DISINI s3s2001
❙♦✉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❝❡♣t 1/x.
✸✳ ■❢ x = 1 t❤❡r❡ ✐s ♥♦t❤✐♥❣ t♦ ♣r♦✈❡✱ s♦ ❧❡t x1919 = k + t, x1960 = m + t ❛♥❞ x2001 = n + t,
✇❤❡r❡ k, m ❛♥❞ n ❛r❡ ❞✐st✐♥❝t ✐♥t❡❣❡rs ❛♥❞ 0 6 t < 1. ❚❤❡♥
x41 =
m+t
n+t
=
.
k+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❤❛t 0 = 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
t❤❛t v = 1.
✹✳ ▲❡t R ❛♥❞ B ❜❡ t❤❡ s❡ts ♦❢ r❡❞ ❛♥❞ ❜❧✉❡ ♣♦✐♥ts r❡s♣❡❝t✐✈❡❧②✳ ❋♦r ❡❛❝❤ ♦♥❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥
f : R → B, ❞❡☞♥❡ t❤❡ ❧❡♥❣t❤ L(f) ♦❢ t❤❡ ♣❛✐r✐♥❣ f t♦ ❜❡ t❤❡ s✉♠ ♦❢ t❤❡ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥
❝♦rr❡s♣♦♥❞✐♥❣ ♣♦✐♥ts✳
▼♦r❡ ♣r❡❝✐s❡❧②✱ ✐❢ d(r, f(r)) ✐s t❤❡ ❞✐st❛♥❝❡ ❜❡✇❡❡♥ ❝♦rr❡s♣♦♥❞✐♥❣
P
♣♦✐♥ts✱ L(f) = r∈R d(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)) ❛♥❞ (r2, g✖(r2)) ❛r❡ ♦♣♣♦s✐t❡ s✐❞❡s ♦❢ t❤❡ ❝♦♥✈❡① q✉❛❞r✐❧❛t❡r❛❧ ✇✐t❤ ❞✐❛❣♦♥❛❧s
(r1, g
(r1, g(r1)) ❛♥❞ (r2, g(r2)). ❆s s❤♦✇♥ ✐♥ Pr♦❜❧❡♠ ✶✱
✖(r1)) + d(r2, g✖(r2)) < d(r1, g(r1)) + d(r2, g(r2)).
d(r1, g
❍❡♥❝❡ 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 ❞✐✈✐❞❡s S ✐♥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❣♦♥ C t❤❛t ❝♦♥t❛✐♥s ❛❧❧ t❤❡ ❣✐✈❡♥ ♣♦✐♥ts✳ ❊❛❝❤ ♦❢ ✐ts
✈❡rt✐❝❡s ♦❜✈✐♦✉s❧② ❝♦✐♥❝✐❞❡s ✇✐t❤ s♦♠❡ ❣✐✈❡♥ ♣♦✐♥t✳ ■❢ ❛♥② t✇♦ ✈❡rt✐❝❡s A ❛♥❞ 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✳
b = α, DBA
b = β, ACB
b = γ ❛♥❞
✺✳ ❉r❛✇ t❤❡ ❞✐❛❣♦♥❛❧s ♦❢ Q0 = ABCD ❛♥❞ ❧❡t CAD
b = δ. ▲❛❜❡❧ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ q✉❛❞r✐❧❛t❡r❛❧ Q1 = A1B1C1D1 s♦ t❤❛t A1BB1,
BDC
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 x^1, x^2, . . . , x^n ❜❡ ❛♥♦t❤❡r s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs t❤❛t s❛t✐s☞❡s t❤❡ ❡q✉❛t✐♦♥s✳ ❲❡ s❤❛❧❧
s❤♦✇✿
✇❤❡♥ n ✐s ❡✈❡♥✱ t❤❡ x^i ❝❛♥♥♦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❤❡ x^i ❝❛♥ 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 ❢♦r di ❛♥❞ s ✐♠♣❧② t❤❛t t❤❡r❡
✐s ❛ ♥✉♠❜❡r h s✉❝❤ t❤❛t
x
^ i − xi =
h, ❢♦r ♦❞❞ i❀
−h, ❢♦r ❡✈❡♥ i✳
✸
❚❤❡ ❝♦♥❞✐t✐♦♥ ❢♦r t ✐♠♣❧✐❡s t❤❛t
0=
n
X
x
^2i − x2i
=
n
X
(^
xi − xi)(^
xi + xi)
i=1
i=1
m
X
=h
(2x2i−1 + h) − h
i=1
m
X
(2x2i − h)
i=1
= mh2 + 2h
m
X
x2i−1 + mh2 − 2h
i=1
m
X
= 2mh2 − 2h
m
X
x2i
i=1
(x2i − x2i−1)
i=1
❚❤❡ ❧❛st ❡q✉❛t✐♦♥ ❤❛s t✇♦ s♦❧✉t✐♦♥s✿ h = 0, t❤❡ ❭tr✉❡✧ s♦❧✉t✐♦♥✱ ❛♥❞
m
X
^= 1
(x2i − x2i−1),
h
m
i=1
t❤❡ ❭s♣✉r✐♦✉s✧ s♦❧✉t✐♦♥✳ ❋r♦♠ t❤❡ ♦r❞❡r r❡❧❛t✐♦♥s✱ ✐t ❢♦❧❧♦✇s t❤❛t h^ > 0, ✐✳❡✳ x^1 > x1. ■❢
t❤❡ x^i s❛t✐s❢② t❤❡ ♦r❞❡r r❡❧❛t✐♦♥s✱ ✇❡ ❝❛♥ st❛rt ✇✐t❤ x^i ❛s t❤❡ ❭tr✉❡✧ s♦❧✉t✐♦♥ ❛♥❞ r❡♣❡❛t
t❤❡ ♣r♦❝❡ss t♦ ♦❜t❛✐♥ ❛ t❤✐r❞ s♦❧✉t✐♦♥ ✐♥ ✇❤✐❝❤
⑦1 > x^1 > x1.
