Problem ideas by Stanley Rabinowitz at a fixed rate of speed but is intelligent and can alter its trajectory. What path should the interceptor mis- Problem 1. sile follow so as to hit the rocket in the minimum time? Problem E3144 [1] shows that given a set A in a Is the path always a straight line? topological space, at most 34 sets can be generated us- ing the operations {complement, interior, boundary}. Problem 6. Moser [2] shows that at most 13 sets can be generated In an ellipse, in which direction is the average from {closure, interior, union}. (See also [4].) Kura- chord length largest? towski [3] has shown that at most 14 sets can be gen- erated from {closure, interior, complement}. Generate Problem 7. and solve other problems of this nature. Let Fn denote the n-th Fibonacci number and Ln REFERENCES denote the n-th Lucas number. For which n do Fn and [1] Jes´us Ferrer, “Solution to Problem E3144 (proposed Ln have the same period (modulo n)? by Edwin Buchman)”, American Mathematical Monthly. 95(1988)353. Problem 8. [2] L. E. Moser, “Closure, interior, and union in finite Find the number of divisors of n!. topological spaces”, Colloq. Math. 38(1977)41– 51. Problem 9. [3] Kuratowski, “Sur l’op´eration A¯ de l’analyses situs”, Can a with sides of lengths a, b, c have Fund. Math. 3(1922)182–199. three concurrent cevians of lengths a, b, c, respectively? [4] Arthur Smith, “Problem 5996”, American Mathe- not respectively? matical Monthly. 81(1974)1034. Problem 10. Problem 2. A triangle with sides of lengths a, b, c has three Chang [1] showed that the sequence an satisfying concurrent cevians of lengths x, y, z, respectively. Find 3 3 2 the recurrence n an − (34n − 51n +27n − 5)an−1 + a symmetrical relationship between a, b, c and x, y, z; 3 (n − 1) an − 2=0with initial conditions a0 =1,a1 = something like 1/a +1/b +1/c =1/x +1/y +1/z. 5, consists only of . Find other cubics for the coefficient of an−1 that have this property. Problem 11. REFERENCES Let A and B be two points on diameter CD of a [1] Derek K. Chang, “Recurrence Relations and circle, γ, with A between B and C. With A as center Sequences”, Utilitas Mathematica. 32(1987)95– and radius AC, draw a circle O internally tangent to 99. γ.IfP is the center of a circle tangent externally to O and internally to γ, and if furthermore, B is the center Problem 3. of γ, then it is easy to show that the locus of P is an Let Fn denote the n-th Fibonacci number and Ln ellipse, since PA+PB = AC+CD.Onthe other hand, denote the n-th Lucas number. Express Ln in terms of suppose O is instead an ellipse tangent internally to γ Fn (only). with points A and B as foci. Now, what is the locus of point P , the center of a circle tangent externally to O Problem 4. and internally to γ? Let P beapoint inside ABC and draw the three cevians from the vertices to the opposite sides through Problem 12. P .How many paths are there from A to B such that Investigate equilateral hexagons (or pentagons) in- no segment is traversed more than once? such that no scribed in ABC with one edge parallel to BC. vertex is gone through more than once? Generalize to a simplex in En. Problem 13. Let P be apoint inside ABC. Consider the Problem 5. proposition: “Let X, Y , Z be the ‘notable point’ in- Points A and B are on a horizontal line. A rocket side BPC, CPA, AP B, respectively. Extend from A is launched upwards toward B following a PX, PY, PZ to meet BC, CA, AB at D, E, F re- parabolic trajectory due to the normal laws of grav- spectively. Then AD, BE, CF concur.” Prove that ity and Newtonian physics. Assume the Earth is flat. this proposition is true when ‘notable point’ is replaced At the same time that this rocket is launched, an inter- by (a) , (b) incenter, (c) symmedian point, (d) ceptor missile is launched from point B which travels isotomic conjugate of (b) or (c).

Page 1 Problem 14. Problem 22. Let P be apoint inside non- ABC. Are there any positive integers, n>1, such that Circles inscribed in triangles PBC, PCA, PAB touch n2 starts with n in the decimal system. BC, CA, AB at points D, E, F respectively. Find all points P such that AD, BE, CF concur. Problem 23. Devise an alphametic to go along with the proverb Problem 15. “Money is the root of all evil”. For example, something like √ Let P be afocus of an ellipse. It is known that 3 2 for all chords passing through P ,ifP divides the chord MONEY = ALLEVIL. into two pieces of lengths a and b, then 1/a +1/b is a constant. Suppose a convex body in the plane has Problem 24. twopoints with this property. Must that body be an Evaluate or simplify ellipse? Generalize to higher dimensions. What can be n n said about convex bodies that have one point with this . k property? k=0

