The Fifty-Eighth William Lowell Putnam Mathematical Competition Saturday, December 6, 1997 ✂✁☎✄✝✆ ✂✁✟✞✟✠✡✠ ✁☎✄☛✞ ✁ ✌✎✍✑✏ ✄ ✆ ✍✑✏ ✌ ✍✑✏ ✠✓✒✕✔✖✒✘✗✙✒✛✚✜✚✛✚✢✒✤✣ are seated around a table, and A–2 Players ☞ , has sides and A–1 A rectangle, A triangle has as the intersection of the altitudes, the center of the circumscribed circle, the midpoint of , and the foot of the altitude from What is the length of ? each has a single penny Player passes a penny to player 2, who then passes two pennies to player Player then passes one penny to Player 4, who passes two pennies to Player 5, and so on, players alternately passing one penny or two to the next player who still has some pennies A player who runs out of pennies drops out of the game and leaves the table Find an infinite set of numbers for which some player ends up with all pennies ✣ ✣ A–3 Evaluate ✥✧✦ ✲✛✲✜✲✤✹ ★ ✩✖✪✬✫ ✪✮✔✰✭ ✯ ✔✳✪✮✲✵✱ ✴ ✫ ✔✳✲✵✪✮✴☎✶ ✲✜✷✸✯ ✠ ✲✜✲✛✲✽✹✿✾ ✚ ✩ ✯ ✪✮✔ ✺✺ ✯ ✔ ✺ ✪✮✲✜✻ ✴ ✺ ✯ ✔ ✺ ✲✜✪✮✴ ✺✼ ✲✛✷ ✺ ✯ ✪ A–4 Let ❀ be a group with identity ❁ and ❂❄❃❅❀❇❆❈❀ a function such that ❂❊❉●❋■❍✛❏✤❂❊❉●❋ ❏✤❂❊❉●❋ ❏ ✞ ❂❑❉▼▲◆❍✜❏✤❂❊❉▼▲ ❏✤❂❊❉❖▲ ❏ ✺ ✞ ❁ ✭ ✞ ▲ ❍ ▲ ▲ Prove ✺ ✭ that there whenever ❋ ❍ ❋ ❋ ✺ ✭ ❘◗❙❀ such✺ that ✭ ❚✎❉ ✪ ❏ ✞ ✖❂❊❉ ✪ ❏ exists an element ✞ ❚✳❉ ✪ ❏✽❚✎❉ ❯ ❏ for all is✒ a homomorphism (i.e ❚✎❉ ✪✙❯ ❏ ) ❱ ◗ ❀ ✪ ❯ ✣ A–5 Let ❲✑❳ denote the number of ordered ✠❫-tuples ✒ ✛ ✒ ✜ ✚ ✛ ✚ ❩ ✚ ✒ ❪ ❍ ✯ of✠❴❪ posi❨ ❉ ❬ ❭ ❳ ❏ ✯ tive integers such that ❍ ✚✛✚✛✚ ✯ ✠❫❪ ❳ ✞❵✠ Determine ✺ ✺ whether ❲❛❍ ★ is even or odd ✣ A–6 For a positive integer✞❡❞ and any number ❜ ❞ , define ✞✝✠ ,real ★ recursively by , and for ❢❤❣ , ❍ ✪◆❝ ✪ ✪ ✣ ✞ ❜ ❍ ❉ ❢❭❏ ✚ ✪ ❝❥✐ ✺ ✪✢❝❥✐ ✫ ❢ ✯ ✠ ✫ ✪✢❝ ✣ and then take ❜ to be the largest value for which Fix ✞❦ ❞ ✣ ✠☎❧ ❧♠✣ ✪ ❳ ✐ ❍ Find ✪ ❝ in terms of and ❢ , ❢ ♥ ✪♣♦ ✣ B–1 Let denote the distance between the real number and the nearest integer For each positive integer , evaluate ✪ (Here ❷ ✈①✇⑨② ✆ ❳ ✞ ✼ r ❳✖q ❍ ❉④♥⑥✷⑦⑤ ✣ ✒ ♥⑧✗✓⑤ ✣ ❏ ✚ ♦ ♦ s✉t ❍❑✈①✇③② ❉❖ ✒❶⑩ ❏ denotes the minimum of and ⑩ ) B–2 Let be a twice-differentiable real-valued function satisfying ❸❷ ❉ ✪ ❏ ✯ ✢❷ ❹ ❹❺❉ ✪ ❏ ✞ ✫❅✪ ❋❻❉ ✪ ❏✘❷✢❹❺❉ ✪ ❏ ✒ where ❻ ❋ ❉ ✪ ❏✧❣ ❞ for all real ✪ Prove that ❼ ❷❸❉ ✪ ❏✛❼ is bounded ✣ ❳ ✠❴❪ B–3 For each positive integer , write❿ the sum ❽ s✉t ❍ ⑦ ❪ ❿ ⑤ in the form ❾ ❳ ❳ , where ❾ ❳ and ✣ ❳ are relatively prime positive❿ integers Determine all such that does not divide ❳ ❳ B–4 Let ✠ s✎➀ ❳ denotes the coefficient of ✪ in the expansion ❞ of ❉ ✯ ✪ ✯ ✪✮✺ ❏ Prove that for all [integers] ❢❤❣ , ➆ ➆➆ ❸ ➄ ➅ ❞❛❧✰➁✤r ➆ ➂▼➃ ❉ ✠ ❏ q ➀ ❧❦✠✡✚ t★ ✫ ❝ ✣ ✔ B–5 Prove that for ❣ , ✣ terms ✣ ✠ terms ✫ ➇ ➈✵➉ ➊ ➇ ➈❥➉ ➊ ✔ ➂➍➌ ✔ ➂ ❉ ✣ ❏✚ ✺✵➋ ➋ ➋ ✺✛➋ ➋ ➋ ✈①➎✖➏ B–6 The dissection of the 3–4–5 triangle shown below (into four congruent right triangles similar to the original) has diameter Find the least diameter of a dissection of this triangle into four parts (The diameter of a dissection is the least upper bound of the distances between pairs of points belonging to the same part.) ☞ ❪⑦✔