x
❇✉t t❤❡r❡ ❛r❡ ♦♥❧② t✇♦ s♦❧✉t✐♦♥s✳ ❚❤✐s ❝♦♥tr❛❞✐❝t✐♦♥ s❤♦✇s t❤❛t t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t
t❤❡ x^i s❛t✐s❢② t❤❡ ♦r❞❡r r❡❧❛t✐♦♥s ✐s ✉♥t❡♥❛❜❧❡✳
❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✇❤❡♥ n = 2m − 1 ✭m > 2✮ ♦♥❡ ❝❛♥ ❡❛s✐❧② ☞♥❞ ❛ ❝♦✉♥t❡r❡①❛♠♣❧❡✳
▲❡t xi = m − i + hi ✇❤❡r❡
h
m−1 ❢♦r ❡✈❡♥ i❀
hi =
−h
m ❢♦r ♦❞❞ i❀
✇❤❡r❡ h ✐s ❛♥② ♥♦♥③❡r♦ r❡❛❧ ♥✉♠❜❡r s✉❝❤ t❤❛t xi < xi+1 ❤♦❧❞s✱ ❡✳❣✳ h = 12 . ❍❡r❡ s = 0
❛♥❞ ❛❧❧ di = 1.P◆♦✇ t❛❦❡ x^i = −x2m−i.P❈❧❡❛r❧② t❤❡ P
x
^i s❛t✐s❢② t❤❡ ♦r❞❡r ❝♦♥❞✐t✐♦♥s✱ ❛❧❧
2m−1
2m−1 2
2m−1 2
x
^i+2 − x
^i = 1, i=1
x
^i = −s = 0 ❛♥❞ i=1
x
^i = i=1
xi = t. ❇✉t x
^m = −xm ❛♥❞
xm ✐s ♥♦♥③❡r♦✱ t❤❡r❡❢♦r❡ x
^m 6= xm ❛♥❞ t❤❡ x
^i ❢♦r♠ ❛ ❞✐☛❡r❡♥t ✈❛❧✐❞ s♦❧✉t✐♦♥✳
✹
❙✉✐❞✲❆❢r✐❦❛❛♥s❡ ❲✐s❦✉♥❞❡✲♦❧✐♠♣✐❛❞❡
❉❡r❞❡ ❘♦♥❞❡ ✷✵✵✶✿ ❖♣❧♦ss✐♥❣s
✶✳ ❱✐r ❞✐❡ ❜♦❣r❡♥s✱ ❣❡❜r✉✐❦ ❞✐❡ ❞r✐❡❤♦❡❦s♦♥❣❡❧②❦❤❡✐❞ ✐♥ ❞r✐❡❤♦❡❦❡
AB + BC > AC ❡♥ AD + DC > AC,
2p > 2AC + 2BD.
❞✉s
p > 2AC.
◆❡t s♦✱
ABC ❡♥ ADC. ❉❛♥
p > 2BD. ❉❛❛r♦♠
AC ❡♥ BD ✐♥ E s♥②✳ ❖♠❞❛t ABCD ❦♦♥✈❡❦s ✐s✱ ✐s E ❜✐♥♥❡ ❞✐❡
AE + EC = AC ❡♥ BE + ED = BD. ●❡❜r✉✐❦ ❞✐❡ ❞r✐❡❤♦❡❦s♦♥❣❡✲
❱✐r ❞✐❡ ♦♥❞❡r❣r❡♥s✱ ❧❛❛t
✈✐❡r❤♦❡❦✱ ❡♥ ❞❛❛r♦♠
❧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
(x + y)/(xy − 1). ❉✐❡ ✈❡r❡✐st❡ tr✐♣❧❡tt❡ ✐s ❞✉s (x, y, (x + y)/(xy − 1))
❣❡t❛❧ ✐s✱ ❡♥ y ❡♥✐❣❡ r❡⑧
❡❧❡ ❣❡t❛❧ ❜❡❤❛❧✇❡ 1/x.
x = 1 ✐s ❞❛❛r ♥✐❦s ♦♠ t❡
2001
x
= n + t, ✇❛❛r k, m ❡♥ n
✸✳ ❆s
✇❛❛r
x41 =
❡♥✐❣❡ r❡⑧
❡❧❡
x1919 = k + t, x1960 = m + t
❤❡❡❧t❛❧❧❡ ✐s✱ ❡♥ 0 6 t < 1. ❉❛♥
❜❡✇②s ♥✐❡❀ ❧❛❛t ❞✉s
❛❧♠❛❧ ✈❡rs❦✐❧❧❡♥❞❡
x
z =
❡♥
m+t
n+t
=
.
k+t
m+t
❉✉s
0 = (m + t)2 − (k + t)(n + t) = (m2 − kn) − (k − 2m + n)t.
k − 2m + n = 0, ❞❛♥ ♦♦❦ m2 − kn = 0 ❡♥ ❞✐t ✈♦❧❣ ❞❛t 0 = m2 − k(2m − k) = (m − k)2
2
41 ❡♥ x1919
✇❛t k 6= m ✇❡❡rs♣r❡❡❦✳ ❉✉s ✐s t = (m − kn)/(k − 2m + n) r❛s✐♦♥❛❛❧✱ ❡♥ x
✐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
❆s
x = x41a+1919b = (x41)a + (x1919)b
r❛s✐♦♥❛❛❧✱ s❫
❡
x = u/v
♠❡t
❉❛♥ ✐s ❞✐❡ ♥♦❡♠❡rs ✈❛♥
♦♠❞❛t
x1960 − x2001
✬♥
v>0
❡♥
u
❡♥
v
♦♥❞❡r❧✐♥❣ ♣r✐❡♠✳
x1960 ❡♥ x2001, ✉✐t❣❡❞r✉❦ ✐♥ ❦❧❡✐♥st❡ t❡r♠❡✱ v1960 ❡♥ v2001. ▼❛❛r
❤❡❡❧t❛❧ ✐s✱ ♠♦❡t ❤✐❡r❞✐❡ ♥♦❡♠❡rs ❣❡❧②❦ ✇❡❡s✳ ❉✐t ✈♦❧❣ ❞❛t v = 1.
