❙♦✉t❤ ❆❢r✐❝❛♥ ▼❛t❤❡♠❛t✐❝s ❖❧②♠♣✐❛❞
❚❤✐r❞ ❘♦✉♥❞ ✷✵✵✶✿ ❙♦❧✉t✐♦♥s
✶✳ ❋♦r t❤❡ ✉♣♣❡r ❜♦✉♥❞✱ ✉s❡ t❤❡ tr✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t② ✐♥ tr✐❛♥❣❧❡s ABC ❛♥❞ ADC. ❚❤❡♥ AB+BC > AC ❛♥❞ AD+DC > AC, ❣✐✈✐♥❣p > 2AC. ❙✐♠✐❧❛r❧②✱p > 2BD. ❚❤❡r❡❢♦r❡ 2p > 2AC+2BD.
❋♦r t❤❡ ❧♦✇❡r ❜♦✉♥❞✱ ❧❡t AC ❛♥❞BD ♠❡❡t ✐♥E. ❙✐♥❝❡ABCD✐s ❝♦♥✈❡①✱ E✐s ✐♥s✐❞❡ t❤❡ q✉❛❞r✐❧❛t❡r❛❧✱ ❛♥❞ t❤❡r❡❢♦r❡ AE+EC=AC ❛♥❞ BE+ED=BD. ◆♦✇ ✉s❡ t❤❡ tr✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t② ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦✉r s♠❛❧❧ tr✐❛♥❣❧❡s✱ ❣✐✈✐♥❣
p=AB+BC+CD+DA
<(AE+EB) + (BE+EC) + (CE+ED) + (DE+EA) =2AE+2BE+2CE+2DE=2(AC+DB).
✷✳ ▲❡t x+y+z=xyz=a, xy+yz+zx=b.▼✉❧t✐♣❧②✐♥❣ ♦✉t t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✱ ✇❡ ☞♥❞ x−xy2−xz2+xy2z2+y−yz2−yx2+yz2x2+ +z−zx2−zy2+zx2y2
= (x+y+z) − (xy2+zy2+xyz) − (yx2+zx2+xyz)− − (yz2+yx2+xyz) +3xyz+xyz(yz+zx+xy) =4a−xb−yb−zb+ab
=4a− (x+y+z)b+ab
=4a.
❚❤❡r❡❢♦r❡ t❤❡ ☞rst ❡q✉❛t✐♦♥ ✐s r❡❞✉♥❞❛♥t✳ ❋r♦♠ t❤❡ s❡❝♦♥❞ ❡q✉❛t✐♦♥ ✇❡ ♦❜t❛✐♥ z = (x+y)/(xy−1). ❚❤❡ r❡q✉✐r❡❞ tr✐♣❧❡s ❛r❡ t❤❡r❡❢♦r❡ (x, y,(x+y)/(xy−1))✇❤❡r❡ x ✐s ❛♥② r❡❛❧ ♥✉♠❜❡r ❛♥❞ y❛♥② r❡❛❧ ♥✉♠❜❡r ❡①❝❡♣t1/x.
✸✳ ■❢x=1 t❤❡r❡ ✐s ♥♦t❤✐♥❣ t♦ ♣r♦✈❡✱ s♦ ❧❡tx1919=k+t, x1960=m+t❛♥❞ x2001=n+t, ✇❤❡r❡ k, m❛♥❞ n ❛r❡ ❞✐st✐♥❝t ✐♥t❡❣❡rs ❛♥❞06t < 1.❚❤❡♥
x41= m+t
k+t = n+t m+t. ❚❤❡r❡❢♦r❡
0= (m+t)2− (k+t)(n+t) = (m2−kn) − (k−2m+n)t.
■❢k−2m+n=0,t❤❡♥ ❛❧s♦m2−kn=0❛♥❞ ✐t ❢♦❧❧♦✇s t❤❛t0=m2−k(2m−k) = (m−k)2
✇❤✐❝❤ ❝♦♥tr❛❞✐❝ts k 6= m.❚❤❡r❡❢♦r❡ t = (m2−kn)/(k−2m+n) ✐s r❛t✐♦♥❛❧✱ ❛♥❞ x41
❛♥❞ x1919 ❛r❡ ❜♦t❤ r❛t✐♦♥❛❧✳ ❙✐♥❝❡ ✹✶ ✐s ♣r✐♠❡ ❛♥❞ 1919= 19×101✐s ♥♦t ❞✐✈✐s✐❜❧❡ ❜② ✹✶✱ ✐t ❢♦❧❧♦✇s t❤❛t ✹✶ ❛♥❞ ✶✾✶✾ ❛r❡ ♠✉t✉❛❧❧② ♣r✐♠❡✱ ❛♥❞ ✇❡ ❝❛♥ ☞♥❞ ✐♥t❡❣❡rs a ❛♥❞ b s✉❝❤ t❤❛t 41a+1919b=1.❚❤❡r❡❢♦r❡