Problem 25. Problem 16. If f(x, y, z)=x + y + z − xyz, then it is known Investigate properties of a circle that cuts each side that f(tan A, tan B,tan C) ≡ 0 whenever A, B, C are of a triangle at the same . the of a triangle. Find a polynomial, g(x, y, z), such that g(sin A, sin B,sin C) ≡ 0 when A, B, C are Problem 17. the angles of a triangle. Do the same thing for cos. Let D be apoint on side BC of ABC. Let C1 be the incircle of ABD and let C2 be the incircle of Problem 26. ADC. Let C be the incircle of ABC. Determine Number an n×n square with the integers from 1 to when C is orthogonal to both C1 and C2. What if C is n2 such that two consecutive integers (1 and n2 should just orthogonal to C1? be considered consecutive) fall as far apart as possible (taxicab metric). Problem 18. Characterize polynomials all of whose derivatives Problem 27. (and itself) are unimodal. A polynomial is unimodal if Find a magic square on an n×n torus consisting of it’s coefficients first increase, then decrease. the integers from 1 to n2 such that the numbers on all straight lines through n numbers sum to the magic con- Problem 19. stant. In other words, the opposite sides of the square Find numbers that are 10 times their number of are to be identified. divisors. For example, 180 has 18 divisors. Generalize. Problem 28. Let P be inside an ellipse and let R beavariable Problem 20. point on the ellipse. When does the radial distance, { } Find n such that any subdivision of 1, 2,...,n PR,have extrema with R not being at a vertex? What into two pieces contains a piece containing an n-term is the average of the radial distances? For related in- Fibonacci sequence (second-order linear recurrence). formation, see [1]. REFERENCE Problem 21. [1] Ali R. Amir-Mo´ez, “The Focal Distance of a Conic You have n pairs of socks. Each pair is colored Section”, Pi Mu Epsilon Journal. 8(1986)246– with a different color. The colors are numbered from 1 249. to n. The colors are not completely distinguishable. To your inexperienced eye, color k looks exactly like colors Problem 29. k ± 1. The socks are mixed up and you proceed to pull Foratriangle with inradius r and circumradius R, ≥ { R } { R } out pairs, pairing two socks if they appear to have the we have R 2r or inf r =2. Find inf r where same color. What is the probability that all the socks the inf is taken over all quadrilaterals. In this case, r will get paired up in this manner (i.e. that you are not is the radius of the largest circle contained within the left with a sock that does not match any other sock left quadrilateral and R is the radius of the smallest circle over)? containing the quadrilateral.

Page 2 Problem 30. Rationalize this formula. Let s be the side of an inscribed REFERENCE in ABC. Find the minimum and maximum values for [1] V. A. Krechmar, A Problem Book in Algebra. Mir s. Let S be the side of an equilateral triangle circum- Publishers. Moscow: 1974. { S } scribed about ABC. Find inf s . Problem 31. Problem 38. Find the of the largest/smallest equilateral Is there an explicit formula for the finite continued triangle circumscribing a 3-4-5 right triangle. fraction 1 1 1 1 ... ? Problem 32. 1+ 2+ 3+ n Let P be apoint inside A1A2A3. Line segments PA , PA , PA divide A A A into three triangles. If not, can it be expressed rationally in terms of Hn, 1 2 3 1 2 3 1 1 1 the n-th harmonic number? (Hn = + + ···+ .) Let Ti be the triangle opposite vertex Ai.Investigate 1 2 n REFERENCES properties of the lines from Ai to the orthocenter of Ti. What happens if we replace “orthocenter” by “incen- [1] Stanley Rabinowitz, “Problem E3264”, American ter” or “centroid”? Mathematical Monthly. 95(1988)352. [2] B. Buchberger, G. E. Collins, and R. Loos, Com- Problem 33. puter Algebra: Symbolic and Algebraic Compu- Let P beapoint inside ABC. Cevians AD, tation, second edition. Springer-Verlag. New BE, CF all pass through P . Let C1 be the incircle York: 1983. of AP F . Let C2 be the incircle of BPD. Let C3 [3] J. C. Lafon, Summation in Finite Terms, pages 71– be the incircle of CPE.Investigate these three cir- 77 in [2]. cles. Can they all have the same radius? [4] M. Karr, “Summation in Finite Terms”, Journal of the Association for Computing Machinery. Problem 34. 28(1981)305–350. For which n is 10n a non-trivial binomial coeffi- [5] R. W. Gosper, jr., “Decision Procedure for In- cient? definite Hypergeometric Summation”, Proceed- Problem 35. ings of the National Academy of Sciences of the When are binomial coefficients perfect squares? United States of America. 75(1978)40–42. triangular numbers? Fibonacci numbers? Problem 39. Problem 36. Find the number of vertices of the smallest poly- An arbelos is formed from semicircles with radii a, tope in 4-space that has the 5 Platonic solids as facets. b, and a + b along a horizontal line segment of length 2a +2b. The semicircles are drawn downward and the diameters are not drawn in. This forms a type of “cup” Problem 40. with two concave portions for holding water. Assume Given two unit squares on top of each other in par- b>aand fill the semicircle of radius b with water. Tip allel planes, d units apart. The center of the top square the arbelos until water spills from the big hole to the lies directly over the center of the bottom square. A little hole. Eventually the water will drip out of the side of one square is inclined by an angle of θ to the cor- smaller hole. If responding side of the lower square. The corresponding a π +3 = , vertices are joined. Find the volume of the resulting b 5π − 3 solid in terms of d and θ. This is a generalization of at what angle is the arbelos tilted when the water first problem 7B12 on page 88 of [1]. drips out? [Answer: 15◦.] REFERENCE [1] L. Lines, Solid .Dover Publications, Inc. Problem 37. New York: 1965. From Krechmar [1], page 24, we have that if A + B + C = π, then sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C.Ifwelet x = sin A, y = sin B, Problem 41. z = sin C, then this gives Given integers p and q (p not divisible by 3), is there an integer n with sum of digits q that is divisible 2x 1 − x2 +2y 1 − y2 +2z 1 − z2 =4xyz. by p?