✹✳ ▲❛❛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❡✳ P
▼❡❡r ♣r❡s✐❡s✱ ❛s d(r, f(r)) ❞✐❡ ❛❢st❛♥❞ t✉ss❡♥
♦♦r❡❡♥st❡♠♠❡♥❞❡ ♣✉♥t❡ ✐s✱ ❞❛♥ ✐s L(f) = r∈R d(r, f(r)). ▲❡t ♦♣✿ ❞❛❛r ✐s ♥❡t ✬♥ ❡✐♥❞✐❣❡
❛❛♥t❛❧ s✉❧❦❡ ❢✉♥❦s✐❡s✱ ❞✉s ✐s ❞❛❛r t✉ss❡♥ ❛❧ ❞✐❡ L(f) ✬♥ ❦❧❡✐♥st❡ ❡❡♥✳ ◆♦❡♠ ❞✐t L(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❡❧❢❞❡ ❛s g ✇❡❡s✱ ❜❡❤❛❧✇❡ ❞❛t ❞✐❡ ❡✐♥❞♣✉♥t❡ r1 ❡♥ r2 ♦♠❣❡r✉✐❧ ✐s✳ ❉✉s ✐s (r1, g✖(r1)) ❡♥
✖(r2)) t❡❡♥♦♦rst❛❛♥❞❡ s②❡ ✈❛♥ ❞✐❡s❡❧❢❞❡ ❦♦♥✈❡❦s❡ ✈✐❡r❤♦❡❦ ♠❡t ❞✐❛❣♦♥❛❧❡ (r1, g(r1))
(r2, g
❡♥ (r2, g(r2)). ❙♦♦s ✐♥ Pr♦❜❧❡❡♠ ✶ ❛❛♥❣❡t♦♦♥✱
✖(r1)) + d(r2, g✖(r2)) < d(r1, g(r1)) + d(r2, g(r2)).
d(r1, g
❉✉s ✐s L(g✖) < L(g)✱ ✬♥ t❡❡♥s♣r❛❛❦✱ ✇❛♥t L(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❧✐❦
✇❛♥♥❡❡r n = 1. ❉✐❡ ✐❞❡❡ ✐s ♦♠ ✬♥ r❡❣✉✐t ❧②♥ t❡ ✈✐♥❞ ✇❛t S ✐♥ ❣❡❜❛❧❛♥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 ❤❫❡ ♥✐❡✳
b = α, DBA
b = β, ACB
b = γ ❡♥
✺✳ ❚r❡❦ ❞✐❡ ❤♦❡❦❧②♥❡ ✈❛♥ Q0 = ABCD ❡♥ ❧❛❛t CAD
b
BDC = δ. ▼❡r❦ ❞✐❡ ♦♠❣❡s❦r❡✇❡ ✈✐❡r❤♦❡❦ Q1 = A1B1C1D1 ♦♣ s♦ ✬♥ ♠❛♥✐❡r ❞❛t A1BB1,
B1CC1, C1DD1 ❡♥ D1AA1 s② s②❡ ✐s✳ ❍❡r❤❛❛❧❞❡ ❣❡❜r✉✐❦ ✈❛♥ ❞✐❡ r❛❛❦❧②♥✲❦♦♦r❞✲st❡❧❧✐♥❣
❣❡❡ ❞❛t ❞✐❡ ❤♦❡❦❡ ✈❛♥ A1B1C1D1 ♦♥❞❡rs❦❡✐❞❡❧✐❦ 2α, 2β, 2γ ❡♥ 2δ ✐s✳ ❆s A1B1C1D1 ♦♦❦
s✐❦❧✐❡s ✐s✱ ✈♦❧❣ ❞✐t ❞❛t
α + γ = 90◦ = β + δ.
❚r❡❦ ❞✐❡ ❤♦❡❦❧②♥❡ ✈❛♥ A1B1C1D1 ❡♥ ❧❛❛t α1, 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 + β1 + 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◦ ).
0 < α4n < 180◦ ❤❫❡✳ ▼❛❛r t❡♥s② α = 45◦ ✭✇❛♥♥❡❡r ABCD
◦
✬♥ ✈✐❡r❦❛♥t ✐s✮ ✈♦r♠ α4n − 45 ✬♥ ❞✐✈❡r❣❡♥t❡ ♠❡❡t❦✉♥❞✐❣❡ r②✱ ✇❛t ♦♣ ❞✐❡ ♦✉ ❡♥t ❜✉✐t❡
◦
◦
❞✐❡ ✐♥t❡r✈❛❧ t✉ss❡♥ −45 ♠♦❡t 135 ✈❛❧✱ ❡♥ ❞✉s ✐s ❞✐❡ r② ❡✐♥❞✐❣✳
❆s
Q4n
✻✳ ▲❛❛t
❦♦♥✈❡❦s ✐s✱ ♠♦❡t ♦♥s
x
^1, x
^2, . . . , x
^n ♥♦❣ ✬♥ r② ❣❡t❛❧❧❡ ✇❡❡s ✇❛t ❞✐❡ ✈❡r❣❡❧②❦✐♥❣s ❜❡✈r❡❞✐❣✳
❖♥s s❛❧ ❛❛♥t♦♦♥
❞❛t✿
x
^i
❛s
n
❡✇❡ ✐s✱ ❦❛♥ ❞✐❡
❛s
n
♦♥❡✇❡ ✐s✱ ✐s ❞✐t ♠♦♦♥t❧✐❦ ♦♠
♥✐❡ ❞✐❡ ♦r❞❡♥✐♥❣ ❜❡✈r❡❞✐❣ ♥✐❡❀
xi
t❡ ❦r② s♦❞❛t
❉✉s ✐s ❞✐❡ ❛♥t✇♦♦r❞ ♦♣ ❞✐❡ ♣r♦❜❧❡❡♠✿ ❛❧❧❡ ❡✇❡
❖♥s ❜❡s❦♦✉ ❡❡rs ❞✐❡ ❣❡✈❛❧
❣❡t❛❧
h
n = 2m.
