2016 Polish MO Finals

Part of Polish MO Finals

Subcontests

(6)

1

Geometry

I

be an incenter of

\triangle ABC

D, \ S \neq A

intersections of

AI

BC, \ O(ABC)

respectively. Let

K, \ L

be incenters of

\triangle DSB, \ \triangle DCS

P

be a reflection of

I

with the respect to

KL

BP \perp CP

1

Polish Algebra

There are given two positive real number

a<b

. Show that there exist positive integers

p, \ q, \ r, \ s

satisfying following conditions:

1

a< \frac{p}{q} < \frac{r}{s} < b

2.

p^2+q^2=r^2+s^2

1

Number Theory

k, n

be odd positve integers greater than

1

. Prove that if there a exists natural number

a

k|2^a+1, \ n|2^a-1

, then there is no natural number

b

k|2^b-1, \ n|2^b+1

1

Combinatorics

a, \ b \in \mathbb{Z_{+}}

f(a, b)

the number sequences

s_1, \ s_2, \ ..., \ s_a

s_i \in \mathbb{Z}

|s_1|+|s_2|+...+|s_a| \le b

f(a, b)=f(b, a)

1

Planimetry

ABCD

be a quadrilateral circumscribed on the circle

\omega

I

\angle BAD+ \angle ADC <\pi

M, \ N

be points of tangency of

\omega

AB, \ CD

respectively. Consider a point

K \in MN

AK=AM

ID

bisects the segment

KN

1

Polynomial

p

be a certain prime number. Find all non-negative integers

n

for which polynomial

P(x)=x^4-2(n+p)x^2+(n-p)^2

may be rewritten as product of two quadratic polynomials

P_1, \ P_2 \in \mathbb{Z}[X]