available.

PROBLEMS 11474. Proposed by Cezar Lupu, student, University of Bucharest, Bucharest, Romania, and Valentin Vornicu, Aops-MathLinks forum, San Diego, CA. Show that when x, y, and z are greater than 1, 0(x)x

2 +2yz

0(y) y

2 +2zx

+ 0(z)z

2 +2x y

≥ (0(x)0(y)0(z))x y+yz+zx .

¨ 11475. Proposed by Omer Pk E˘g1ecio˘glu, University of California Santa Barbara, Santa Barbara, CA. Let h k = j =1 j , and let Dn be the determinant of the (n + 1) × (n + 1) Hankel matrix with (i, j) entry h i+ j +1 for 0 ≤ i, j ≤ n. (Thus, D1 = −5/12 and D2 = 1/216.) Show that for n ≥ 1, Qn n 4 X (−1) j (n + j + 1)!(n + 1)h j +1 i=1 i! . Dn = Q2n+1 · j!( j + 1)!(n − j)! i! j =0 i=1

11476. Proposed by Panagiote Ligouras, “Leonardo da Vinci” High School, Noci, Italy. Let a, b, and c be the side-lengths of a triangle, and let r be its inradius. Show a 2 bc b2 ca c2 ab + + ≥ 18r 2 . (b + c)(b + c − a) (c + a)(c + a − b) (a + b)(a + b − c)

11477. Proposed by Antonio Gonz´alez, Universidad de Sevilla, Seville, Spain, and Jos´e Heber Nieto, Universidad del Zulia, Maracaibo, Venezuela. Several boxes sit in a row, numbered from 0 on the left to n on the right. A frog hops from box to box, starting at time 0 in box 0. If at time t, the frog is in box k, it hops one box to the left with probability k/n and one box to the right with probability 1 − k/n. Let pt (k) be the probability that the frog launches its (t + 1)th hop from box k. Find limi→∞ p2i (k) and limi→∞ p2i+1 (k). doi:10.4169/000298910X475032

86

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 °

11478. Proposed by Marius Cavachi, “Ovidius” University of Constanta, Constanta, Romania. Let K be a field of characteristic zero, and let f and g be relatively prime polynomials in K [x] with deg(g) < deg( f ). Suppose that for infinitely many λ in K there is a sublist of the roots of f + λg (counting multiplicity) that sums to 0. Show that deg(g) < deg( f ) − 1 and that the sum of all the roots of f (again counting multiplicity) is 0. 11479. Proposed by Vitaly Stakhovsky, National Center for Biotechnological Information, Bethesda, MD. Two circles are given. The larger circle C has center O and radius R. The smaller circle c is contained in the interior of C, and has center o and radius r . Given an initial point P on C, we construct a sequence hPk i (the Poncelet trajectory for C and c starting at P) of points on C: Put P0 = P, and for j ≥ 1, let P j be the point on C to the right of o as seen from P j −1 on a line through P j −1 and tangent to c. For j ≥ 1, let ω j be the radian measure of the angle counterclockwise along C from P j −1 to P j . Let k 1 X ωj. k→∞ 2πk j =1

Ä(C, c, P) = lim

(a) Show that Ä(C, c, P) exists for all allowed choices of C, c, and P, and that it is independent of P. (b) Find a formula for Ä(C, c, P) in terms of r , R, and the distance d between O and o. 11480. Proposed by Omran Kouba, Higher Institute for Applied Sciences and Technology, Damascus, Syria. Let a, b, and c be the lengths of the sides opposite vertices A, B, and C, respectively, in a nonobtuse triangle. Let h a , h b , and h c be the corresponding lengths of the altitudes. Show that µ ¶2 µ ¶2 µ ¶2 hb hc ha 9 + + ≥ , a b c 4 and determine the cases of equality.

SOLUTIONS Powerful Polynomials 11348 [2008, 262]. Proposed by Richard P. Stanley, Massachusetts Institute of Technology, Cambridge, MA. A polynomial f over a field K is powerful if every irreducible factor of f has multiplicity at least 2. When q is a prime or a power of a prime, let Pq (n) denote the number of monic powerful polynomials of degree n over the finite field Fq . Show that for n ≥ 2, Pq (n) = q bn/2c + q bn/2c−1 − q b(n−1)/3c . Solution by Richard Stong, Center for Communications Research, San Diego, CA. Let Aq (n) and Sq (n) be the numbers of monic and monic square-free polynomials of degree n over Fq , respectively. Introduce the ordinary generating functions:

Aq (x) = January 2010]

∞ X n=0

n

Aq (n)x ,

Pq (x) =

∞ X

n

Pq (n)x ,

n=0

PROBLEMS AND SOLUTIONS

Sq (x) =

∞ X

Sq (n)x n .

n=0

87

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Normalized average angular increment Ω

Figure 5. Normalized Average Angular Increment Ω per Tangent Throw in a Poncelet Trajectory: Primitive Throw, SLoLN, and Richard Stong Methods Compared 0.48 0.46 0.44 0.42 0.4 0.38 0.36 0.34 0.32 0.3 -1

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Inner circle center displacement d/R from master center O Thu Jun 09 14:26:37 2011 r/R=0.1, GLQ vis-a-vis I/J (RS) r/R=0.1, GLQ vis-a-vis Eq. (11) r/R=0.1, primitive average r/R=0.2, GLQ vis-a-vis I/J (RS) r/R=0.2, GLQ vis-a-vis Eq. (11) r/R=0.2, primitive average

r/R=0.3, GLQ vis-a-vis I/J (RS) r/R=0.3, GLQ vis-a-vis Eq. (11) r/R=0.3, primitive average r/R=0.4, GLQ vis-a-vis I/J (RS) r/R=0.4, GLQ vis-a-vis Eq. (11) r/R=0.4, primitive average

1