x
^i
❞✐❡ ♦r❞❡♥✐♥❣ ❜❡✈r❡❞✐❣✳
n > 4.
❉✐❡ ✈♦♦r✇❛❛r❞❡s ✈✐r
❜❡st❛❛♥ s♦❞❛t
x
^ i − xi =
h, ✈✐r ♦♥❡✇❡ i❀
−h, ✈✐r ❡✇❡ i✳
✼
di
❡♥
s
❜r✐♥❣ ♠❡❡ ❞❛t ❞❛❛r ✬♥
❉✐❡ ✈♦♦r✇❛❛r❞❡ ✈✐r t ❜r✐♥❣ ♠❡❡ ❞❛t
0=
n
X
x
^2i − x2i
=
n
X
(^
xi − xi)(^
xi + xi)
i=1
i=1
m
X
=h
(2x2i−1 + h) − h
i=1
m
X
(2x2i − h)
i=1
= mh2 + 2h
m
X
x2i−1 + mh2 − 2h
i=1
m
X
= 2mh2 − 2h
m
X
x2i
i=1
(x2i − x2i−1)
i=1
❉✐❡ ❧❛❛st❡ ✈❡r❣❡❧②❦✐♥❣ ❤❡t t✇❡❡ ♦♣❧♦ss✐♥❣s✿ h = 0, ❞✐❡ ❭✇❛r❡✧ ♦♣❧♦ss✐♥❣✱ ❡♥
m
X
^= 1
(x2i − x2i−1),
h
m
i=1
^ > 0, ❞✳✇✳s✳ x
❞✐❡ ❭✈❛❧s✧ ♦♣❧♦ss✐♥❣✳ ❯✐t ❞✐❡ ♦r❞❡♥✐♥❣ ✈♦❧❣ ❞❛t h
^1 > x1. ❆s ❛❧ ❞✐❡ x
^i
❞✐❡ ♦r❞❡♥✐♥❣ ❜❡✈r❡❞✐❣✱ ❦❛♥ ♦♥s ♠❡t x
^i ❛s ❞✐❡ ❭✇❛r❡✧ ♦♣❧♦ss✐♥❣ ❜❡❣✐♥ ❡♥ ❞✐❡ ❤❡❧❡ ♣r♦s❡s
❤❡r❤❛❛❧ ♦♠ ✬♥ ❞❡r❞❡ ♦♣❧♦ss✐♥❣ ♠❡t
⑦1 > x
x
^ 1 > x1
t❡ ✈❡r❦r②✳ ▼❛❛r ❞❛❛r ✐s ♥❡t t✇❡❡ ♦♣❧♦ss✐♥❣s✳ ❍✐❡r❞✐❡ t❡❡♥s♣r❛❛❦ t♦♦♥ ❛❛♥ ❞❛t ❞✐❡
❛❛♥♥❛♠❡ ❞❛t x
^i ❞✐❡ ♦r❞❡♥✐♥❣ ❜❡✈r❡❞✐❣✱ ♦♥❤♦✉❞❜❛❛r ✐s✳
❆❛♥ ❞✐❡ ❛♥❞❡r ❦❛♥t✱ ❛s n = 2m − 1 ✭m > 2✮ ❦❛♥ ✬♥ ♠❡♥s ♠❛❦❧✐❦ ✬♥ t❡❡♥✈♦♦r❜❡❡❧❞ ❦r②✳
▲❛❛t xi = m − i + hi ✇❛❛r
h
m−1 ✈✐r ❡✇❡ i❀
hi =
−h
m ✈✐r ♦♥❡✇❡ i❀
✇❛❛r h ❡♥✐❣❡ ♥✐❡✲♥✉❧ r❡⑧❡❧❡ ❣❡t❛❧ ✐s ✇❛❛r✈♦♦r xi < xi+1 ❣❡❧❞✱ ❜✈✳ h = 12 . ❍✐❡r ✐s s = 0
❡♥ ❛❧❧❡ di = 1.
♥♦✉ x
^i = −x2m−i
. ❉✉✐❞❡❧✐❦P❜❡✈r❡❞✐❣ x
^i ❞✐❡ ♦r❞❡♥✐♥❣✱ ✐s ❛❧❧❡
P◆❡❡♠
P2m−1
2m−1
2m−1 2
x
^i+2 − x
^i = 1, i=1
x
^i = −s = 0 ❡♥ i=1
x
^2i = i=1
xi = t. ▼❛❛r x
^m = −xm ❡♥
xm ✐s ♥✐❡✲♥✉❧✱ ❞✉s ✐s x
^m 6= xm ❡♥ x
^i ✈♦r♠ ✬♥ t✇❡❡❞❡ ❣❡❧❞✐❣❡ ♦♣❧♦ss✐♥❣✳
✽
❚❤✐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❝❡♣t 1/x.