Page 3 Problem 42. Problem 49. Let P and Q be two points inside ABC. The Can we place distinct positive integers on Z2 cevians through P and Q determine 6 points on the such that orthogonally adjacent integers are relatively of the triangle. It is known that these 6 prime? [Yes, place all the primes.] Can it also be done points lie on an ellipse (see Carnot’s Theorem, page if every positive integer must appear at some lattice 271 in [2], combined with Ceva’s Theorem, page 108 in point? Can we place all the positive integers on Z2 [2]; see also [1]). What can be said about the center of such that if d(a, b) ≤ k, then a and b are relatively this ellipse in relationship to P and Q? Where are the prime? foci with respect to P and Q? Consider also the cases when P and Q are notable points. Are the foci of the Problem 50. ellipse located at any notable points? Evaluate √ ∞   REFERENCES (−1) n . [1] Stanley Rabinowitz, “Problem 903”, Crux Mathe- n n=1 maticorum. 10(1984)19. [2] Luigi Cremona, Elements of Projective Geometry. See G¨unter and Kusmin, volume II, problem 294 which Dover Publicatrions, Inc. New York: 1960. discusses convergence.√ Failing this, evaluate some sum involving (−1) n. Problem 43. Problem 51. Given n points in the plane such that any three Given n points in the plane. Suppose any three are within  of some line. Must they all be within  of are within  of some point. Must they all be within  some line? [No.] What can be said? of some point? [No.] What can be said?

Problem 44. Problem 52. Given n points in the plane. Suppose any three Find the centroid, orthocenter, incenter, etc. for form a triangle with area less than . Must the area of the triangle whose vertices are the roots of x3 + ax2 + their convex hull be less than ? [No.] What can be bx + c =0in the complex plane. said? Problem 53. Problem 45. Find a condition so that the roots of a quartic Is there a polygonal line with all slopes different equation (in the complex plane) are cyclic. that passes through all the lattice points in the plane? Problem 54. Problem 46. Let Fi be the i-th Fibonacci number. Let f(x) Given n coins, all weighing different, except for two beapolynomial of degree n such that f(i)=Fi for of them. What is the minimum number of weighings i =0, 1, 2,...,n. What can be said about f? What needed to find the two that weigh the same using a pan can be said about f(n + 1)? balance? Problem 55. Find the number of 3 × 3 magic squares in Z . Problem 47. n Among n balls are k radioactive ones. A machine Problem 56. detects radioactivity. What is the maximum number of Let A be a given point inside a fixed ellipse in the machine uses needed to locate the k radioactive ones? plane. Vary points P and Q on the boundary of the REFERENCE ellipse so that  PAQ remains constant (and equal to [1] C. D. Cox, ed., “Problem 291”, Parabola. θ). When is PQ the largest? the smallest? 12(1976)28. Problem 57. Problem 48. Which chord of a given length of a fixed ellipse Is there a polynomial f(x, y)ofdegree n>1, such subtends the largest angle at a focus? that f(f(x, y),f(y, x)) has degree less than n? has de- gree less than n2?Howabout Problem 58. Find the number of triangles whose sides are a, b, a g(g(a, b, c),g(b, c, a),g(c, a, b))? and b with a and b integers and a>b. Page 4 Problem 59. √ √ Problem 69. Find the square root of 9 + 2 2+2 3. Generalize Napoleon’s Theorem to 3-space. Given an arbitrary tetrahedron, suppose we describe an Problem 60. isosceles tetrahedron outwardly on each face. What Do the angle bisectors of a triangle always form can be said about the centers of these 4 tetrahedra? a triangle? If so, find an inequality for its area. Same question for other notable cevians such as the altitudes. Problem 70. Let f(n)bethe smallest positive integer such that Problem 61. x2 − x +2divides xn + x + f(n). Find a formula for If A, B, C are lattice points and the inradius of f(n). ABC is a rational number, r,must r be integral? REFERENCE [1] A. M. Gleason, R. E. Greenwood, and L. M. Problem 62. Kelly, The William Lowell Putnam Mathemat- Does there exist an infinite set of points such that ical Competition Problems and Solutions: 1938– all “small” distances between them fall into k classes? 1964. The Mathematical Association of Amer- ica. USA: 1980, page 577. Problem 63. Let A, B, C be convex sets (resp. circles, etc.). Problem 71. Let dist(P, A) denotes the distance from a point P to Find the point on the parabola x2+···+4y2+···= the nearest point in the set A. Consider the set of all 0 that is closest to the origin. points P such that {dist(P, A), dist(P, B), dist(P, C)} are not the lengths of the sides of a triangle. What can Problem 72. be said about the shape of this set? (e.g. is it convex?) Two circles are internally tangent. A chord of the outer circle is divided into three parts by the inner cir- Problem 64. cle. Let the three parts (in order) be x, y, and z. Let Let Fn denote the n-th Fibonacci number and Ln the radii of the circles be r and R (R>r). Given r, R, denote the n-th Lucas number. Investigate FLn and x, and y, find z.

LFn . Problem 73. Problem 65. From page 507 of [1] we know that the number of Investigate the triangles whose vertices are at ways of picking a 3-term arithmetic progression from − 2 (Fk,Lm), (Fk+1,Lm+1), and (Fk+2,Lm+2), where Fi { }  (n 1)  1, 2,...,n is 4 . What about a 4-term arith- and Li are Fibonacci and Lucas numbers, respectively. metic progression? Find the area of this triangle. REFERENCE [1] S. Barnard and J. M. Child, Higher Algebra. Problem 66. MacMillan and Co. limited. New York: 1955. Find the volume of a tetrahedron whose edges have lengths n+i as i varies from 0 to 5 (with n fixed). Does Problem 74. the order matter? [Yes.] Does the tetrahedron exist? A subset of {1, 2,...,n} is chosen at random. Which one has largest volume? Which one has smallest What is the probability that it forms an arithmetic pro- volume? gression?