x=x41a+1919b
= (x41)a+ (x1919)b
✐s r❛t✐♦♥❛❧✱ s❛② x=u/v✇✐t❤v > 0 ❛♥❞u❛♥❞ v ♠✉t✉❛❧❧② ♣r✐♠❡✳
❚❤❡♥ t❤❡ ❞❡♥♦♠✐♥❛t♦rs ♦❢ x1960 ❛♥❞ x2001, ❡①♣r❡ss❡❞ ✐♥ ❧♦✇❡st t❡r♠s✱ ❛r❡ v1960 ❛♥❞
v2001.❇✉t s✐♥❝❡x1960−x2001✐s ❛♥ ✐♥t❡❣❡r✱ t❤❡s❡ ❞❡♥♦♠✐♥❛t♦rs ♠✉st ❜❡ ❡q✉❛❧✳ ■t ❢♦❧❧♦✇s
✹✳ ▲❡tR❛♥❞B❜❡ t❤❡ s❡ts ♦❢ r❡❞ ❛♥❞ ❜❧✉❡ ♣♦✐♥ts r❡s♣❡❝t✐✈❡❧②✳ ❋♦r ❡❛❝❤ ♦♥❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ f:R→B,❞❡☞♥❡ t❤❡ ❧❡♥❣t❤L(f)♦❢ t❤❡ ♣❛✐r✐♥❣ft♦ ❜❡ t❤❡ s✉♠ ♦❢ t❤❡ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ ❝♦rr❡s♣♦♥❞✐♥❣ ♣♦✐♥ts✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✐❢ d(r, f(r))✐s t❤❡ ❞✐st❛♥❝❡ ❜❡✇❡❡♥ ❝♦rr❡s♣♦♥❞✐♥❣
♣♦✐♥ts✱ L(f) = Pr∈Rd(r, f(r)). ◆♦t❡ t❤❛t t❤❡r❡ ❛r❡ ♦♥❧② ☞♥✐t❡❧② ♠❛♥② s✉❝❤ ❢✉♥❝t✐♦♥s✱
s♦ ❛♠♦♥❣st ❛❧❧ t❤❡ L(f) t❤❡r❡ ✐s ❛ ❧❡❛st ♦♥❡✳ ❈❛❧❧ ✐t L(g)✳ ❲❡ ❝❧❛✐♠ t❤❛t g ❤❛s ♥♦ ✐♥t❡rs❡❝t✐♥❣ s❡❣♠❡♥ts✳
■❢ ♥♦t✱ ✇❡ ❤❛✈❡ ❛ ♣❛✐r ♦❢ ♦☛❡♥❞✐♥❣ ✐♥✐t✐❛❧ ♣♦✐♥ts r1 ❛♥❞ r2, ✇❤♦s❡ s❡❣♠❡♥ts ❝r♦ss✳ ◆♦✇
❧❡t ✖g ❜❡ t❤❡ s❛♠❡ ❛s g❡①❝❡♣t t❤❛t t❤❡ ❡♥❞ ♣♦✐♥ts ♦❢ r1 ❛♥❞ r2❛r❡ s✇✐t❝❤❡❞✳ ❚❤❡r❡❢♦r❡
(r1,g✖(r1)) ❛♥❞ (r2,g✖(r2))❛r❡ ♦♣♣♦s✐t❡ s✐❞❡s ♦❢ t❤❡ ❝♦♥✈❡① q✉❛❞r✐❧❛t❡r❛❧ ✇✐t❤ ❞✐❛❣♦♥❛❧s
(r1, g(r1))❛♥❞ (r2, g(r2)).❆s s❤♦✇♥ ✐♥ Pr♦❜❧❡♠ ✶✱
d(r1,g✖(r1)) +d(r2,g✖(r2))< d(r1, g(r1)) +d(r2, g(r2)).
❍❡♥❝❡ L(g✖)< L(g)✱ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✱ s✐♥❝❡L(g)✇❛s s✉♣♣♦s❡❞❧② ❧❡❛st✳
Second solution: ❈❛❧❧ ❛ s❡t ♦❢ ♣♦✐♥ts ❜❛❧❛♥❝❡❞ ✐❢ ✐t ❤❛s ❛s ♠❛♥② r❡❞ ❛s ❜❧✉❡ ♣♦✐♥ts✳ ❚❤❡ ♣r♦♦❢ ✐s ❜② str♦♥❣ ✐♥❞✉❝t✐♦♥ ♦♥ n. ❚❤❡ ♣❛✐r✐♥❣ ✐s tr✐✈✐❛❧❧② ♣♦ss✐❜❧❡ ✇❤❡♥ n = 1. ❚❤❡ ✐❞❡❛ ✐s t♦ ☞♥❞ ❛ str❛✐❣❤t ❧✐♥❡ t❤❛t ❞✐✈✐❞❡sS✐♥t♦ ❜❛❧❛♥❝❡❞ s✉❜s❡ts ♦❢2k❛♥❞2n−2k ♣♦✐♥ts✱ ✇❤❡r❡1 < k < n.❚❤❡s❡ s✉❜s❡ts ❝❛♥ ❜❡ ♣❛✐r❡❞ ❜② t❤❡ ✐♥❞✉❝t✐♦♥ ❤②♣♦t❤❡s✐s✱ ❛♥❞ t❤❡ ❧✐♥❡ s❡❣♠❡♥ts ❝❛♥♥♦t ❝r♦ss t❤❡ ❞✐✈✐❞✐♥❣ ❧✐♥❡✱ s♦ ♥❡✐t❤❡r s✉❜s❡t ❝❛♥ ✐♥t❡r❢❡r❡ ✇✐t❤ t❤❡ ♦t❤❡r✳