✸✳ ■❢ x = 1 t❤❡r❡ ✐s ♥♦t❤✐♥❣ t♦ ♣r♦✈❡✱ s♦ ❧❡t x1919 = k + t, x1960 = m + t ❛♥❞ x2001 = n + t,
✇❤❡r❡ k, m ❛♥❞ n ❛r❡ ❞✐st✐♥❝t ✐♥t❡❣❡rs ❛♥❞ 0 6 t < 1. ❚❤❡♥
x41 =
m+t
n+t
=
.
k+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❤❛t 0 = 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
t❤❛t v = 1.
✹✳ ▲❡t R ❛♥❞ B ❜❡ t❤❡ s❡ts ♦❢ r❡❞ ❛♥❞ ❜❧✉❡ ♣♦✐♥ts r❡s♣❡❝t✐✈❡❧②✳ ❋♦r ❡❛❝❤ ♦♥❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥
f : R → B, ❞❡☞♥❡ t❤❡ ❧❡♥❣t❤ L(f) ♦❢ t❤❡ ♣❛✐r✐♥❣ f t♦ ❜❡ t❤❡ s✉♠ ♦❢ t❤❡ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥
❝♦rr❡s♣♦♥❞✐♥❣ ♣♦✐♥ts✳
▼♦r❡ ♣r❡❝✐s❡❧②✱ ✐❢ d(r, f(r)) ✐s t❤❡ ❞✐st❛♥❝❡ ❜❡✇❡❡♥ ❝♦rr❡s♣♦♥❞✐♥❣
P
♣♦✐♥ts✱ L(f) = r∈R d(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)) ❛♥❞ (r2, g✖(r2)) ❛r❡ ♦♣♣♦s✐t❡ s✐❞❡s ♦❢ t❤❡ ❝♦♥✈❡① q✉❛❞r✐❧❛t❡r❛❧ ✇✐t❤ ❞✐❛❣♦♥❛❧s
(r1, g
(r1, g(r1)) ❛♥❞ (r2, g(r2)). ❆s s❤♦✇♥ ✐♥ Pr♦❜❧❡♠ ✶✱
✖(r1)) + d(r2, g✖(r2)) < d(r1, g(r1)) + d(r2, g(r2)).
d(r1, g
❍❡♥❝❡ 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 ❞✐✈✐❞❡s S ✐♥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❣♦♥ C t❤❛t ❝♦♥t❛✐♥s ❛❧❧ t❤❡ ❣✐✈❡♥ ♣♦✐♥ts✳ ❊❛❝❤ ♦❢ ✐ts
✈❡rt✐❝❡s ♦❜✈✐♦✉s❧② ❝♦✐♥❝✐❞❡s ✇✐t❤ s♦♠❡ ❣✐✈❡♥ ♣♦✐♥t✳ ■❢ ❛♥② t✇♦ ✈❡rt✐❝❡s A ❛♥❞ 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✳
b = α, DBA
b = β, ACB
b = γ ❛♥❞
✺✳ ❉r❛✇ t❤❡ ❞✐❛❣♦♥❛❧s ♦❢ Q0 = ABCD ❛♥❞ ❧❡t CAD
b = δ. ▲❛❜❡❧ t❤❡ ❝✐r❝✉♠s❝r✐❜❡❞ q✉❛❞r✐❧❛t❡r❛❧ Q1 = A1B1C1D1 s♦ t❤❛t A1BB1,
BDC
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 x^1, x^2, . . . , x^n ❜❡ ❛♥♦t❤❡r s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs t❤❛t s❛t✐s☞❡s t❤❡ ❡q✉❛t✐♦♥s✳ ❲❡ s❤❛❧❧
s❤♦✇✿
✇❤❡♥ n ✐s ❡✈❡♥✱ t❤❡ x^i ❝❛♥♥♦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❤❡ x^i ❝❛♥ 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 ❢♦r di ❛♥❞ s ✐♠♣❧② t❤❛t t❤❡r❡
✐s ❛ ♥✉♠❜❡r h s✉❝❤ t❤❛t
x
^ i − xi =
h, ❢♦r ♦❞❞ i❀
−h, ❢♦r ❡✈❡♥ i✳
✸
❚❤❡ ❝♦♥❞✐t✐♦♥ ❢♦r t ✐♠♣❧✐❡s t❤❛t
0=
n
X
x
^2i − x2i
=
n
X
(^
xi − xi)(^
xi + xi)
i=1
i=1
m
X
=h
(2x2i−1 + h) − h
i=1
m
X
(2x2i − h)
i=1
= mh2 + 2h
m
X
x2i−1 + mh2 − 2h
i=1
m
X
= 2mh2 − 2h
m
X
x2i
i=1
(x2i − x2i−1)
i=1
❚❤❡ ❧❛st ❡q✉❛t✐♦♥ ❤❛s t✇♦ s♦❧✉t✐♦♥s✿ h = 0, t❤❡ ❭tr✉❡✧ s♦❧✉t✐♦♥✱ ❛♥❞
m
X
^= 1
(x2i − x2i−1),
h
m
i=1
t❤❡ ❭s♣✉r✐♦✉s✧ s♦❧✉t✐♦♥✳ ❋r♦♠ t❤❡ ♦r❞❡r r❡❧❛t✐♦♥s✱ ✐t ❢♦❧❧♦✇s t❤❛t h^ > 0, ✐✳❡✳ x^1 > x1. ■❢
t❤❡ x^i s❛t✐s❢② t❤❡ ♦r❞❡r r❡❧❛t✐♦♥s✱ ✇❡ ❝❛♥ st❛rt ✇✐t❤ x^i ❛s t❤❡ ❭tr✉❡✧ s♦❧✉t✐♦♥ ❛♥❞ r❡♣❡❛t
t❤❡ ♣r♦❝❡ss t♦ ♦❜t❛✐♥ ❛ t❤✐r❞ s♦❧✉t✐♦♥ ✐♥ ✇❤✐❝❤
⑦1 > x^1 > x1.
