The 60th William Lowell Putnam Mathematical Competition Saturday, December 4, 1999 ✂✁☎✄✝✆ ☎✁ ✄✝✆ ,✞ , A-1 Find polynomials ✄ such that for all , and ✁☎✄✠✆ ✟ ✓✔ ☞✘✗ ✕ ✡ ✂✁☛✄✝✆ ✡✌☞✍✡ ✁☎✄✠✆ ✏✡ ✎ ✁☎✄✝✆✒✑ ✔✖ ✛ ✄ ✢ ✎ ✜ ✞ ✟ ☞✧✜ ✄ ✢ ✎ ✜ , if they exist, ✄✚✙ ☞✘✗ if ☞✘✗✤✣ ✄ if ✄✚★ ✥ if ✣✦✥ ✁☎✄✠✆ A-2 Let ✩ ✄ be a polynomial that is nonnegative for all real Prove ✌✫✬✁☎✄✠✆✮✭✏✯✰✯✰✯✱✭✲ ✌✳✴✁☎✄ that for some ✪ , there are polynomials ) such that ✳ ✵ ✁☎✄✝✆✒✑ ✹✁ ✶ ✁☎✄✠✆✺✆✼✻✽✯ ✩ ✸✶ ✷ ✫ ✑ B-1 Right triangle ❲✧❳❨❍ has right angle at ❍ ✡ and✡ ✑ ❩❬❳❭ ✡ ❲❪❍ ✡ ✑ ❫ ; the point ❴ is chosen on ❪ ❲ ❳ so that ❵ ❲ ❍ ❲✧❴ ✑❞❫ ✗ ; the point ❛ is chosen on ❳❭❍ so that ❩❜❍✤❴❝❛ ❭ ❳ ❍ ❛ ❪ ❲ ❳ ❡ The perpendicular to at meets at Evalu✡ ✡ ate ❢❤❣❤✐❦❥♠❧ ❃ ❛♥❡ ☛✁ ✄✝✆ ✁☎✄✠✆♣✑ ✁☎✄✠be✆ a polynomial q❦✁☎✄✠✆ of degree ❇ such that ♦ is a quadratic and ✁☛✄✝♦✘ ✆ rsr , where ✁☎✄✠polynomial ✆ ♦✘r ✁☎r ✄✠✆ is the second derivative of ♦ Show that if ♦ has at least two distinct roots then it must have ❇ B-2 Let q❦✁☎✄✠♦ ✆ distinct roots ✑✉t✈✁☛✄▼✭✺✇①✆❬② ✥③✣ ✄▼✭✸✇④✙ ✗⑥⑤ B-3 Let ❲ ⑨✾✁☛✄▼✭✸✇⑦✆⑩✑ A-3 Consider the power series expansion ✗ ✵ ✗✾☞✿✜ ✄ ☞ ✄ ✻ ✑❁❀ ❂ ✄ ❂ ✯ ❂ ✷❄❃❆❅ ✥ Prove that, for each integer ❇❉❈ , there is an integer ❊ such that ✻ ✎ ❂ ❅ A-4 Sum the series ✵❀ ● ❅ ✻ ✫ ✑ ❂✌❋ ❅✴● ✯ ✻ ✯ ❊ ✎❇ ✁ ✛ ✛ ✛ ✆ ❊ ❂ ✷ ✫ ❂✷ ✫ ● ❇ ● A-5 Prove that there is a constant ❍ such that, if ✩ polynomial of degree 1999, then A-6 ✌❷❶ ❸❬❹ ❺✒❸ ✻ ✄ ● ✁☛✄▼✭✸✇⑦✆✾⑧ ❲ , let ✇ ❂ ✭ ✁ ✭ ✆ where the sum ranges over all pairs ❊ ❇ of positive integers satisfying the indicated inequalities Evaluate ✁ ✗➅☞ ✄➆✇ ✻ ✆➁✁ ✗✾☞ ✄ ✻ ✇①✆✺⑨✾✁☛✄▼✭✸✇⑦✆➁✯ ❻❽❼✌❾ ❿➁➀ ❧ ❻ ✫ ❢❤❣❤❾ ✐✫ ➀➂❾s❻❽❼✌❾ ❿➁➀➂➃✌➄ ✵❀ ✵ For ✁☎✄✠✆ is a ✫ ✡ ✁ ✥ ✆ ✡✴✣ ✡ ✁☎✄✝✆ ✡✼▲ ✄▼✯ ✩ ❍❏■ ✫ ✩ ❑ ✁ ❂ ✆ ❂✴◆ ✫ ✫❖✑ ✗ ✭ ✻ ✑ The is◗ defined by ✜ ✭ sequence ✑ ✜❘◗ ✭ ❅ ❅ and,❅ for ❇❙❈ , ❅✴ ✻ ✫ ❂ ☞✿❱ ❂ ❑ ✫ ✻❂ ✻ ✯ ❂ ✑❯❚ ❅ ❂ ❑ ❅ ❑ ❅ ❑ ❂ ❑ ✻ ❂ ❑❅ ❅ ❅ ❅ Show that, for all n, ❂ is an integer multiple of ❇ ❅ B-4 Let be a real ✂✁☎function ✄✠✆✮✭♠ ✁☛✄✝✆✮with ✭♠ ✁☛a✄✝continuous ✆➁✭✲ ✁☛✄✝✆ third derivative✄ such that r ✁☎✄✠✆ ✣ rsr ✂✁☎✄✝✆ rsr r are✄ positive for for all Show that all ✁☛✄✝.✆❜Suppose ✙ ✜ ✂✁☛✄✝✆ that r rs✄r for all r ✛ ❫✉✑ ✜❘➈➊➉ B-5 For an integer ❇➇❈ , let ✎ ❇ Evaluate the determinant of the ❇➌➋✢❇ matrix ✑➏ ➍ ✁ ❲ ✳ ,✆ where ➍ is ✶ has entries the✳ ❇✍ ➋➎❇ identity matrix and ❲ ❅ ✶ ✑➑➐✰➒✽➓✰✁❤➔✽❫ ✎ ✪ ❫✈✆ for all ➔✽✭ ✪ ❅ ⑨ B-6 Let be a finite set of integers, each greater than ⑧➣1.⑨ ❇ → Suppose that for each integer there is some ✗ ➐✏↕➙✁ ✭ ✆➛✑ ➐✰↕▼✁ ✭ ✆➛✑ such that ↔ ✭✸➜✒→ ⑧✚❇ ⑨ or ↔ ➐✏↕➙✁ → ✭✺❇ ➜✺✆ → Show that there exist → such that ↔ → is prime