❈♦♥s✐❞❡r t❤❡ s♠❛❧❧❡st ❝♦♥✈❡① ♣♦❧②❣♦♥ Ct❤❛t ❝♦♥t❛✐♥s ❛❧❧ t❤❡ ❣✐✈❡♥ ♣♦✐♥ts✳ ❊❛❝❤ ♦❢ ✐ts ✈❡rt✐❝❡s ♦❜✈✐♦✉s❧② ❝♦✐♥❝✐❞❡s ✇✐t❤ s♦♠❡ ❣✐✈❡♥ ♣♦✐♥t✳ ■❢ ❛♥② t✇♦ ✈❡rt✐❝❡sA❛♥❞B♦❢C❛r❡ ♦❢ ❞✐☛❡r❡♥t ❝♦❧♦✉r✱ ♠❛t❝❤ t❤❡♠✳ ◆♦♥❡ ♦❢ t❤❡ ♦t❤❡r ♣♦✐♥ts ❝❛♥ ❧✐❡ ♦♥ t❤❡ ❧✐♥❡ AB ✭t❤❛t ✇♦✉❧❞ ❣✐✈❡ t❤r❡❡ ❝♦❧❧✐♥❡❛r ♣♦✐♥ts✮ ❛♥❞ t❤❡r❡❢♦r❡ t❤❡② ❛❧❧ ❧✐❡ t♦ t❤❡ s❛♠❡ s✐❞❡ ♦❢ t❤❡ ❧✐♥❡ AB, ❣✐✈✐♥❣ ❛ ❜❛❧❛♥❝❡❞ s❡t ♦❢ 2n−2 ♣♦✐♥ts ✇❤✐❝❤ ❝❛♥ ❜❡ ♣❛✐r❡❞✳
■♥ t❤❡ ❝❛s❡ t❤❛t ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ C ❛r❡ ♦❢ t❤❡ s❛♠❡ ❝♦❧♦✉r✱ s❛② r❡❞✱ ❞r❛✇ ❛ ❧✐♥❡ ♥♦t ♣❛ss✐♥❣ t❤r♦✉❣❤ ❛♥② ❣✐✈❡♥ ♣♦✐♥t✱ ✇✐t❤ s❧♦♣❡ ❞✐☛❡r❡♥t ❢r♦♠ ❛♥② ♣♦ss✐❜❧❡ ❧✐♥❡ ❝♦♥♥❡❝t✐♥❣ t✇♦ ❣✐✈❡♥ ♣♦✐♥ts✱ s❡♣❛r❛t✐♥❣ t❤❡ s❡t ✐♥t♦ t✇♦ ♥♦♥✲❡♠♣t② s✉❜s❡ts✱ ❡❛❝❤ ✇✐t❤ ❛♥ ❡✈❡♥ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts✳ ❚❤✐s ✐s ♣♦ss✐❜❧❡ ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ ♦♥❧② ☞♥✐t❡❧② ♠❛♥② s❧♦♣❡s✳
■❢ t❤❡s❡ s✉❜s❡ts ❛r❡ ♥♦t ❜❛❧❛♥❝❡❞✱ ♦♥❡ ♦❢ t❤❡♠ ❤❛s ❛♥ ❡①❝❡ss ♦❢ ❜❧✉❡ ♣♦✐♥ts✳ ▼♦✈❡ t❤❡ ❧✐♥❡ ♣❛r❛❧❧❡❧ t♦ ✐ts❡❧❢ s♦ t❤❛t t✇♦ ♣♦✐♥ts ❛r❡ tr❛♥s❢❡rr❡❞ ❢r♦♠ t❤❛t s✉❜s❡t t♦ t❤❡ ♦t❤❡r✳ ❈♦♥t✐♥✉❡ ❞♦✐♥❣ s♦ ✉♥t✐❧ t❤❡ s✉❜s❡ts ❛r❡ ❜❛❧❛♥❝❡❞✳
❚❤✐s ♠✉st ❤❛♣♣❡♥ ❜❡❝❛✉s❡ ✇❤❡♥ ♦♥❧② t✇♦ ♣♦✐♥ts ❛r❡ ❧❡❢t✱ ♦♥❡ ♦❢ t❤❡♠ ✐s ❛ ✈❡rt❡① ♦❢ C ❛♥❞ t❤❡r❡❢♦r❡ r❡❞✳ ❙♦ t❤❛t ❧❛st ♣❛✐r ❝❛♥♥♦t ❤❛✈❡ ❛ ❜❧✉❡ ❡①❝❡ss✳
✺✳ ❉r❛✇ t❤❡ ❞✐❛❣♦♥❛❧s ♦❢ Q0 = ABCD ❛♥❞ ❧❡t CbAD = α, DbBA = β, ACBb = γ ❛♥❞
BDCb = δ. ▲❛❜❡❧ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ q✉❛❞r✐❧❛t❡r❛❧ Q1 = A1B1C1D1 s♦ t❤❛t A1BB1,
B1CC1, C1DD1❛♥❞ D1AA1❛r❡ ✐ts s✐❞❡s✳ ❘❡♣❡❛t❡❞ ✉s❡ ♦❢ t❤❡ t❛♥✲❝❤♦r❞ t❤❡♦r❡♠ ❣✐✈❡s
t❤❛t t❤❡ ❛♥❣❧❡s ♦❢ A1B1C1D1 ❛r❡2α, 2β, 2γ ❛♥❞ 2δ r❡s♣❡❝t✐✈❡❧②✳ ■❢ A1B1C1D1 ✐s ❛❧s♦
❝②❝❧✐❝✱ ✐t ❢♦❧❧♦✇s t❤❛t
α+γ=90◦
=β+δ.
