2012 Balkan Report Gabriel

8/12/2019 2012 Balkan Report Gabriel

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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

2012 Balkan Report Gabriel

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Transcript of 2012 Balkan Report Gabriel