x
❇✉t t❤❡r❡ ❛r❡ ♦♥❧② t✇♦ s♦❧✉t✐♦♥s✳ ❚❤✐s ❝♦♥tr❛❞✐❝t✐♦♥ s❤♦✇s t❤❛t t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t
t❤❡ x^i s❛t✐s❢② t❤❡ ♦r❞❡r r❡❧❛t✐♦♥s ✐s ✉♥t❡♥❛❜❧❡✳
❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✇❤❡♥ n = 2m − 1 ✭m > 2✮ ♦♥❡ ❝❛♥ ❡❛s✐❧② ☞♥❞ ❛ ❝♦✉♥t❡r❡①❛♠♣❧❡✳
▲❡t xi = m − i + hi ✇❤❡r❡
h
m−1 ❢♦r ❡✈❡♥ i❀
hi =
−h
m ❢♦r ♦❞❞ i❀
✇❤❡r❡ h ✐s ❛♥② ♥♦♥③❡r♦ r❡❛❧ ♥✉♠❜❡r s✉❝❤ t❤❛t xi < xi+1 ❤♦❧❞s✱ ❡✳❣✳ h = 12 . ❍❡r❡ s = 0
❛♥❞ ❛❧❧ di = 1.P◆♦✇ t❛❦❡ x^i = −x2m−i.P❈❧❡❛r❧② t❤❡ P
x
^i s❛t✐s❢② t❤❡ ♦r❞❡r ❝♦♥❞✐t✐♦♥s✱ ❛❧❧
2m−1
2m−1 2
2m−1 2
x
^i+2 − x
^i = 1, i=1
x
^i = −s = 0 ❛♥❞ i=1
x
^i = i=1
xi = t. ❇✉t x
^m = −xm ❛♥❞
xm ✐s ♥♦♥③❡r♦✱ t❤❡r❡❢♦r❡ x
^m 6= xm ❛♥❞ t❤❡ x
^i ❢♦r♠ ❛ ❞✐☛❡r❡♥t ✈❛❧✐❞ s♦❧✉t✐♦♥✳
✹
❙✉✐❞✲❆❢r✐❦❛❛♥s❡ ❲✐s❦✉♥❞❡✲♦❧✐♠♣✐❛❞❡
❉❡r❞❡ ❘♦♥❞❡ ✷✵✵✶✿ ❖♣❧♦ss✐♥❣s
✶✳ ❱✐r ❞✐❡ ❜♦❣r❡♥s✱ ❣❡❜r✉✐❦ ❞✐❡ ❞r✐❡❤♦❡❦s♦♥❣❡❧②❦❤❡✐❞ ✐♥ ❞r✐❡❤♦❡❦❡
AB + BC > AC ❡♥ AD + DC > AC,
2p > 2AC + 2BD.
❞✉s
p > 2AC.
◆❡t s♦✱
ABC ❡♥ ADC. ❉❛♥
p > 2BD. ❉❛❛r♦♠
AC ❡♥ BD ✐♥ E s♥②✳ ❖♠❞❛t ABCD ❦♦♥✈❡❦s ✐s✱ ✐s E ❜✐♥♥❡ ❞✐❡
AE + EC = AC ❡♥ BE + ED = BD. ●❡❜r✉✐❦ ❞✐❡ ❞r✐❡❤♦❡❦s♦♥❣❡✲
❱✐r ❞✐❡ ♦♥❞❡r❣r❡♥s✱ ❧❛❛t
✈✐❡r❤♦❡❦✱ ❡♥ ❞❛❛r♦♠
❧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
(x + y)/(xy − 1). ❉✐❡ ✈❡r❡✐st❡ tr✐♣❧❡tt❡ ✐s ❞✉s (x, y, (x + y)/(xy − 1))
❣❡t❛❧ ✐s✱ ❡♥ y ❡♥✐❣❡ r❡⑧
❡❧❡ ❣❡t❛❧ ❜❡❤❛❧✇❡ 1/x.
x = 1 ✐s ❞❛❛r ♥✐❦s ♦♠ t❡
2001
x
= n + t, ✇❛❛r k, m ❡♥ n
✸✳ ❆s
✇❛❛r
x41 =
❡♥✐❣❡ r❡⑧
❡❧❡
x1919 = k + t, x1960 = m + t
❤❡❡❧t❛❧❧❡ ✐s✱ ❡♥ 0 6 t < 1. ❉❛♥
❜❡✇②s ♥✐❡❀ ❧❛❛t ❞✉s
❛❧♠❛❧ ✈❡rs❦✐❧❧❡♥❞❡
x
z =
❡♥
m+t
n+t
=
.
k+t
m+t
❉✉s
0 = (m + t)2 − (k + t)(n + t) = (m2 − kn) − (k − 2m + n)t.
k − 2m + n = 0, ❞❛♥ ♦♦❦ m2 − kn = 0 ❡♥ ❞✐t ✈♦❧❣ ❞❛t 0 = m2 − k(2m − k) = (m − k)2
2
41 ❡♥ x1919
✇❛t k 6= m ✇❡❡rs♣r❡❡❦✳ ❉✉s ✐s t = (m − kn)/(k − 2m + n) r❛s✐♦♥❛❛❧✱ ❡♥ x
✐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
❆s
x = x41a+1919b = (x41)a + (x1919)b
r❛s✐♦♥❛❛❧✱ s❫
❡
x = u/v
♠❡t
❉❛♥ ✐s ❞✐❡ ♥♦❡♠❡rs ✈❛♥
♦♠❞❛t
x1960 − x2001
✬♥
v>0
❡♥
u
❡♥
v
♦♥❞❡r❧✐♥❣ ♣r✐❡♠✳
x1960 ❡♥ x2001, ✉✐t❣❡❞r✉❦ ✐♥ ❦❧❡✐♥st❡ t❡r♠❡✱ v1960 ❡♥ v2001. ▼❛❛r
❤❡❡❧t❛❧ ✐s✱ ♠♦❡t ❤✐❡r❞✐❡ ♥♦❡♠❡rs ❣❡❧②❦ ✇❡❡s✳ ❉✐t ✈♦❧❣ ❞❛t v = 1.
