❉✐❡ ❙✉✐❞✲❆❢r✐❦❛❛♥s❡ ❲✐s❦✉♥❞❡✲❖❧✐♠♣✐❛❞❡
❉❡r❞❡ ❘♦♥❞❡ ✷✵✵✺
❙❡♥✐♦r ❆❢❞❡❧✐♥❣ ✭●r❛❞❡ ✶✵ t♦t ✶✷✮
❚②❞✿ ✹ ✉✉r
✶✳ ❱②❢ ❣❡t❛❧❧❡ ✇♦r❞ ✉✐t ❞✐❡ ❞✐❛❣r❛♠ ❤✐❡r♦♥❞❡r ❣❡❦✐❡s✱ s♦❞❛t ❞❛❛r ♥✐❡ t✇❡❡ ❣❡t❛❧❧❡ ✉✐t ❞✐❡s❡❧❢❞❡ r② ♦❢ ✉✐t ❞✐❡s❡❧❢❞❡ ❦♦❧♦♠ ❣❡❦✐❡s ✇♦r❞ ♥✐❡✳ ❚♦♦♥ ❛❛♥ ❞❛t ❤✉❧❧❡ s♦♠ ❛❧t②❞ ❞✐❡s❡❧❢❞❡ ✐s✳
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73
✷✳ ▲❛❛t F❞✐❡ ✈❡rs❛♠❡❧✐♥❣ ✈❛♥ ❛❧❧❡ ❜r❡✉❦❡m/n ✇❡❡s✱ ✇❛❛rm ❡♥n ♣♦s✐t✐❡✇❡ ❤❡❡❧t❛❧❧❡ ✐s ♠❡t m+n62005✳ ❱✐♥❞ ❞✐❡ ❣r♦♦tst❡ ❣❡t❛❧a✐♥ Fs♦❞❛ta < 16/23✳
✸✳ ✬♥ P❛❦❤✉✐s ❜❡✈❛t175♥♦♠♠❡r✲8st❡✇❡❧s✱175♥♦♠♠❡r✲9st❡✇❡❧s✱ ❡♥200♥♦♠♠❡r✲10st❡✇❡❧s✳
❱❛♥ ❤✐❡r❞✐❡550st❡✇❡❧s ✐s250❧✐♥❦❡rst❡✇❡❧s ❡♥300r❡❣t❡rst❡✇❡❧s✳ ▲❛❛tnst❛❛♥ ✈✐r ❞✐❡ t♦t❛❧❡
❛❛♥t❛❧ ❜r✉✐❦❜❛r❡ ♣❛r❡ st❡✇❡❧s ✐♥ ❞✐❡ ♣❛❦❤✉✐s✳ ✭✬♥ ❇r✉✐❦❜❛r❡ ♣❛❛r ❜❡st❛❛♥ ✉✐t ✬♥ ❧✐♥❦❡r✲ ❡♥ r❡❣t❡rst❡✇❡❧ ✈❛♥ ❞✐❡s❡❧❢❞❡ ❣r♦♦tt❡✳✮
✭❛✮ ■s n=50 ♠♦♦♥t❧✐❦❄ ✭❜✮ ■sn=51 ♠♦♦♥t❧✐❦❄
✹✳ ❉✐❡ ✐♥❣❡s❦r❡✇❡ s✐r❦❡❧ ✈❛♥ ❞r✐❡❤♦❡❦ ABC r❛❛❦ ❞✐❡ s②❡BC, CA ❡♥ AB ❜② D, E❡♥ F ♦♥❞❡r✲
s❦❡✐❞❡❧✐❦✳ ▲❛❛t Q st❛❛♥ ✈✐r ❞✐❡ ❛♥❞❡r s♥②♣✉♥t ✈❛♥ AD ❡♥ ❞✐❡ ✐♥❣❡s❦r❡✇❡ s✐r❦❡❧✳ ❇❡✇②s
❞❛t ❞✐❡ ✈❡r❧❡♥❣✐♥❣ ✈❛♥ EQ ❞❡✉r ❞✐❡ ♠✐❞❞❡❧♣✉♥t ✈❛♥AF❣❛❛♥ ❛s ❡♥ s❧❡❣s ❛sAC=BC✳
✺✳ ▲❛❛t x1, x2, . . . , xn ♣♦s✐t✐❡✇❡ ❣❡t❛❧❧❡ ✇❡❡s ♠❡t ♣r♦❞✉❦ ❣❡❧②❦ ❛❛♥ ✶✳ ❇❡✇②s ❞❛t ❞❛❛r ✬♥
k∈{1, 2 . . . , n}❜❡st❛❛♥ s♦❞❛t
xk
k+x1+x2· · ·+xk
>1− 1
n
√
2.