❉r❛✇ t❤❡ ❞✐❛❣♦♥❛❧s ♦❢ A1B1C1D1 ❛♥❞ ❧❡t α1, β1 ❡t❝✳ ❜❡ t❤❡ ❛♥❛❧♦❣♦✉s ❛♥❣❧❡s t♦ α, β
❡t❝✳ ■❢ t❤❡ s❡q✉❡♥❝❡ ✐s t♦ ❝♦♥t✐♥✉❡ t♦ ❛ t❤✐r❞ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧✱ ❜② t❤❡ ❛❜♦✈❡ ❛r❣✉♠❡♥t ❛♣♣❧✐❡❞ t♦ A1B1C1D1✇❡ r❡q✉✐r❡ t❤❛t
α1+γ1=90◦ =β1+δ1.
❇② ❝♦♥s✐❞❡r❛t✐♦♥ ♦❢ t❤❡ ❛♥❣❧❡s ✐♥ △A1B1D1✇❡ ☞♥❞ t❤❛t
180◦=
2α+β1+γ1
=2α+ (2β−α1) + (90◦−α1)
⇒α1=α+β−45◦;
β1=β−α+45◦.
❘❡✇r✐t❡ t❤❡s❡ ❡q✉❛t✐♦♥s ❛s
(α1−45◦) = (α−45◦) + (β−45◦);
(β1−45◦) = (β−45◦) − (α−45◦).
❈♦♥t✐♥✉❡ t♦ Q2:
(α2−45◦) = (α1−45◦) + (β1−45◦) =2(β−45◦);
(β2−45◦) = (β1−45◦) − (α1−45◦) = −2(α−45◦);
❛♥❞ t♦ Q4:
(α4−45◦) =2(β2−45◦) = −4(α−45◦).
◆♦✇ ♠♦✈❡ ✐♥ st❡♣s ♦❢ ❢♦✉r t♦ ☞♥❞
(α4n−45◦) = (−4) n
(α−45◦) .
■❢ Q4n ✐s ❝♦♥✈❡①✱ ✇❡ ♠✉st ❤❛✈❡ 0 < α4n< 180◦. ❇✉t ✉♥❧❡ss α = 45◦ ✭✇❤❡♥ ABCD ✐s
❛ sq✉❛r❡✮ t❤❡ q✉❛♥t✐t② α4n−45◦ ✐s ❛ ❞✐✈❡r❣❡♥t ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡ ❛♥❞ ✇✐❧❧ ❡✈❡♥t✉❛❧❧②
♠♦✈❡ ♦✉ts✐❞❡ t❤❡ r❛♥❣❡−45◦ t♦135◦,❛♥❞ t❤❡r❡❢♦r❡ t❤❡ s❡q✉❡♥❝❡ ✐s ☞♥✐t❡✳
✻✳ ▲❡t^x1,^x2, . . . ,^xn❜❡ ❛♥♦t❤❡r s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs t❤❛t s❛t✐s☞❡s t❤❡ ❡q✉❛t✐♦♥s✳ ❲❡ s❤❛❧❧
s❤♦✇✿
✇❤❡♥n✐s ❡✈❡♥✱ t❤❡ ^xi❝❛♥♥♦t s❛t✐s❢② t❤❡ ♦r❞❡r r❡❧❛t✐♦♥s❀
✇❤❡♥ n ✐s ♦❞❞✱ ✐t ✐s ♣♦ss✐❜❧❡ t❤❛t t❤❡ xi ❛r❡ s✉❝❤ t❤❛t t❤❡ ^xi ❝❛♥ s❛t✐s❢② t❤❡ ♦r❞❡r
r❡❧❛t✐♦♥s✳
❚❤❡r❡❢♦r❡ t❤❡ ❛♥s✇❡r t♦ t❤❡ ♣r♦❜❧❡♠ ✐s✿ ❛❧❧ ❡✈❡♥ n>4.
❲❡ ☞rst ❝♦♥s✐❞❡r t❤❡ ❝❛s❡ ✇❤❡r❡ n=2m.❚❤❡ ❝♦♥❞✐t✐♦♥s ❢♦rdi❛♥❞s ✐♠♣❧② t❤❛t t❤❡r❡
✐s ❛ ♥✉♠❜❡rh s✉❝❤ t❤❛t
^
xi−xi=
h, ❢♦r ♦❞❞i❀
−h, ❢♦r ❡✈❡♥i✳
❙✉✐❞✲❆❢r✐❦❛❛♥s❡ ❲✐s❦✉♥❞❡✲♦❧✐♠♣✐❛❞❡
❉❡r❞❡ ❘♦♥❞❡ ✷✵✵✶✿ ❖♣❧♦ss✐♥❣s
✶✳ ❱✐r ❞✐❡ ❜♦❣r❡♥s✱ ❣❡❜r✉✐❦ ❞✐❡ ❞r✐❡❤♦❡❦s♦♥❣❡❧②❦❤❡✐❞ ✐♥ ❞r✐❡❤♦❡❦❡ ABC ❡♥ ADC. ❉❛♥ AB + BC > AC ❡♥ AD+ DC > AC, ❞✉s p > 2AC. ◆❡t s♦✱ p > 2BD. ❉❛❛r♦♠ 2p > 2AC+2BD.
