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18.03.2013

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, x > 0 nN :

: n .

: n , : x1 = x2 = ... = xn-1 = 1, xn = 1 + x.


: .

: 2 ≤ x ≤ 5.


, x ∈ [0; π/2] xcosx < .

: : sinxx x ≥ 0 cosx = sin(π/2 - x).

: x ≥ 0 sinxx; x ∈ [0; π/2] :
xcosx = xsin(π/2 - x) ≤ x(π/2 - x) ≤ π2/16.
π < 3.2, > 1.4, π2 < 10.24, 8 > 11.2 .
xcosxπ2/16 < 10.24/16 < 11.2/16 < .


2[x] = 1 + 2x, [x] - x.

: , [x] ≤ -2 . , [x] ≥ 4 .

: x = -1/4, x = 0, x = 7/2
: 2[x] - 1 = 1/2 + [x] + {x}, {x} = x - [x].
[x] ≤ -2, 2[x] - 1 > 1/2 + [x] + {x}(*), . . [x], [x] ≥ 4 (*) .
: [x] = 4, 24 - 1 = 8 > 51/2 > 1/2 + 4 + {x}. .
: [x] 2[x] - 1 = 1/2 + [x] + {x}.
2[x] + 1 - 1 = 1/2 + [x] + 1 + {x}.
: 2[x] = 2(1/2 + [x] + {x}) = 1/2 + [x] + {x} + 1/2 + [x] + {x} > 1/2 + [x] + {x} + 4 > 1/2 + [x] + 1 + {x}. .
, [x] = -1, 0, 1, 2, 3. , : x = -1/4, x = 0, x = 7/2.


a, b, c, d - . ,
((a - c)2 + (b - d)2)(a2 + b2) - (ad - bc)2 .

: , ((a - c)2 + (b - d)2)(a2 + b2) - (ad - bc)2 = (a2 + b2 - ac - bd)2.


k m n, mk + n nk + m k- ?

: k = 1.

: k = 1, m n . k ≥ 2 . .
. p ≥ 1,
mk + n = (m + p)k. :
n = (m + p)k - mk = ((m + p) - m)((m + p)k - 1 + (m + p)k - 2m + ... + (m + p)mk - 2 + mk - 1) >
p(mk - 1 + mk - 1 + ... + mk - 1) = pkmk - 1 > m. .. n > m.
, , , m > n. . : k = 1.


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