✻✳ ❇❡s❦♦✉ ❞✐❡ st②❣❡♥❞❡ r②1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, 19, . . .✈❛♥ ♣♦s✐t✐❡✇❡ ❤❡❡❧t❛❧❧❡✱ ✈❡r❦r②
❞❡✉r ♦♠ ❞✐❡ ❜❡✉rt ❜❧♦❦❦❡ {1},{2, 4},{5, 7, 9},{10, 12, 14, 16}, . . . ✈❛♥ ♦♥❡✇❡ ❡♥ ❡✇❡ ❣❡t❛❧❧❡
❛❛♥♠❡❦❛❛r t❡ ❧❛s✳ ❊❧❦❡ ❜❧♦❦ ❜❡✈❛t ❡❡♥ ❡❧❡♠❡♥t ♠❡❡r ❛s ❞✐❡ ✈♦r✐❣❡ ❡❡♥✱ ❡♥ ❞✐❡ ❡❡rst❡ ❡❧❡♠❡♥t ✈❛♥ ❡❧❦❡ ❜❧♦❦ ✐s ❡❡♥ ♠❡❡r ❛s ❞✐❡ ❧❛❛st❡ ❡❧❡♠❡♥t ✈❛♥ ❞✐❡ ✈♦r✐❣❡ ❜❧♦❦✳ ❇❡✇②s ❞❛t ❞✐❡
n✲❞❡ ❡❧❡♠❡♥t ✈❛♥ ❞✐❡ r② ❣❡❣❡❡ ✇♦r❞ ❞❡✉r
2n−
1+√8n−7 2
.
❚❤❡ ❙♦✉t❤ ❆❢r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❖❧②♠♣✐❛❞
❚❤✐r❞ ❘♦✉♥❞ ✷✵✵✺
❙❡♥✐♦r ❉✐✈✐s✐♦♥ ✭●r❛❞❡s ✶✵ t♦ ✶✷✮
❚✐♠❡ ✿ ✹ ❤♦✉rs
✶✳ ❋✐✈❡ ♥✉♠❜❡rs ❛r❡ ❝❤♦s❡♥ ❢r♦♠ t❤❡ ❞✐❛❣r❛♠ ❜❡❧♦✇✱ s✉❝❤ t❤❛t ♥♦ t✇♦ ♥✉♠❜❡rs ❛r❡ ❝❤♦s❡♥ ❢r♦♠ t❤❡ s❛♠❡ r♦✇ ♦r ❢r♦♠ t❤❡ s❛♠❡ ❝♦❧✉♠♥✳ Pr♦✈❡ t❤❛t t❤❡✐r s✉♠ ✐s ❛❧✇❛②s t❤❡ s❛♠❡✳
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73
✷✳ ▲❡tF❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ❢r❛❝t✐♦♥sm/n✇❤❡r❡m❛♥❞n❛r❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ✇✐t❤m+n62005✳
❋✐♥❞ t❤❡ ❧❛r❣❡st ♥✉♠❜❡ra✐♥ Fs✉❝❤ t❤❛t a < 16/23✳
✸✳ ❆ ✇❛r❡❤♦✉s❡ ❝♦♥t❛✐♥s175❜♦♦ts ♦❢ s✐③❡8✱175❜♦♦ts ♦❢ s✐③❡9❛♥❞ 200❜♦♦ts ♦❢ s✐③❡10✳ ❖❢
t❤❡s❡550❜♦♦ts✱250❛r❡ ❢♦r t❤❡ ❧❡❢t ❢♦♦t ❛♥❞300❢♦r t❤❡ r✐❣❤t ❢♦♦t✳ ▲❡tn❞❡♥♦t❡ t❤❡ t♦t❛❧