Problem 67. Problem 75. Find an example where two non-congruent tetra- Evaluate n hedra with the same set of integer edge lengths has the Fi same volume. i! i=0

Problem 68. where Fi denotes the i-th Fibonacci number. Find the Start with an arbitrary convex hexagon in the limit of this value as n →∞. plane. Describe an equilateral triangle externally on each side of the hexagon. Going around in order, let Problem 76. the centers of these 6 equilateral triangles be A, Z, B, Find distinct primes p and q and an integer, d, Y , C, X. What is the relationship, if any, between (d>2), so that pn + qn is divisible by d for all positive ABC and XY Z? Are they similar? integers n.

Page 5 Problem 77. Problem 85. Find integers a, b, c, d, e, f,sothat 357|3an+b + Can the first n2 perfect squares form a magic 5cn+d +7en+f for all positive integers n. square? How about cubes? triangular numbers?

Problem 78. Problem 86. 2 Investigate properties of expressions such as 3n + Find all functions whose graphs are congruent to 5an+b. the graphs of their derivatives. Same question but re- place “congruent” by “similar”. Problem 79. Problem 87. An n-sided die is a homogeneous solid convex poly- Foratriangle of given area, which triangle has the hedron with n faces numbered from 1 to n.Intuitively, largest Morley triangle? Napoleon triangle? it is said to be a fair die if whenever you roll it, the probability that it comes to rest on any given face is 1 . n Problem 88. (a) Give a precise definition of fair. Devise a problem with a finite answer with a form (b) For what n do there exist fair n-sided dice? something like the following: Find REFERENCE [1] Persi Diaconis and Joseph B. Keller, “Fair Dice”, 1 American Mathematical Monthly. card2A 96(1989)337–339. as A ranges over all finite subsets of Z+.

Problem 80. Problem 89. Prove that 3 In the game of Dox (invented by Monroe H. Berg), an(b − c) twenty five counters are arranged to form five rows and 1 and five columns of five counters each. Two players alternate in removing counters. At his turn, a player 3 − is divisible by 1(c b) and find the quotient. Answer: must remove from one to five counters which lie in the same row or column. (They need not be adjacent.) aibjck. Whoever removes the last counter loses. Find a winning i+j+k=n−2 strategy for this game. How does the strategy change if the center counter is removed before the start of the This generalizes problem 188 in [1]. game? Suppose, instead, that the winner is the person REFERENCE removing the last counter. [1] H. S. Hall and S. R. Knight, Higher Algebra. MacMilland & Co. Ltd. London: 1957, page Problem 90. × 509. Let A and B be opposite vertices of an n n square which is divided up into n2 unit squares by lines paral- lel to the sides. How many non-self-intersecting paths Problem 81. are there from A to B travelling along the lines? (A Find max{sin A sin 2B sin 4C} subject to the con- path can go backwards, but must not cross itself.) How straint that A, B, C are the angles of a triangle. Same many such paths are of length k? Generalize to higher question but replace “4” by “3”. dimensions.

Problem 82. Problem 91. Find max{cos A + cos B + cos C} subject to the ◦ Find all functions f(x, y) such that constraint that A + B + C = 120 . Same question but ◦ ◦ replace 120 by 60 . a = f(x, y) x = f(a, b) ⇒ . b = f(y, x) y = f(b, a) Problem 83. Prove that you can’t quadrangulate a triangle. For example, x f(x, y)= 2 2 . Problem 84. x + y Given n points, no 5 cyclic, what is the maximum Add any necessary conditions about continuity or dif- number of cyclic quadrilaterals that can be formed? ferentiability.

