2012 balkan report gabriel
TRANSCRIPT
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3 = 2 1 = 0
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S
W N R
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G N R = R W = W G = g G g / W g S \W
W R = S S \ W S \ R = g / S \ R g R g / N
W G = W G = C
C
P ABC AP BC D BP AC E CP AB F
AF
F B
BD
DC
C E
EA = 1
AF FB + BD DC + CE EA = 0
a b c
a b c a5 bc a4 c3
n n + 1 n = 3
a b c D
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D
a 3 a2 b + ab2 b2 c + bc2 c3
a
1
1 1b c
a
1b c
a c
R R \ Q
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0
0
20
20
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R 2 R 0
1 12 n 1 1 12 n
33 22
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ABC B C T T AB = CAM M BC
ABC M BC H D
AH MH BC X XD Y
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AY
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A B C O ABC > 90 D
AB AC C D AO E
AC F D E BF E CF D
F
x,y,z R +
(x + y) x + y y + z
4(xy + xz + yz)
a2 = x + y b2 = x + z c2 = y + z
a b c d + e = a d + f = b e + f = c
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P n = {2n , 2n1 3, . . . , 2nr 3r , . . . , 3n } P n S X S = 0 y R 0 y S Y
3n +1
2n +1
P n
0 yS Y < 2n
Z + f : Z + Z +
f (n!) = f (n)!
n Z +
m n | f (m) f (n) m = n Z +
f (n) = 1 f (n) = 2 f (n) = n
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AB P AB Q QPA = 90
QP B
AB G AG = AQ
P ABC APD BP E CP F AF F B + BD DC + CE EA = 0
a b c P (x,y,z ) a(z y)(y + x)(x + z) + b(x z)(z + y)(y + x)+ c(yx)(x + z)(z + y) = 0
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tan A +tan B +tan C =tan A tan B tan C A + B + C = 180 sin2A + sin 2 B + sin 2 C =4sin A sin B sin C
3 a 2 b2 c2 sin A sin B sin C 2
sin A +sin B +sin C sin2A +sin2 B +sin2 C DEF DEF
R sin2A R sin2B R sin2C 2R sin A 2R sin B 2R sin C BC F D + DE
sin x cosx x 0 90
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P P ABC DEF D E F
P
[DEF ] = R2 OP 2
4R 2 [ABC ]
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E
A O
sin A cos B +sin B cos A sin A cos A +sin B cos B sin A cos C + sin B cos B +sin C cosA sin A cos A +sin B cos B + sin C cosC
sin +cos = 0 tan = 1
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p p(x) x x
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12
n< 1
1000000
log 14
12
n< log 1
4
11000000
n 0
f (x) = 1 /x 2 lim0 2 lnx
4 r r R
lim0
f (x + ) f (x )2
= lim0
f (x + r ) f (x )(1 + r ) r R
x+ y+ z = 0
x(x + 2)2x2 + 1 0
1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, 1, . . .
f (n) n
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f (1) = 1 f (2) = 3 f (3) = 9
f (n)
f (10)
g(1) g(64)
ABC P P AB =PBC = PCA = cot A + cot B + cot C = cot
30
a2 b(bc)a + b
0
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ABC =CBD = DBA = 90 [ABC ]2 +[ CBD ]2 +[ DBA ]2 = [ACD ]2
ABCD P AP 4 + BP 4 + CP 4 + DP 4
BC AB A1 AC A2B 1 B2 C 1 C 2
AA1 AA2 a2
+ b2
+ c2
9
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[ABC ]2 = r r A r B r C ABC
I AI B I C
n
0
2 3 n
a 1 + 1
a2 + 1a 3 + 1a 4 +
1...
