Language: English

Day: 1

r② ②

Pr♦♠ ❲ s② tt ♥t st S ♦ ♣♦♥ts ♥ t ♣♥ s ♥ ♦r ♥② t♦ r♥t♣♦♥ts A ♥ B ♥ S tr s ♣♦♥t C ♥ S s tt AC = BC ❲ s② tt S s ♥trr ♦r ♥② tr r♥t ♣♦♥ts A B ♥ C ♥ S tr s ♥♦ ♣♦♥t P ♥ S s tt PA = PB = PC

♦ tt ♦r ♥trs n > 3 tr ①sts ♥ st ♦♥sst♥ ♦ n ♣♦♥ts

tr♠♥ ♥trs n > 3 ♦r tr ①sts ♥ ♥trr st ♦♥sst♥ ♦ n♣♦♥ts

Pr♦♠ tr♠♥ tr♣s (a, b, c) ♦ ♣♦st ♥trs s tt ♦ t ♥♠rs

ab− c, bc− a, ca− b

s ♣♦r ♦ 2

♣♦r ♦ 2 s ♥ ♥tr ♦ t ♦r♠ 2n r n s ♥♦♥♥t ♥tr

Pr♦♠ t ABC ♥ t tr♥ t AB > AC t Γ ts r♠r H ts♦rt♦♥tr ♥ F t ♦♦t ♦ t tt r♦♠ A t M t ♠♣♦♥t ♦ BC t Q t♣♦♥t ♦♥ Γ s tt ∠HQA = 90 ♥ t K t ♣♦♥t ♦♥ Γ s tt ∠HKQ = 90 ss♠tt t ♣♦♥ts A B C K ♥ Q r r♥t ♥ ♦♥ Γ ♥ ts ♦rr

Pr♦ tt t r♠rs ♦ tr♥s KQH ♥ FKM r t♥♥t t♦ ♦tr

♥ ♥s ♠ ♦rs ♥ ♠♥ts

♣r♦♠ s ♦rt ♣♦♥ts

Language: English

Day: 2

tr② ②

Pr♦♠ r♥ ABC s r♠r Ω ♥ r♠♥tr O r Γ t ♥tr A

♥trsts t s♠♥t BC t ♣♦♥ts D ♥ E s tt B D E ♥ C r r♥t ♥ ♦♥ ♥ BC ♥ ts ♦rr t F ♥ G t ♣♦♥ts ♦ ♥trst♦♥ ♦ Γ ♥ Ω s tt A F B C ♥ G ♦♥ Ω ♥ ts ♦rr t K t s♦♥ ♣♦♥t ♦ ♥trst♦♥ ♦ t r♠r ♦tr♥ BDF ♥ t s♠♥t AB t L t s♦♥ ♣♦♥t ♦ ♥trst♦♥ ♦ t r♠r ♦tr♥ CGE ♥ t s♠♥t CA

♣♣♦s tt t ♥s FK ♥ GL r r♥t ♥ ♥trst t t ♣♦♥t X Pr♦ tt X s ♦♥t ♥ AO

Pr♦♠ t R t st ♦ r ♥♠rs tr♠♥ ♥t♦♥s f : R → R sts②♥ tqt♦♥

f(

x+ f(x+ y))

+ f(xy) = x+ f(x+ y) + yf(x)

♦r r ♥♠rs x ♥ y

Pr♦♠ sq♥ a1, a2, . . . ♦ ♥trs stss t ♦♦♥ ♦♥t♦♥s

1 6 aj 6 2015 ♦r j > 1

k + ak 6= ℓ+ aℓ ♦r 1 6 k < ℓ

Pr♦ tt tr ①st t♦ ♣♦st ♥trs b ♥ N s tt

∣

∣

∣

∣

∣

n∑

j=m+1

(aj − b)

∣

∣

∣

∣

∣

6 10072

♦r ♥trs m ♥ n sts②♥ n > m > N

♥ ♥s ♠ ♦rs ♥ ♠♥ts

♣r♦♠ s ♦rt ♣♦♥ts