❱✐r ❞✐❡ ♦♥❞❡r❣r❡♥s✱ ❧❛❛t AC ❡♥ BD ✐♥ Es♥②✳ ❖♠❞❛t ABCD❦♦♥✈❡❦s ✐s✱ ✐s E ❜✐♥♥❡ ❞✐❡ ✈✐❡r❤♦❡❦✱ ❡♥ ❞❛❛r♦♠ AE+EC= AC ❡♥ BE+ED = BD. ●❡❜r✉✐❦ ❞✐❡ ❞r✐❡❤♦❡❦s♦♥❣❡✲ ❧②❦❤❡✐❞ ✐♥ ❡❧❦ ✈❛♥ ❞✐❡ ✈✐❡r ❦❧❡✐♥ ❞r✐❡❤♦❡❦❡✱ ✇❛❛r✉✐t
p=AB+BC+CD+DA
<(AE+EB) + (BE+EC) + (CE+ED) + (DE+EA) =2AE+2BE+2CE+2DE
=2(AC+DB).
✷✳ ▲❛❛t x+y+z=xyz=a, xy+yz+zx=b. ❆s ♦♥s ❞✐❡ ❧✐♥❦❡r❦❛♥t ✉✐t♠❛❛❧✱ ❦r② ♦♥s x−xy2−xz2+xy2z2+y−yz2−yx2+yz2x2+z−zx2−zy2+zx2y2
= (x+y+z) − (xy2+zy2+xyz) − (yx2+zx2+xyz)− − (yz2+yx2+xyz) +3xyz+xyz(yz+zx+xy) =4a−xb−yb−zb+ab
=4a− (x+y+z)b+ab
=4a.
❉✐❡ ❡❡rst❡ ✈❡r❣❡❧②❦✐♥❣ ✐s ❞✉s ♦♦r❜♦❞✐❣✳✳ ❯✐t ❞✐❡ t✇❡❡❞❡ ✈❡r❣❡❧②❦✐♥❣ ✈❡r❦r② ♦♥s z = (x+y)/(xy−1).❉✐❡ ✈❡r❡✐st❡ tr✐♣❧❡tt❡ ✐s ❞✉s(x, y,(x+y)/(xy−1))✇❛❛rx❡♥✐❣❡ r❡⑧❡❧❡ ❣❡t❛❧ ✐s✱ ❡♥ y❡♥✐❣❡ r❡⑧❡❧❡ ❣❡t❛❧ ❜❡❤❛❧✇❡ 1/x.
✸✳ ❆s x = 1 ✐s ❞❛❛r ♥✐❦s ♦♠ t❡ ❜❡✇②s ♥✐❡❀ ❧❛❛t ❞✉s x1919 = k+ t, x1960 = m+t ❡♥
x2001=n+t,✇❛❛r k, m❡♥ n❛❧♠❛❧ ✈❡rs❦✐❧❧❡♥❞❡ ❤❡❡❧t❛❧❧❡ ✐s✱ ❡♥ 06t < 1. ❉❛♥
x41= m+t
k+t = n+t m+t. ❉✉s
0= (m+t)2− (k+t)(n+t) = (m2−kn) − (k−2m+n)t.
❆sk−2m+n=0,❞❛♥ ♦♦❦m2−kn=0❡♥ ❞✐t ✈♦❧❣ ❞❛t0=m2−k(2m−k) = (m−k)2
✇❛t k6= m✇❡❡rs♣r❡❡❦✳ ❉✉s ✐s t= (m2−kn)/(k−2m+n) r❛s✐♦♥❛❛❧✱ ❡♥ x41 ❡♥ x1919
✐s ❛❧❜❡✐ r❛s✐♦♥❛❛❧✳ ❖♠❞❛t ✹✶ ♣r✐❡♠ ✐s ❡♥ 1919 = 19×101♥✐❡ ❞❡❡❧❜❛❛r ❞❡✉r ✹✶ ✐s ♥✐❡✱ ✈♦❧❣ ❞❛t ✹✶ ❡♥ ✶✾✶✾ ♦♥❞❡r❧✐♥❣ ♣r✐❡♠ ✐s✳ ❖♥s ❦❛♥ ❞✉s ❤❡❡❧❣❡t❛❧❧❡ a ❡♥ b ✈✐♥❞ s♦❞❛t 41a+1919b=1.❉✉s ✐s
x=x41a+1919b= (x41)a+ (x1919)b
r❛s✐♦♥❛❛❧✱ s❫❡x=u/v♠❡t v > 0❡♥ u❡♥ v ♦♥❞❡r❧✐♥❣ ♣r✐❡♠✳
❉❛♥ ✐s ❞✐❡ ♥♦❡♠❡rs ✈❛♥x1960❡♥ x2001,✉✐t❣❡❞r✉❦ ✐♥ ❦❧❡✐♥st❡ t❡r♠❡✱v1960❡♥ v2001.▼❛❛r