♥✉♠❜❡r ♦❢ ✉s❛❜❧❡ ♣❛✐rs ♦❢ ❜♦♦ts ✐♥ t❤❡ ✇❛r❡❤♦✉s❡✳ ✭❆ ✉s❛❜❧❡ ♣❛✐r ❝♦♥s✐sts ♦❢ ❛ ❧❡❢t ❛♥❞ ❛ r✐❣❤t ❜♦♦t ♦❢ t❤❡ s❛♠❡ s✐③❡✳✮
✭❛✮ ■s n=50 ♣♦ss✐❜❧❡❄ ✭❜✮ ■sn=51♣♦ss✐❜❧❡❄
✹✳ ❚❤❡ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡ ♦❢ tr✐❛♥❣❧❡ ABC t♦✉❝❤❡s t❤❡ s✐❞❡s BC, CA ❛♥❞ AB ❛t D, E ❛♥❞ F
r❡s♣❡❝t✐✈❡❧②✳ ▲❡t Q❞❡♥♦t❡ t❤❡ ♦t❤❡r ♣♦✐♥t ♦❢ ✐♥t❡rs❡❝t✐♦♥ ♦❢AD ❛♥❞ t❤❡ ✐♥s❝r✐❜❡❞ ❝✐r❝❧❡✳
Pr♦✈❡ t❤❛tEQ ❡①t❡♥❞❡❞ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♠✐❞♣♦✐♥t ♦❢ AF✐❢ ❛♥❞ ♦♥❧② ✐❢ AC=BC✳
✺✳ ▲❡tx1, x2, . . . , xn❜❡ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs ✇✐t❤ ♣r♦❞✉❝t ❡q✉❛❧ t♦ ✶✳ Pr♦✈❡ t❤❛t t❤❡r❡ ❡①✐sts ❛
k∈{1, 2 . . . , n}s✉❝❤ t❤❛t
xk
k+x1+x2· · ·+xk
>1− 1
n
√
2.
✻✳ ❈♦♥s✐❞❡r t❤❡ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, 19, . . .♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs✱
♦❜t❛✐♥❡❞ ❜② ❝♦♥❝❛t❡♥❛t✐♥❣ ❛❧t❡r♥❛t✐♥❣ ❜❧♦❝❦s {1},{2, 4},{5, 7, 9},{10, 12, 14, 16}, . . . ♦❢ ♦❞❞
❛♥❞ ❡✈❡♥ ♥✉♠❜❡rs✳ ❊❛❝❤ ❜❧♦❝❦ ❝♦♥t❛✐♥s ♦♥❡ ♠♦r❡ ❡❧❡♠❡♥t t❤❛♥ t❤❡ ♣r❡✈✐♦✉s ♦♥❡ ❛♥❞ t❤❡ ☞rst ❡❧❡♠❡♥t ✐♥ ❡❛❝❤ ❜❧♦❝❦ ✐s ♦♥❡ ♠♦r❡ t❤❛♥ t❤❡ ❧❛st ❡❧❡♠❡♥t ♦❢ t❤❡ ♣r❡✈✐♦✉s ♦♥❡✳ Pr♦✈❡ t❤❛t t❤❡ n✲t❤ ❡❧❡♠❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡ ✐s ❣✐✈❡♥ ❜②
2n−
1+√8n−7 2
.