Language: Romanian · Fie K al doilea punct de intersecµie a cercului circumscris triunghiului BDF...
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Language: Romanian Day: 1 S A B S C S AC = BC S AB C S P S PA = PB = PC n 3 n n 3 n (a, b, c) ab - c, bc - a, ca - b 2 2 2 n n ABC AB > AC Γ H F A M BC Q Γ ∠HQA = 90 ◦ K Γ ∠HKQ = 90 ◦ AB C K Q Γ KQH FKM
Transcript of Language: Romanian · Fie K al doilea punct de intersecµie a cercului circumscris triunghiului BDF...
Language: Romanian
Day: 1
❱♥r
Pr♦♠ ♣♥♠ ♦ ♠♠ ♥t S ♣♥t ♥ ♣♥ st rt ♣♥tr ♦r♦ ♣♥t rt A B ♥ S ①st ♥ ♣♥t C ♥ S st î♥ât AC = BC ♣♥♠ S
st ♥♥trt ♣♥tr ♦r tr ♣♥t rt A B C ♥ S ♥ ①st ♥♥ ♣♥t P ♥S st î♥ât PA = PB = PC
rt ♣♥tr ♦r î♥tr n > 3 ①st ♦ ♠♠ rt tt ♥ n ♣♥t
tr♠♥ t♦ î♥tr n > 3 ♣♥tr r ①st ♦ ♠♠ rt ♥♥trt tt♥ n ♣♥t
Pr♦♠ tr♠♥ t♦t tr♣t î♥tr strt ♣♦③t (a, b, c) st î♥ât r ♥tr♥♠r
ab− c, bc− a, ca− b
s ♦ ♣tr 2
♣tr 2 st ♥ î♥tr ♦r♠ 2n ♥ n st ♥ î♥tr ♥♥t
Pr♦♠ ABC ♥ tr♥ st♥ AB > AC Γ r s r♠srs H♦rt♦♥tr s F ♣♦r î♥♠ ♥ A M ♠♦ s♠♥t BC Q ♣♥t ♣Γ ♣♥tr r ∠HQA = 90 K ♣♥t ♣ Γ ♣♥tr r ∠HKQ = 90 Prs♣♥♠ ♣♥t A B C K Q s♥t rt ♦ ât ♦ s♥t stt ♣ Γ î♥ st ♦r♥
♠♦♥str tr♥r KQH FKM rr r♠srs t♥♥t î♥tr
♥ ♦♠♥♥ ♠♣ ♦r ♠♥t
r ♣r♦♠ ♦r③ ♣♥t
Language: Romanian
Day: 2
â♠t
Pr♦♠ r♥ ABC r r r♠srs Ω â♥ ♥tr O ❯♥ r Γ ♥tr A♥trst③ s♠♥t BC î♥ ♣♥t D E st î♥ât B D E C s♥t rt ♦ ât♦ s♥t stt ♣ r♣t BC î♥ st ♦r♥ F G ♣♥t ♥trs rr♦rΓ Ω st î♥ât A F B C G s ♣ Ω î♥ st ♦r♥ K ♦ ♣♥t ♥trs r r♠srs tr♥ BDF s♠♥t AB L ♦ ♣♥t ♥trs r r♠srs tr♥ CGE s♠♥t CA
Prs♣♥♠ r♣t FK GL s♥t rt s t î♥ ♣♥t X ♠♦♥str X s ♣ r♣t AO
Pr♦♠ R ♠♠ ♥♠r♦r r tr♠♥ t♦t ♥ f : R → R r rr
f(
x+ f(x+ y))
+ f(xy) = x+ f(x+ y) + yf(x)
♣♥tr ♦r ♥♠r r x y
Pr♦♠ ➇r a1, a2, . . . ♥♠r î♥tr sts ♦♥
1 6 aj 6 2015 ♣♥tr ♦r j > 1
k + ak 6= ℓ+ aℓ ♣♥tr ♦r 1 6 k < ℓ
♠♦♥str ①st ♦ î♥tr strt ♣♦③t b N st î♥ât
∣
∣
∣
∣
∣
n∑
j=m+1
(aj − b)
∣
∣
∣
∣
∣
6 10072
♣♥tr ♦r î♥tr m n r sts r n > m > N
♥ ♦♠♥♥ ♠♣ ♦r ♠♥t
r ♣r♦♠ ♦r③ ♣♥t