✹✳ ▲❛❛t R ❡♥ B♦♥❞❡rs❦❡✐❞❡❧✐❦ ❞✐❡ ✈❡rs❛♠❡❧✐♥❣s r♦♦✐ ❡♥ ❜❧♦✉ ♣✉♥t❡ ✇❡❡s✳ ❱✐r ❡❧❦❡ ❡❡♥✲t♦t✲ ❡❡♥ ❢✉♥❦s✐❡ f : R → B, ❞❡☞♥✐❡❡r ❞✐❡ ❧❡♥❣t❡ L(f) ✈❛♥ ❞✐❡ ❛❢♣❛r✐♥❣ f ❛s ❞✐❡ s♦♠ ✈❛♥ ❞✐❡ ❛❢st❛♥❞❡ t✉ss❡♥ ♦♦r❡❡♥st❡♠♠❡♥❞❡ ♣✉♥t❡✳ ▼❡❡r ♣r❡s✐❡s✱ ❛s d(r, f(r))❞✐❡ ❛❢st❛♥❞ t✉ss❡♥
♦♦r❡❡♥st❡♠♠❡♥❞❡ ♣✉♥t❡ ✐s✱ ❞❛♥ ✐sL(f) =Pr∈Rd(r, f(r)).▲❡t ♦♣✿ ❞❛❛r ✐s ♥❡t ✬♥ ❡✐♥❞✐❣❡
❛❛♥t❛❧ s✉❧❦❡ ❢✉♥❦s✐❡s✱ ❞✉s ✐s ❞❛❛r t✉ss❡♥ ❛❧ ❞✐❡L(f)✬♥ ❦❧❡✐♥st❡ ❡❡♥✳ ◆♦❡♠ ❞✐tL(g)✳ ❖♥s
❜❡✇❡❡r ❞❛t g ❣❡❡♥ s❡❣♠❡♥t❡ ❤❡t ✇❛t ♠❡❦❛❛r s♥② ♥✐❡✳
❆♥❞❡rs ❤❡t ♦♥s ✬♥ ♣❛❛r ❤✐♥❞❡r❧✐❦❡ ❜❡❣✐♥♣✉♥t❡r1❡♥r2,✇✐❡ s❡ s❡❣♠❡♥t❡ ❦r✉✐s✳ ▲❛❛t ♥♦✉ ✖g
❞✐❡s❡❧❢❞❡ ❛sg✇❡❡s✱ ❜❡❤❛❧✇❡ ❞❛t ❞✐❡ ❡✐♥❞♣✉♥t❡r1❡♥r2♦♠❣❡r✉✐❧ ✐s✳ ❉✉s ✐s(r1,g✖(r1))❡♥
(r2,g✖(r2))t❡❡♥♦♦rst❛❛♥❞❡ s②❡ ✈❛♥ ❞✐❡s❡❧❢❞❡ ❦♦♥✈❡❦s❡ ✈✐❡r❤♦❡❦ ♠❡t ❞✐❛❣♦♥❛❧❡(r1, g(r1))
❡♥ (r2, g(r2)).❙♦♦s ✐♥ Pr♦❜❧❡❡♠ ✶ ❛❛♥❣❡t♦♦♥✱
d(r1,g✖(r1)) +d(r2,g✖(r2))< d(r1, g(r1)) +d(r2, g(r2)).
❉✉s ✐sL(g✖)< L(g)✱ ✬♥ t❡❡♥s♣r❛❛❦✱ ✇❛♥tL(g) ✇❛s t♦❣ ❞✐❡ ❦❧❡✐♥st❡✳
Tweede oplossing: ◆♦❡♠ ✬♥ ✈❡rs❛♠❡❧✐♥❣ ❣❡❜❛❧❛♥s❡❡r❞ ❛s ❞✐t ❡✇❡ ✈❡❡❧ r♦♦✐ ❡♥ ❜❧♦✉ ♣✉♥t❡ ❜❡✈❛t✳ ❉✐❡ ❜❡✇②s ✇❡r❦ ♠❡t st❡r❦ ✐♥❞✉❦s✐❡ ♦♣n.❉✐❡ ❛❢♣❛r✐♥❣ ✐s tr✐✈✐❛❛❧ ♠♦♦♥t❧✐❦ ✇❛♥♥❡❡rn=1.❉✐❡ ✐❞❡❡ ✐s ♦♠ ✬♥ r❡❣✉✐t ❧②♥ t❡ ✈✐♥❞ ✇❛tS✐♥ ❣❡❜❛❧❛♥s❡❡r❞❡ s✉❜✈❡rs❛♠❡❧✲ ✐♥❣s ♠❡t 2k ❡♥ 2n−2k ♣✉♥t❡ ♦♣❞❡❡❧✱ ✇❛❛r 1 < k < n. ❍✐❡r❞✐❡ s✉❜✈❡rs❛♠❡❧✐♥❣s ❦❛♥ ✈♦❧❣❡♥s ❞✐❡ ✐♥❞✉❦s✐❡❤✐♣♦t❡s❡ ❛❢❣❡♣❛❛r ✇♦r❞✱ ❡♥ ❞✐❡ ❧②♥s❡❣♠❡♥t❡ ❦❛♥ ♥✐❡ ❞✐❡ s❦❡✐❞s❧②♥ ❦r✉✐s ♥✐❡✱ ❞✉s ❦❛♥ ❞✐❡ t✇❡❡ s✉❜✈❡rs❛♠❡❧✐♥❣s ♥✐❡ ♠❡❦❛❛r ♣❧❛ ♥✐❡✳