an ak ak+1 k = 1
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ABC P D E F
AF F B + BD DC + CE EA = 0
r R x y z xr + yr + zr
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f f : R R \Q f 1 : R \Q R f 1
P P P f f 1 (P )
f 1 (P ) f 1 (P ) f 1 (P )
P P
12 n
0
D AB DT = BT B
= D
DT = BT = CT DT AC E BCA = T BA
T BA + BCA = 180T BA = T BD T BD = BDT BDT
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BCA = BDT ABC AED
T EC = ABC ABC = ACT
ACT = ECT ECT = T EC T E = T C T C = T B = T D T E = T D T DE
ABC AED AMC ATD
T AD = CAM
G AO M AB 2ACB = AOB = 2AOM ACB = AOM
AMO 90 AOM = 90 MAO = 180 DGA DAG = ADG BDE = BCE BDCE
DCE = 90
EBD = 180 90 = 90
ACF + FBA = 180 F BA = 180 ACF = ACD ACF = F CD EF N F D P EBF FCD O1 O2
2F BE = F O 1 E = 2FO 1 N 2F BE = 2F CD =FO 2 D = 2F O 2 P FO1 N = F O2 P O1 N
P O 2 O1 N O2 P O1 FO 2
a b c
a 2 bc + ab2 c + abc2 2a2 b2 + 2 b2 c2 + 2 a 2 c2 a 4 b4 c4
a 2 bc + ab2 c + abc2 + a 4 + b4 + c4 2a 2 b2 + 2 b2 c2 + 2 a 2 c2
xt (x y)(x z) 0
a2 (a b)(a c) 0
a 4 + b4 + c4 + a 2 bc + ab2 c + abc2
a 3 b
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a 3 b
a2 b2
a + b > c
a2 bc + ab2 c + abc2
(a 2 + b2 c2 )(a2 + c2 b
2 )
a2 + b2 c2 = 2 abcosC a 2 bc+ ab2 c + abc2 4a
2 bccos B cos C + 4 ab2 c cos A cos C + 4 abc2 cos A cos B
2/ sin C = ab
42
sin A sin B sin C
sin A+sin B +sin C 4sin A cos B cos C +4sin B cos A cos C +4sin C cos A cos B
2sin A cos B cos C +2sin B cos A cos C = 2
cos C (sin A cos B +sin B cos A)
sin A cos B + sin B cos A = sin( A + B) sin(A + B ) = sin(180 A B ) = sin C
2sin A cos A
sin A + sin B + sin C
sin 2A + sin2 B + sin2 C
ABC A B 2sin A 2sin B
cos A cos B 2(sin A cos B + sin B cos A) 2sin A cos A + 2 sin B cos B
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2sin C sin2A + sin2 B
sin A + sin B + sin C sin 2A + sin2 B + sin2 C
x + y 2 xy
(x + y)z 2z xy
z2 + ( x + y)z + xy z2 + 2 z xy + xy
z2 + xy
(z + x)(z + y) (z + xy)2
z + x z + y z + xy
(x + y) z + x z + y (x + y)(z + xy)
(x + y)(z + xy)
2
xy +
(x + y) xy
x + y xy
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2
xy + 2
xy xy
4
xy
a b c
(a + b + c)abc = 2
a 2 b2
a 4
(a + b+ c)(a + bc)(a + cb)(b+ ca) a b c
2abc(a + b + c)/ 2 (a + b + c)(a + bc)(a + c b)(b + c a)
2abc(a + b + c)/ 2 162
abc4 R rs
2abcs 16abcrs
4R
R 2r
n
n + 1
n = 1P n a b
a y b Y P n 0 y S Y 2n P n 3n +1 2n +1
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P n S P n = 3n +1 2n +1
S P n = a1r n +11r = 2
n 132n +1
132= 3 n +1 2n +1 .
P n U 0 , U 1 , U 2 , . . . , U m U i P n S U i
U 0 = U m = P n S U j S U j 1 2n 0 < j m
y = S U j 1 y S U j 1 < 2n S U j 1 S U j 1 < 2n
S U j S U j 1 < 2n . P n
S U j 1 y < S U j S U j 1 0 < y S U j 1 < S U j S U j 1 0 yS U j 1 2n .P n1 P n n 2
X 0 , X 1 , . . . X m P n1 V 0 , V 1 , . . . V m P n X
W 0 , W 1 , . . . W m 3n
9 3n2 8 2n2 > 0(n 2)2 (9 3n2 ) 8 2n2 > 9 3n22 (3
n
2n
) > 3n
2 (S X m ) > 3n P S V m > S W 0 V i W i P n
X i P n
P 1
, {2}, {3}, {2, 3} P n S Y P 1 0 y 32 22 = 5
P n
f (n) = k n n 0 k
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f (n) = k n n0 > 1
n 0 Z+
n 0 1 3
k = f ((n 0 1)!) = f (n 0 1)!
n0 1 > n 0 n 0 1 3 f (n 0 !) = k n! n n f (n 0 )! = k f (n 0 ) = k f (n 0 )! = k! k! = k k = f (n 0 1)! k! = f (n 0 1)! k = f (n 0 1) n0
f (n) = k n n0 n0 1 3 n 0 1 < 3 n0 1 (n 0 1)! = n 0 1
f (n 0 1)! = f (n 0 1) f (n 0 1)f (n 0 +1) f (n 0 1) f (n 0 +1) = k k f (n 0 1) (mod 2) f (n 0 1) 1 = 2 (mod 2) k = f (n 0 1) n0 > 1
n 0 1 n0 = 1 f (n) = k n
k
f (n)
f n,k Z+ f (n + k) =f (n) f ((n + k)!) f (n!)