✹✳ ▲❛❛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❡✳ P
▼❡❡r ♣r❡s✐❡s✱ ❛s d(r, f(r)) ❞✐❡ ❛❢st❛♥❞ t✉ss❡♥
♦♦r❡❡♥st❡♠♠❡♥❞❡ ♣✉♥t❡ ✐s✱ ❞❛♥ ✐s L(f) = r∈R d(r, f(r)). ▲❡t ♦♣✿ ❞❛❛r ✐s ♥❡t ✬♥ ❡✐♥❞✐❣❡
❛❛♥t❛❧ s✉❧❦❡ ❢✉♥❦s✐❡s✱ ❞✉s ✐s ❞❛❛r t✉ss❡♥ ❛❧ ❞✐❡ L(f) ✬♥ ❦❧❡✐♥st❡ ❡❡♥✳ ◆♦❡♠ ❞✐t L(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❡❧❢❞❡ ❛s g ✇❡❡s✱ ❜❡❤❛❧✇❡ ❞❛t ❞✐❡ ❡✐♥❞♣✉♥t❡ r1 ❡♥ r2 ♦♠❣❡r✉✐❧ ✐s✳ ❉✉s ✐s (r1, g✖(r1)) ❡♥
✖(r2)) t❡❡♥♦♦rst❛❛♥❞❡ s②❡ ✈❛♥ ❞✐❡s❡❧❢❞❡ ❦♦♥✈❡❦s❡ ✈✐❡r❤♦❡❦ ♠❡t ❞✐❛❣♦♥❛❧❡ (r1, g(r1))
(r2, g
❡♥ (r2, g(r2)). ❙♦♦s ✐♥ Pr♦❜❧❡❡♠ ✶ ❛❛♥❣❡t♦♦♥✱
✖(r1)) + d(r2, g✖(r2)) < d(r1, g(r1)) + d(r2, g(r2)).
d(r1, g
❉✉s ✐s L(g✖) < L(g)✱ ✬♥ t❡❡♥s♣r❛❛❦✱ ✇❛♥t L(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❧✐❦
✇❛♥♥❡❡r n = 1. ❉✐❡ ✐❞❡❡ ✐s ♦♠ ✬♥ r❡❣✉✐t ❧②♥ t❡ ✈✐♥❞ ✇❛t S ✐♥ ❣❡❜❛❧❛♥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 ❤❫❡ ♥✐❡✳
b = α, DBA
b = β, ACB
b = γ ❡♥
✺✳ ❚r❡❦ ❞✐❡ ❤♦❡❦❧②♥❡ ✈❛♥ Q0 = ABCD ❡♥ ❧❛❛t CAD
b
BDC = δ. ▼❡r❦ ❞✐❡ ♦♠❣❡s❦r❡✇❡ ✈✐❡r❤♦❡❦ Q1 = A1B1C1D1 ♦♣ s♦ ✬♥ ♠❛♥✐❡r ❞❛t A1BB1,
B1CC1, C1DD1 ❡♥ D1AA1 s② s②❡ ✐s✳ ❍❡r❤❛❛❧❞❡ ❣❡❜r✉✐❦ ✈❛♥ ❞✐❡ r❛❛❦❧②♥✲❦♦♦r❞✲st❡❧❧✐♥❣
❣❡❡ ❞❛t ❞✐❡ ❤♦❡❦❡ ✈❛♥ A1B1C1D1 ♦♥❞❡rs❦❡✐❞❡❧✐❦ 2α, 2β, 2γ ❡♥ 2δ ✐s✳ ❆s A1B1C1D1 ♦♦❦
s✐❦❧✐❡s ✐s✱ ✈♦❧❣ ❞✐t ❞❛t
α + γ = 90◦ = β + δ.
❚r❡❦ ❞✐❡ ❤♦❡❦❧②♥❡ ✈❛♥ A1B1C1D1 ❡♥ ❧❛❛t α1, 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 + β1 + 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◦ ).
0 < α4n < 180◦ ❤❫❡✳ ▼❛❛r t❡♥s② α = 45◦ ✭✇❛♥♥❡❡r ABCD
◦
✬♥ ✈✐❡r❦❛♥t ✐s✮ ✈♦r♠ α4n − 45 ✬♥ ❞✐✈❡r❣❡♥t❡ ♠❡❡t❦✉♥❞✐❣❡ r②✱ ✇❛t ♦♣ ❞✐❡ ♦✉ ❡♥t ❜✉✐t❡
◦
◦
❞✐❡ ✐♥t❡r✈❛❧ t✉ss❡♥ −45 ♠♦❡t 135 ✈❛❧✱ ❡♥ ❞✉s ✐s ❞✐❡ r② ❡✐♥❞✐❣✳
❆s
Q4n
✻✳ ▲❛❛t
❦♦♥✈❡❦s ✐s✱ ♠♦❡t ♦♥s
x
^1, x
^2, . . . , x
^n ♥♦❣ ✬♥ r② ❣❡t❛❧❧❡ ✇❡❡s ✇❛t ❞✐❡ ✈❡r❣❡❧②❦✐♥❣s ❜❡✈r❡❞✐❣✳
❖♥s s❛❧ ❛❛♥t♦♦♥
❞❛t✿
x
^i
❛s
n
❡✇❡ ✐s✱ ❦❛♥ ❞✐❡
❛s
n
♦♥❡✇❡ ✐s✱ ✐s ❞✐t ♠♦♦♥t❧✐❦ ♦♠
♥✐❡ ❞✐❡ ♦r❞❡♥✐♥❣ ❜❡✈r❡❞✐❣ ♥✐❡❀
xi
t❡ ❦r② s♦❞❛t
❉✉s ✐s ❞✐❡ ❛♥t✇♦♦r❞ ♦♣ ❞✐❡ ♣r♦❜❧❡❡♠✿ ❛❧❧❡ ❡✇❡
❖♥s ❜❡s❦♦✉ ❡❡rs ❞✐❡ ❣❡✈❛❧
❣❡t❛❧
h
n = 2m.