❇❡s❦♦✉ ❞✐❡ ❦❧❡✐♥st❡ ❦♦♥✈❡❦s❡ ✈❡❡❧❤♦❡❦C✇❛t ❛❧ ❞✐❡ ❣❡❣❡✇❡ ♣✉♥t❡ ♦♠s❧✉✐t✳ ❊❧❦❡ ❤♦❡❦♣✉♥t ❞❛❛r✈❛♥ ✈❛❧ ❦❧❛❛r❜❧②❦❧✐❦ s❛❛♠ ♠❡t ✬♥ ❣❡❣❡✇❡ ♣✉♥t✳ ❆s t✇❡❡ ❤♦❡❦♣✉♥t❡ A❡♥ B✈❛♥ C ✈❡rs❦✐❧❧❡♥❞❡ ❦❧❡✉r❡ ❤❡t✱ ♣❛❛r ❤✉❧❧❡ ❛❢✳ ◆✐❡ ❡❡♥ ✈❛♥ ❞✐❡ ❛♥❞❡r ♣✉♥t❡ ❦❛♥ ♦♣ ❞✐❡ ❧②♥ AB ❧❫❡ ♥✐❡ ✭❞✐t s♦✉ ❞r✐❡ s❛❛♠❧②♥✐❣❡ ♣✉♥t❡ ❣❡❡✮ ❡♥ ❞❛❛r♦♠ ❧❫❡ ❤✉❧❧❡ ❛❧♠❛❧ ❛❛♥ ❞✐❡s❡❧❢❞❡ ❦❛♥t ✈❛♥ ❞✐❡ ❧②♥ AB ❡♥ ✈♦r♠ ✬♥ ❣❡❜❛❧❛♥s❡❡r❞❡ ✈❡rs❛♠❡❧✐♥❣ ✈❛♥ 2n−2 ♣✉♥t❡ ✇❛t ❛❢❣❡♣❛❛r ❦❛♥ ✇♦r❞✱
■♥❣❡✈❛❧ ❛❧ ❞✐❡ ❤♦❡❦♣✉♥t❡ ✈❛♥ C ❞✐❡s❡❧❢❞❡ ❦❧❡✉r ❤❡t✱ s❫❡ r♦♦✐✱ tr❡❦ ✬♥ ❧②♥ ✇❛t ♥✐❡ ❞❡✉r ❡♥✐❣❡ ❣❡❣❡✇❡ ♣✉♥t ❣❛❛♥ ♥✐❡✱ ♠❡t ❤❡❧❧✐♥❣ ✈❡rs❦✐❧❧❡♥❞ ✈❛♥ ❡♥✐❣❡ ♠♦♦♥t❧✐❦❡ ❧②♥ t✉ss❡♥ t✇❡❡ ❣❡❣❡✇❡ ♣✉♥t❡✱ s♦❞❛t ❞✐t ❞✐❡ ✈❡rs❛♠❡❧✐♥❣ ✐♥ t✇❡❡ ♥✐❡✲❧❡⑧❡ ✈❡rs❛♠❡❧✐♥❣s ♦♣❞❡❡❧✱ ❡❧❦ ♠❡t ✬♥ ❡✇❡ ❛❛♥t❛❧ ♣✉♥t❡✳ ❉✐t ✐s ♠♦♦♥t❧✐❦✱ ✇❛♥t ❞❛❛r ✐s ♥❡t ✬♥ ❡✐♥❞✐❣❡ ❛❛♥t❛❧ ❤❡❧❧✐♥❣s ♦♠ t❡ ✈❡r♠②✳
❆s ❞✐❡ t✇❡❡ s✉❜✈❡rs❛♠❡❧✐♥❣s ♥✐❡ ❣❡❜❛❧❛♥s❡❡r❞ ✐s ♥✐❡✱ ❤❡t ❡❡♥ ✈❛♥ ❤✉❧❧❡ ✬♥ ♦♦r♠❛❛t ❜❧♦✉ ♣✉♥t❡✳ ❙❦✉✐❢ ❞✐❡ ❧②♥ ♣❛r❛❧❧❡❧ ❛❛♥ ❤♦♠s❡❧❢ s♦❞❛t t✇❡❡ ♣✉♥t❡ ✈❛♥ ❞❛❛r❞✐❡ ✈❡rs❛♠❡❧✐♥❣ ♥❛ ❞✐❡ ❛♥❞❡r ♦♦r❣❡❞r❛ ✇♦r❞✳ ❍♦✉ ❛❛♥ ♦♠ ❞✐t t❡ ❞♦❡♥ t♦t❞❛t ❞✐❡ s✉❜✈❡rs❛♠❡❧✐♥❣s ❣❡❜❛❧✲ ❛♥s❡❡r❞ ✐s✳
❉✐t ♠♦❡t ❣❡❜❡✉r✱ ✇❛♥t ❛s ❞❛❛r ♥❡t t✇❡❡ ♣✉♥t❡ ♦♦r ✐s✱ ✐s ❡❡♥ ✈❛♥ ❤✉❧❧❡ ✬♥ ❤♦❡❦♣✉♥t ✈❛♥ C ❡♥ ❞✉s r♦♦✐✳ ❉❛❛r❞✐❡ ❧❛❛st❡ ♣❛❛r ❦❛♥ ♥✐❡ ✬♥ ❜❧♦✉ ♦♦r♠❛❛t ❤❫❡ ♥✐❡✳
✺✳ ❚r❡❦ ❞✐❡ ❤♦❡❦❧②♥❡ ✈❛♥ Q0 = ABCD ❡♥ ❧❛❛t CADb = α, DBAb = β, ACBb = γ ❡♥
BDCb =δ.▼❡r❦ ❞✐❡ ♦♠❣❡s❦r❡✇❡ ✈✐❡r❤♦❡❦Q1=A1B1C1D1♦♣ s♦ ✬♥ ♠❛♥✐❡r ❞❛tA1BB1,
B1CC1, C1DD1 ❡♥ D1AA1 s② s②❡ ✐s✳ ❍❡r❤❛❛❧❞❡ ❣❡❜r✉✐❦ ✈❛♥ ❞✐❡ r❛❛❦❧②♥✲❦♦♦r❞✲st❡❧❧✐♥❣
❣❡❡ ❞❛t ❞✐❡ ❤♦❡❦❡ ✈❛♥A1B1C1D1♦♥❞❡rs❦❡✐❞❡❧✐❦2α, 2β, 2γ❡♥2δ✐s✳ ❆sA1B1C1D1♦♦❦
s✐❦❧✐❡s ✐s✱ ✈♦❧❣ ❞✐t ❞❛t
α+γ=90◦ =β+δ.