Page 6 Problem 92. Problem 97. What is the envelope of all normals to an ellipse? A multigrade of order n is an identity of the form k k Problem 93. ai = bi Let [1..n] denote the infinite repeating [1, 2,...,n]. Is there a simple formula for which is true for k =1, 2,...,n.For example, a multi- [1..n]? Here is some data. grade of order 6 is √ 1+ 5 k k k k k k k k k k [1] = 7 +18 +23 +29 +42 +44 +48 +54 +67 +73 = 2√ 1+ 3 8k +14k +27k +33k +37k +39k +52k +58k +63k +74k [1, 2] = 2√ which is true for k =1, 2,...6 (problem 100 in [1]). 4+ 37 [1..3] = Typically, the number of terms in a multigrade of order 7 √ n is about 2(n − 1) on each side. However, the number 9+2 39 of terms can be smaller, as can be seen in [1..4] = 15 √ k k k k k k 195 + 65029 7 +9 +25 +29 +45 +47 = [1..5] = 314 √ 5k +15k +17k +37k +39k +49k 103+2 4171 [1..6] = 162√ which is true for k =1, 2,...,5 (problem 253 in [1]) but 4502 + 29964677 has only 6 terms on each side. For a given n, find the [1..7] = 6961 smallest number of terms in a multigrade of order n. √ REFERENCE 9280 + 3 13493990 [1..8] = [1] A. S. Moiseenko, 1017 Problems. Privately pub- 14165√ lished. Belleville, NJ: 1987. 684125 + 635918528029 [1..9] = 1033802 Problem 98. Problem 94. Find the maximum number of points of intersec- An ellipse moves around in the first quadrant so tion of a square and an ellipse. A square and a parabola. that it remains tangent to both axes. It is known that An n-gon and an ellipse. the locus of the center of the ellipse is a quadrant of a Problem 99. circle with center at the origin. What is the envelope of Construct an “alphametic” chess problem. Some- all these ellipses? Fix one focus in mind. Find the locus thing like: each piece is replaced by a letter; the same of the focus. Find the envelope of the latera recta. kind of piece is replaced by the same letter. Colors are Problem 95. shown. White has a mate in two. Determine which Find piece each letter stands for. xn − 1 dx. (x − 1)(xn +1) Problem 100. Suppose n people observe a tortoise over Problem 96. aperiod of 1 minute, such that there is to- Solve the diophantine equation x2 + y2 = z2 sub- tal coverage. If the average speed observed is ject to the constraint that xyz is of the form 2t3u5v7w. less than x mi./hr. for all observers, what The case w =0is of special interest. The case where is the farthest the tortoise could have travelled? the diophantine equation is x + y = z is given in [1]. In REFERENCE particular, find all Pythagorean triangles whose sides [1] C. D. Cox, ed., “Solution to Problem O218”, have lengths a, b, and c such that abc has at most 3 Parabola. 9(Sept. 1973)36. prime factors. [There must be at least 3 prime factors, since it is known that 2, 3, and 5 are all divisors of a Problem 101. side of a Pythagorean triangle.] Let P move along an ellipse. Let Q be a fixed point REFERENCE on the ellipse. Let R be the point such that PQR is [1] L. Alex, “Diophantine Equations Related to Finite an equilateral triangle (described clockwise). What is Groups”, Comm. Algebra. 4(1976)77–100. the locus of point R?

Page 7 Problem 102. Problem 111. At each point P on an ellipse, draw a line segment Find all determinants which have the property that PQ into the ellipse such that the length of PQ is twice each element is equal to its the radius of curvature of the ellipse at point P . Find (a) minor the locus of Q.Isthis the envelope of the circles of (b) cofactor. curvature? Problem 112. Problem 103. A set, S,ofpoints in the plane has the property A square is dissected into regular n-gons. Must that any unit circle in the plane contains at least k of they all be squares? these points. What is the largest possible value that

M = inf {d(x, y)} Problem 104. x,y∈S Find a “simple” function, f(x), such that the in- teger n is a Fibonacci number if and only if f(n)=0. can have? Here, d(x, y) denotes the distance from x to y. Problem 105. Problem 113. Find a tetrahedron with integer volume such that n 35 the lengths of the edges form an arithmetic progression. Does any row or column of sum to a 42 perfect square? Problem 106. Can an n×n square be covered by n+1 dominoes, Problem 114. each an (n − 1) × 1 rectangle, plus one 1 × 1 square not Can we put n2 consecutive positive integers into an located in a corner of the original square? n × n matrix such that the determinant of this matrix REFERENCE is 1? [1] George Berzsenyi, “In Search of Colorations”, Jour- nal of Recreational Mathematics. 8(1975– 1976)191–194. Problem 115. Let ABCD beasquare with center O. What hap- pens to PA+ PC − PB − PD as P varies on a circle Problem 107. with center O? Let k and n be given positive integers. Find the largest value for j so that k white queens and j black Problem 116. queens can be placed on distinct squares of an n × n Fix a line L in the plane. Look at all triangles with chessboard such that no white queen attacks a black area K and with L as their Euler line. What can be queen. said about these triangles? Are they all congruent?

Problem 108. Problem 117. Look for a Helly-like theorem for pairs of convex Let A =(0, 0, 0), B =(1, 1, 0), C =(1, 1, 1), sets. For example, find a theorem like “if any 2 meet D =(2, 3, 3), E =(1, 2, 3), and F =(3, 1, 0). Find of n, then some 3 meet” or “if any k meet (of n) then the equation of a circle in 3-space that is tangent to all n meet”. the lines AB, CD, and EF.

Problem 109. Problem 118. In the plane, exhibit n convex sets such that any Find the polygon with the smallest number of sides two meet but no three meet. [Solution: just take n lines for which two diagonals colline. [Answer: n = 6.] Sup- in general position.] Suppose that any k meet but not pose that we want k diagonals to colline? k +1. Problem 119. Problem 110. A crossing diagonal of a non-convex polygon is a Let A be a convex set in the plane with width w. diagonal that meets an edge of the polygon, not at a Prove that A is the intersection of all sets of constant vertex. What is the maximum number of crossing di- width w containing A. agonals that a simple polygon of n sides can have?

Page 8 Problem 120. REFERENCES What is the envelope of the lines that bisect the [1] Stanley Rabinowitz, “Problem 1292”, Journal of perimeter of a given triangle? bisect the area of the Recreational Mathematics. 16(1983–1984)137. triangle? [2] Joseph Rosebbaum, “Problem E332”, American Mathematical Monthly. 46(1939). Problem 121. Characterize all numbers such that in their base 10 Problem 129. representation, the number is a multiple of the number Investigate the center of a circle which bisects the formed by removing one of its digits. For example, circumferences of the excircles of a given triangle. (wxyz) = k(wxy) . 10 10 REFERENCE [1] Howard Eves, A Survey of Geometry, revised edi- Problem 122. tion. Allyn and Bacon, Inc. Boston, MA: 1972, Find the locus of the center of all unit equilateral page 96, problem 7. triangles that lie inside a unit square.