x
^i
❞✐❡ ♦r❞❡♥✐♥❣ ❜❡✈r❡❞✐❣✳
n > 4.
❉✐❡ ✈♦♦r✇❛❛r❞❡s ✈✐r
❜❡st❛❛♥ s♦❞❛t
x
^ i − xi =
h, ✈✐r ♦♥❡✇❡ i❀
−h, ✈✐r ❡✇❡ i✳
✼
di
❡♥
s
❜r✐♥❣ ♠❡❡ ❞❛t ❞❛❛r ✬♥
❉✐❡ ✈♦♦r✇❛❛r❞❡ ✈✐r t ❜r✐♥❣ ♠❡❡ ❞❛t
0=
n
X
x
^2i − x2i
=
n
X
(^
xi − xi)(^
xi + xi)
i=1
i=1
m
X
=h
(2x2i−1 + h) − h
i=1
m
X
(2x2i − h)
i=1
= mh2 + 2h
m
X
x2i−1 + mh2 − 2h
i=1
m
X
= 2mh2 − 2h
m
X
x2i
i=1
(x2i − x2i−1)
i=1
❉✐❡ ❧❛❛st❡ ✈❡r❣❡❧②❦✐♥❣ ❤❡t t✇❡❡ ♦♣❧♦ss✐♥❣s✿ h = 0, ❞✐❡ ❭✇❛r❡✧ ♦♣❧♦ss✐♥❣✱ ❡♥
m
X
^= 1
(x2i − x2i−1),
h
m
i=1
^ > 0, ❞✳✇✳s✳ x
❞✐❡ ❭✈❛❧s✧ ♦♣❧♦ss✐♥❣✳ ❯✐t ❞✐❡ ♦r❞❡♥✐♥❣ ✈♦❧❣ ❞❛t h
^1 > x1. ❆s ❛❧ ❞✐❡ x
^i
❞✐❡ ♦r❞❡♥✐♥❣ ❜❡✈r❡❞✐❣✱ ❦❛♥ ♦♥s ♠❡t x
^i ❛s ❞✐❡ ❭✇❛r❡✧ ♦♣❧♦ss✐♥❣ ❜❡❣✐♥ ❡♥ ❞✐❡ ❤❡❧❡ ♣r♦s❡s
❤❡r❤❛❛❧ ♦♠ ✬♥ ❞❡r❞❡ ♦♣❧♦ss✐♥❣ ♠❡t
⑦1 > x
x
^ 1 > x1
t❡ ✈❡r❦r②✳ ▼❛❛r ❞❛❛r ✐s ♥❡t t✇❡❡ ♦♣❧♦ss✐♥❣s✳ ❍✐❡r❞✐❡ t❡❡♥s♣r❛❛❦ t♦♦♥ ❛❛♥ ❞❛t ❞✐❡
❛❛♥♥❛♠❡ ❞❛t x
^i ❞✐❡ ♦r❞❡♥✐♥❣ ❜❡✈r❡❞✐❣✱ ♦♥❤♦✉❞❜❛❛r ✐s✳
❆❛♥ ❞✐❡ ❛♥❞❡r ❦❛♥t✱ ❛s n = 2m − 1 ✭m > 2✮ ❦❛♥ ✬♥ ♠❡♥s ♠❛❦❧✐❦ ✬♥ t❡❡♥✈♦♦r❜❡❡❧❞ ❦r②✳
▲❛❛t xi = m − i + hi ✇❛❛r
h
m−1 ✈✐r ❡✇❡ i❀
hi =
−h
m ✈✐r ♦♥❡✇❡ i❀
✇❛❛r h ❡♥✐❣❡ ♥✐❡✲♥✉❧ r❡⑧❡❧❡ ❣❡t❛❧ ✐s ✇❛❛r✈♦♦r xi < xi+1 ❣❡❧❞✱ ❜✈✳ h = 12 . ❍✐❡r ✐s s = 0
❡♥ ❛❧❧❡ di = 1.
♥♦✉ x
^i = −x2m−i
. ❉✉✐❞❡❧✐❦P❜❡✈r❡❞✐❣ x
^i ❞✐❡ ♦r❞❡♥✐♥❣✱ ✐s ❛❧❧❡
P◆❡❡♠
P2m−1
2m−1
2m−1 2
x
^i+2 − x
^i = 1, i=1
x
^i = −s = 0 ❡♥ i=1
x
^2i = i=1
xi = t. ▼❛❛r x
^m = −xm ❡♥
xm ✐s ♥✐❡✲♥✉❧✱ ❞✉s ✐s x
^m 6= xm ❡♥ x
^i ✈♦r♠ ✬♥ t✇❡❡❞❡ ❣❡❧❞✐❣❡ ♦♣❧♦ss✐♥❣✳
✽