❚r❡❦ ❞✐❡ ❤♦❡❦❧②♥❡ ✈❛♥ A1B1C1D1 ❡♥ ❧❛❛tα1, β1❡♥s✳ ❞✐❡ ❤♦❡❦❡ ✇❡❡s ✇❛t ♠❡t α, β ❡♥s✳
♦♦r❡❡♥st❡♠✳ ❆s ❞✐❡ r② ♥❛ ✬♥ ✈♦❧❣❡♥❞❡ ✈✐❡r❤♦❡❦ ♠♦❡t ✈♦♦rt❣❛❛♥✱ ❣❡❡ ❤✐❡r❞✐❡ ❛r❣✉♠❡♥t✱
t♦❡❣❡♣❛s ♦♣ A1B1C1D1 ❞❛t
α1+γ1=90◦ =β1+δ1.
❉❡✉r ❞✐❡ ❤♦❡❦❡ ✐♥ △A1B1D1 t❡ ❜❡s❦♦✉✱ ✈✐♥❞ ♦♥s ❞❛t
180◦=2α+β
1+γ1
=2α+ (2β−α1) + (90◦−α1)
⇒α1=α+β−45◦;
β1=β−α+45◦.
❍❡rs❦r②❢ ❤✐❡r❞✐❡ ✈❡r❣❡❧②❦✐♥❣s ❛s
(α1−45◦) = (α−45◦) + (β−45◦);
(β1−45◦) = (β−45◦) − (α−45◦).
●❛❛♥ ❛❛♥ ♥❛ Q2:
(α2−45◦) = (α1−45◦) + (β1−45◦) =2(β−45◦);
(β2−45◦) = (β1−45◦) − (α1−45◦) = −2(α−45◦);
❡♥ ♥❛ Q4:
(α4−45◦) =2(β2−45◦) = −4(α−45◦).
❇❡✇❡❡❣ ♥♦✉ ✐♥ st❛♣♣❡ ✈❛♥ ✈✐❡r ♦♠ t❡ ❜❡✈✐♥❞ ❞❛t
(α4n−45◦) = (−4)n(α−45◦).
❆sQ4n ❦♦♥✈❡❦s ✐s✱ ♠♦❡t ♦♥s0 < α4n< 180◦ ❤❫❡✳ ▼❛❛r t❡♥s②α=45◦ ✭✇❛♥♥❡❡rABCD
✬♥ ✈✐❡r❦❛♥t ✐s✮ ✈♦r♠ α4n−45◦ ✬♥ ❞✐✈❡r❣❡♥t❡ ♠❡❡t❦✉♥❞✐❣❡ r②✱ ✇❛t ♦♣ ❞✐❡ ♦✉ ❡♥t ❜✉✐t❡
❞✐❡ ✐♥t❡r✈❛❧ t✉ss❡♥ −45◦ ♠♦❡t135◦ ✈❛❧✱ ❡♥ ❞✉s ✐s ❞✐❡ r② ❡✐♥❞✐❣✳
✻✳ ▲❛❛t^x1,^x2, . . . ,x^n♥♦❣ ✬♥ r② ❣❡t❛❧❧❡ ✇❡❡s ✇❛t ❞✐❡ ✈❡r❣❡❧②❦✐♥❣s ❜❡✈r❡❞✐❣✳ ❖♥s s❛❧ ❛❛♥t♦♦♥
❞❛t✿
❛s n❡✇❡ ✐s✱ ❦❛♥ ❞✐❡x^i♥✐❡ ❞✐❡ ♦r❞❡♥✐♥❣ ❜❡✈r❡❞✐❣ ♥✐❡❀
❛sn ♦♥❡✇❡ ✐s✱ ✐s ❞✐t ♠♦♦♥t❧✐❦ ♦♠xit❡ ❦r② s♦❞❛tx^i❞✐❡ ♦r❞❡♥✐♥❣ ❜❡✈r❡❞✐❣✳
❉✉s ✐s ❞✐❡ ❛♥t✇♦♦r❞ ♦♣ ❞✐❡ ♣r♦❜❧❡❡♠✿ ❛❧❧❡ ❡✇❡n>4.
❖♥s ❜❡s❦♦✉ ❡❡rs ❞✐❡ ❣❡✈❛❧ n=2m.❉✐❡ ✈♦♦r✇❛❛r❞❡s ✈✐r di❡♥ s ❜r✐♥❣ ♠❡❡ ❞❛t ❞❛❛r ✬♥
❣❡t❛❧ h ❜❡st❛❛♥ s♦❞❛t
^
xi−xi=
h, ✈✐r ♦♥❡✇❡i❀
−h, ✈✐r ❡✇❡i✳