Problem 123. Problem 130. Find numbers that are equal to a subset of their Let Z be a fixed point on side AB of ABC. Let base k representation in another base. For example, X beavariable point on side AC and let Y be avariable abcdep = bcdeq. point on side BC.Point X moves on AC toward C at k m.p.h. What speed must Y move at such that the area Problem 124. of XY Z remains constant? You can specify where X Equilateral triangles A1A2A3 and B1B2B3 have and Y start. [probably can’t be done.] sides of lengths a and b respectively. How should they be situated in the plane so that AiBi is minimum? Problem 131. How should they be situated in space to achieve a min- Can you always pass a polygon through itself? imum? Problem 132. Problem 125. Which non-simple n-gon whose vertices are the Two lines are nearly parallel if they are parallel or same as the vertices of a regular n-gon has maximum meet in an angle less than . Find the largest  such perimeter? that there exists a hexagon with no two sides being nearly parallel. Problem 133. 2 2 2 Problem 126. Does the Diophantine equation x +y = nz have What is the smallest possible area of a quadrilat- infinitely many integral solutions for each n? eral with sides a, b, c, and d. The quadrilateral need not be convex. Generalize to n-gons. Problem 134. For each n>1, does there exist an n × n matrix Problem 127. of distinct positive integers such that the determinant Let ≺ bearelation on R+ such that for all x, y, of the matrix is k for any k ∈ Z? and z, ≺ (i) x x Problem 135. ≺ ≺ ⇒ (ii) x y, y x x = y Can you label the edges of a cube with integers ≺ ≺ ⇒ ≺ (iii) x y, y z x z from 1 to 12 so that if edges labelled i and j meet, then and √ |i − j| > 2? Can you change “2” to “3”? ≺ x+y (iv) xy 2 . Must “≺”bethe usual “≤”? Problem 136. Problem 128. What point of a triangle has the Euler Line of the Characterize those n for which the first n perfect triangle as its Simson line? squares can be partitioned into two sets with the same sum. For example, 1 + 4 + 16 + 49 = 9 + 25 + 36. Same Problem 137. question for cubes, p-th powers, Fibonacci numbers, What is the envelope of all Simson lines of a trian- etc. gle?

Page 9 Problem 138. Problem 145. Is there a Heronian triangle with one angle twice Let ABC beafixed triangle. Find the locus of all another? points P such that PA+ PB + PC is a constant. Let REFERENCE the lengths of the perpendiculars dropped from P to [1] Stanley Rabinowitz, “Problem 1193”, Crux Mathe- the sides of the triangle be x, y, z. Find the locus of maticorum. 12(1986)282. points P such that x + y + z is a constant.

Problem 139. Problem 146. Is there a nice recurrence for F2n in a general Make up a problem like: “Let ABC beatriangle second-order linear recurrence? with centroid M and orthocenter H.IfAH + BH + CH = AM + BM + CM, prove that the triangle is Problem 140. equilateral.” What special points must be chosen for For Fibonacci and Lucas numbers, we have F2n = M and H? FnLn.IfUn is the n-th term of a general second-order linear recurrence, is there an analog to the above equa- Problem 147. tion? Which diagonals of a Platonic solid unexpectedly concur? Answer the same question in En for n>3. Problem 141. Let P beapoint inside an acute triangle. Drop Problem 148. perpendiculars from P to the sides of the triangle. The Given two triangulizations by diagonals of a convex feet of the perpendiculars divide the sides into 6 seg- polygon. Must the fields generated by the angles (or the ments of lengths (in order) a, x, b, y, c, z.Itfol- tangents of the angles) be the same? lows from [1], that for an equilateral triangle, we have a + b + c = x + y + z.For an arbitrary triangle, if a + b + c = x + y + z for all choices of P ,must the Problem 149. triangle be equilateral? What about for one choice of What is the smallest non-right Heronian triangle P ? What can be said about the triangle in that case? in the plane? (A Heronian triangle is a triangle with Forhow many different P must the equation be true in integer sides and integer area. order to guarantee that the triangle is equilateral? REFERENCE Problem 150. [1] Stanley Rabinowitz, “Solution to Problem 1518: Let Perpendicular Sums Problem (proposed by 111 Daniel Scher)”, Journal of Recreational Math- Sn = abc . ematics. 19(1987)313–314. an bn cn Factor or simplify S . Find some nice relationship be- Problem 142. n tween the S . Let n beapositive integer. Let A and B be real i numbers such that 0

Page 10 Problem 154. See problem D28 in [1] and its errata. Two curves are said to be equidistant (of length REFERENCES a)fromapoint P if whenever X is on one curve, PX [1] Richard K. Guy, Unsolved Problems in Number intersects the other curve at a unique point, Y , and Theory. Springer-Verlag. New York: 1981. XY = a. Prove that no two curves can be equidistant [2] L. J. Mordell, “Research Problem 6”, Canadian in two different ways. Mathematical Bulletin. 17(1974)149.