(n + k)! n! =
f (n + k)! f (n)!(n + k)! n!
Z
f (n) < n n n < n + k a n! = (n + k)! a n + 1 (a 1)n! = ( n + k)! n! n! | (n + k)! n! (n + k)! n! | f (n + k)! f (n)! n!
| f (n + k)!
f (n)! f (n + k)!
f (n)!
= 0 f (n + k)
= f (n)
f (n + k)! f (n)! n! f (n + k)! f (n)! f (n)! f (n + k)!
n! n! f (n)! n! n! f (n)! n f (n)
f (n + k)! f (n)! n! f (n + k)! > n ! f (n + k) > n n! | f (n + k)! n! | f (n + k)!f (n)! n! | f (n)! n f (n) f (n) n n f (n) f (2) = f (2!) = f (2)! f (2) f (2)
f (2) 2 f (2) = 2 n
3 f (n)
n!
2
f (n!) f (2!)n!2!
Z
f (n!) 2n!2
Z .
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n!2 f (n) n!2 | f (n)! n!2 | 2 n > 3
n!2 > 2
n 3f (n) n!3 f (3) 3!3 = 3 f (3) 3 f (3) = 3f (1) = f (1!) = f (1)! f (1) f (3) f (1)(mod 2) f (1) 1 (mod 2) f (1) = 1f (1) = 1f (2) = 2f (3) = 3
n, n | n! n n! n | f (n!) f (n) n | f (n)! f (n) f (n) n n | f (n)! n | f (n) f (n) = n n
n
f (n) = n n k k 3 f (k + 1)! f (k!)(k + 1)! k!
Z
f (k + 1)! k!(k + 1)! k!
Z
f (k!) = f (k)! = k! (k +1)! k! = (k + 1) k!k! =k(k!) f (k + 1)!
k!
k(k!) Z f (k + 1) 2k (2k)! | f (k + 1)!
k | 2k k! | (2k 1)! k(k!) | 2k(2k 1)! k(k!) | 2k! k(k!) |f (k +1)! k(k!) | k! k(k!) > k ! > 0 f (k + 1) < 2k k + 1 | f (k + 1)
k +1 f (k +1) < 2k k+1 k + 1 k + 1 = f (k + 1)
f f (n) = n f (n) = 1 f (n) = 2
f (n) = n
A AQ P Q B P P B
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B G
G
G = G
AG = AQ
2R = asin A sin A
2R(sin A + sin B + sin C ) = a + b + c
r2
Rr (sin A + sin B + sin C ) = rs = =
R22 (sin2A + sin2 B + sin2 C )
Rr (sin A + sin B + sin C ) = R2
2 (sin2A + sin2 B + sin2 C )
r (sin A + sin B + sin C ) = R2
(sin2A + sin 2 B + sin 2 C )
r R2
r (sin A+sin B +sin C ) = R
2 (sin2A+sin 2 B +sin2 C )
r (sin 2A+sin 2 B +sin2 C )
sin A + sin B + sin C sin 2A + sin2 B + sin2 C
b g b b
2
b 2 g 2
b + g 4 b + g + 1 b + g
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log1 / 4
1
x y z
x + y + z = 0 x + y + z = a a
(x+ y+ z)/a
x + y + z = 0 (x + 1) + ( y + 1) + ( z + 1) = 3
t = x + 1 u = y + 1 v = z + 1
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P
1sin2 A
+ 1
sin2 B +
1sin2 C
= 1sin2
cot 2 A + cot 2 B + cot 2 C = cot 2 2 tan A +tan B +
tan C = tan A tan B tan C
2
cot A cot B = 2
cot 2 A + 2
cot A cot B = cot 2
a3 b3
a3 b2 c 0
a3 b3
a3 b2 c
{a2 b2 , a 2 c2 , b2 c2 } {ab,ac,bc}
a(bc)c(a + b) 0
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a(bc)c(a + b)
+ 1 3
b(a + c)c(a + b) 3