Problem 155. Problem 163. How many integral solutions are there to 3n +4n = In what positions can an equilateral triangle be 5n? Ask the same question about similar examples. inscribed in an ellipse? Note that 3n +6n =2n has the solution n = −1. Problem 164. Problem 156. Find the condition so that the three points of in- How many squares are there whose vertices are all flection of y = x4 + ax3 + bx2 + cx + d colline. [I don’t lattice points with coordinates less than or equal to n think it can happen.] in absolute value? Problem 165. Problem 157. Let P be apoint inside ABC.Itisknown that Can polynomials P (x) and P (x)+1 both have mul- if P is the orthocenter, then the Euler lines of triangles tiple roots? PAB, PBC, PCA concur. Find all P for which this is true. Problem 158. × Find the 3 3 fully magic square of distinct positive Problem 166. integers with the smallest (in magnitude) determinant. Find a simple formula for then-th non-square in- Is it   √ 4113 teger. (Answer: n + n +  n . See [1].)  567 REFERENCES 918 [1] H. W. Gould, “Non-Fibonacci Numbers”, Fi- bonacci Quarterly. 3(1965)177–183. with determinant 260? [2] J. Lambek and L. Moser, “Inverse and Complemen- tary Sequences of Natural Numbers”, American Problem 159. Mathematical Monthly. 61(1954)454–458. If each edge of a triangle has length between x and [3] H. W. Gould, “Problem H-1”, Fibonacci Quarterly. y, what can be said about the area of the triangle? 1(1963)46. Problem 160. Problem 167. Let n beapositive integer. Is there a polynomial Find a Heronian triangle with f(x) such that f(x+k) has k more terms than f(x) for k =1, 2,...,n? (a) r, a, b, c in arithmetic progression (b) a, b, c, K in arithmetic progression Problem 161. (c) a, b, c, R in arithemtic progression For which n can one form a sequence of the first (d) a, b, c in geometric progression. n positive integers so that no two consecutive integers are adjacent and adjacent integers are relatively prime? Problem 168. Example: for n =10wehave 5, 2, 9, 4, 7, 10, 3, 8, 1, Find a Heronian triangle with the length of each 6. side being a perfect square. A triangular number.

Problem 162. Problem 169. Do the following equations always have solutions A triangle with integer sides has one side of length in integers? n. What is the smallest the area can be? 1 1 1 1 + + = a b c abc Problem 170. 1 1 1 1 1 Find the smallest n such that some three vertices + + + = a b c d abcd of a unit n-cube form a triangle with rational area. Page 11 Problem 171. Problem 180. Given a triangle with sides of lengths a, b, c, with Given n lines in general position in the plane. Can a>b>c. Prove that this triangle can be covered by all bounded regions have the same number of sides? an equilateral triangle of side a. Can all bounded regions have the same area?

Problem 172. Problem 181. Given a point P in space outside the plane of a Given four concurrent lines [parallel lines], find the circle, find the maximum value of  AP B where A and condition so that there exists a square with one vertex B lie on the circle. on each line.

Problem 173. Problem 182. Find the condition so that two circles in 3-space What is the smallest number of vertices of a poly- interlock. hedron with k holes?

Problem 174. Problem 183. Is every polynomial the sum of k squares in Z[x]? If the perimeter of a quadrilateral is fixed, what is the minimum and maximum values for the sum of the Problem 175. diagonals? What is the sum of the entries of the Farey se- REFERENCE quence of order n? [1] Murray S. Klamkin, “Diamond Inequalities”, Math- ematics Magazine. 50(1977)96. Problem 176. ↑ ↑↑ Find a matrix, M, such that M 2=M 2, Problem 184. i.e. the square of the matrix results in a matrix whose When can you inscribe a regular hexagon in an elements are the squares of the original elements. ellipse? Regular octagon?

Problem 177. Problem 185. × Let M(n)bethe n n matrix whose elements are Are the roots of x2 + ax + b in GF(2) expressible 2 − the integers from 0 to n 1, entered consecutively, left in terms of radicals? Note that the characteristic of the to right, top to bottom. Let P (n)bethe permanent of field is 2. this matrix. Is there a simple formula for P (n)? Some values are: Problem 186. { | ∈ } P (2) = 2 Find a primitive root in a + bi a, b Zp . P (3) = 144 Problem 187. P (4) = 22432 Find the smallest matrix with integer elements P (5) = 6778800 whose 100th power is 0 (but not its 99th power). P (6) = 3563641440 Problem 188. P (7) = 3001327741440 Find a matrix whose elements are all Egyptian P (8) = 3805251844939776 1 fractions (of the form k ) whose square consists only P (9) = 6921278637502113536 of integer entries. P (10) = 17382962774600072036352 Problem 189. Problem 178. Let N = pq. Can the set {1, 2,...,N} be parti- k n tioned into p sets of q elements each so that the sum of When is 2 > k ? REFERENCE the elements in each set are incongruent modulo p? [1] Stanley Rabinowitz, “Problem to appear”, Ameri- can Mathematical Monthly. 96(1989)??. Problem 190. Let ABC be an equilateral triangle. Let P be a Problem 179. variable point in the plane and let PA = a, PB = b, { n − k} Are all members of the set k 2 for k = PC = c. Find the locus of point P so that ab + bc + ca 0, 1,...,n distinct for a given n? is a constant.

Page 12 Problem 191. Problem 200. Place the integers from 1 to 3n on the vertices of a Let n 2 regular 3n-gon. Prove that some three numbers form an a a a 1 n 2 equilateral triangle and have some nice property (such b b b 1 Dn = n 2 . as their sum is divisible by 3). c c c 1 dn d2 d 1

Problem 192. Prove that D4 = D3(a + b + c + d). Does this relation- How many vertex triangles of a regular octahedron ship hold when we generalize to the appropriate 5 × 5 (dodecahedron, etc.) are equilateral triangles? How determinant? Also, investigate D5, D6, etc. many equivalence classes of shapes are there? REFERENCE [1] J. Heading, Mathematics Problem Book 1: Algebra Problem 193. and Complex Numbers.Pergamon Press. Ox- n Can n straight lines divide a circle up into 2 − 1 ford: 1964, page 15, problem 55. parts of the same area? Problem 201. Problem 194. Suppose I pick a real number that is the root of a Is there a simple formula for polynomial with “small” real coefficients and give you n as many decimal places as you want for it. Devise an n 1 ? algorithm for finding the polynomial. k k k=1 Problem 202. Problem 195. Suppose I pick a real number that is a “simple” Evaluate function of the integers, π, and e, using simple opera- 1 √ π m n tors such as +, −, ×, ÷, , etc. For example, √ . x (x +1) dx. 7− 2 0 I will give you as many decimal places as you want for this number. Devise an algorithm that will determine Problem 196. 2 2 2 the number. Does x + y + z factor over Zp[x, y, z] for any p? REFERENCE Problem 203. [1] Stanley Rabinowitz, “Problem N-3”, AMATYC Re- Let P beavariable point in the plane of a fixed view. 9(1988)71. ellipse. Let m and M be the minimum and maximum distance from P to the ellipse, respectively. Find a Problem 197. non-constant function f(x, y), such that f(m, M)isin- From [1], page 34 we have variant as P varies over all points outside the ellipse. ≡ abc +(b + c)(c + a)(a + b) (a + b + c)(bc + ca + ab). Problem 204. Does this identity generalize? Factor REFERENCE (a + x)n (a + y)n (a + z)n [1] J. C. Burkhill and H. M. Cundy, Mathematical (b + x)n (b + y)n (b + z)n . Scholarship Problems. Cambridge University (c + x)n (c + y)n (c + z)n Press. Cambridge: 1962. Problem 205. Consider an expression of the form Problem 198. 1 If p and q are positive integers less than or equal to 1 p − q x + x+ 1 n, which polynomial, x x +1, has the largest root? x+ 1 f(x) Problem 199. where there are nx’s in the finite continued fraction. Investigate the form that f(x)must take so that this Each of n people, A1 to An, announce “I am Aj.” expression is a constant. For example, when n =1, Can their identities be determined if precisely k are ly- c ing? If not, what additional constraints need we put on f(x)must be of the form 1−cx . When n =2,f(x) cx−1 this problem so that one can determine the identity of must be of the form x−cx2−c . Ask the same question each person (in the presence of some number of liars)? for 1 Suppose, instead, that the n people are arranged in 1 . 1+ x+ 1 a circle and each man also makes the statement “The x2+ 1 x3+ 1 man to my right is Ak”. Analyze this case too. f(x)

Page 13 Problem 206. Problem 214. Evaluate In triangle ABC, inscribe a square with one base on side BC. Let X be the midpoint of the base of the n 1 1 square. Similarly, locate points Y and Z on CA and yn = ≡ . 1+ 1 xi+ AB respectively. Prove that AX, BY , CZ concur. Do x+ 1 i=0    x2+ 1 they concur at any notable point? If X , Y , Z are the x3+··· 1 xn centers of the three squares, prove also that AX, BY , CZ concur. Also find the limit as n →∞. Problem 215. Problem 207. Let P (x)beapolynomial such that P (k)=2k Evaluate the finite continued fraction for k =1, 2,...,n. What can be said about P (0) or P (n + 1)? n 1 . n + Problem 216. i=0 i Let Fn denote the n-th Fibonacci number. Show that if Fa + Fb + Fc =3Fd, then {a, b, c; d} = {n, n + Problem 208. 2,n+3;n+2} or {n, n+1,n+6;n+4} for some integer Find the average distance from a point to a circle. n.

Problem 209. Problem 217. Let P be a fixed point in the plane of the parabola I conjecture (via Macsyma) that if y = x2.AsQ varies on this parabola, find the average 1 xn + xn + ···+ xn + kx x ···x value for PQ. 1 2 n 1 2 n

factors, then either n =2,k = ±2; or n =3,k = −3. Problem 210. The altitudes of a triangle are 3, 4, and 5. Find Problem 218. the area of the triangle. Given ABC, construct three circular arcs out- wardly on the sides so that the resulting curve is Problem 211. smooth. Prove that this curve must be a circle. Ask the Each vertex of a triangle is reflected about the op- same question for an n-gon. How about if we replace posite side to get three new points with coordinates circular arcs by elliptical arcs? (x1,y1), (x2,y2), and (x3,y3). Find the three ver- tices. [It is known that the original triangle is not con- structible with Euclidean tools from the three reflection points.]

Problem 212. Let P beavariable point inside an ellipse with equation x2/a2 + y2/b2 =1. Through P draw two chords with slopes b/a and −b/a respectively. The point P divides these two chords into four pieces of lengths w, x, y, z. Prove that w2 + x2 + y2 + z2 is independent of the location of P and in fact has the value 2(a2 + b2). Generalize to higher dimensions.

Problem 213. Let P be an arbitrary point inside an ellipse. Draw twochords AY and BX through P .If AXB =  AY B, prove that the bisector of  AP B is parallel to the major axis of the ellipse. Equivalently, the slope of AY plus the slope of BX is 0.

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