2015 Baltic Way

Part of Baltic Way

Subcontests

(20)

1

Sets of numbers called balanced sets

S

{1,2,...,n}

is called balanced if for every

a

S

there exists some

b

S

b\neq a

\frac{(a+b)}{2}

S

as well. (a) Let

k > 1

be an integer and let

n = 2k

. Show that every subset

S

{1,2,...,n}

|S| > \frac{3n}{4}

is balanced. (b) Does there exist an

n =2k

k > 1

an integer, for which every subset

S

{1,2,...,n}

|S| >\frac{2n}{3}

1

$v_2(n^{n-1}-1)=2^{2015}$ (Baltic Way 2015)

Find all positive integers

n

n^{n-1} - 1

is divisible by

2^{2015}

2^{2016}

1

Distance of integers

For any integer

n \ge2

A_n

to be the number of positive integers

m

with the following property: the distance from

n

to the nearest multiple of

m

is equal to the distance from

n^3

to the nearest multiple of

m

. Find all integers

n \ge 2

A_n

is odd. (Note: The distance between two integers

a

b

|a -b|

1

Polynomial

f(x)=x^n + a_{n-1}x^{n-1} + ...+ a_0

be a polynomial of degree

n\ge 1

n

(not necessarily distinct) integer roots. Assume that there exist distinct primes

p_0,p_1,..,p_{n-1}

a_i > 1

p_i

i=0,1,..,n-1

. Find all possible values of

n

1

Circles and angles

A circle passes through vertex

B

of the triangle

ABC

, intersects its sides

AB

BC

K

L

, respectively, and touches the side

AC

at its midpoint

M

N

BL

(which does not contain

K

\angle LKN = \angle ACB

\angle BAC

given that the triangle

CKN

is equilateral.

1

Combinatorics

100

members in a ladies' club.Each lady has had tea (in private) with exactly

56

of her lady friends.The Board,consisting of the

50

most distinguished ladies,have all had tea with one another.Prove that the entire club may be split into two groups in such a way that,with in each group,any lady has had tea with any other.

1

Combinatorics

A family wears clothes of three colors: red,blue and green,with a separate,identical laundry bin for each color. At the beginning of the first week,all bins are empty.Each week,the family generates a total of

10 kg

of laundry(the proportion of each color is subject to variation).The laundry is sorted by color and placed in the bins.Next,the heaviest bin(only one of them,if there are several that are heaviest)is emptied and its content swashed.What is the minimal possible storing capacity required of the laundry bins in order for them never to overflow?

1

Combinatorics.

n>2

be an integer. A deck contains

\frac{n(n-1)}{2}

1,2,3,\cdots , \frac{n(n-1)}{2}

Two cards form a magic pair if their numbers are consecutive , or if their numbers are

1

\frac{n(n+1)}{2}

n

is it possible to distribute the cards into

n

stacks in such a manner that, among the cards in any two stacks , there is exactly one magic pair?

1

Geometry

ABC

, the interior and exterior angle bisectors of

\angle BAC

intersect the line

BC

D

E

, respectively. Let

F

be the second point of intersection of the line

AD

with the circumcircle of the triangle

ABC

O

be the circumcentre of the triangle

ABC

D'

be the reflection of

D

O

\angle D'FE =90.

1

Combinatorics

With inspiration drawn from the rectilinear network of streets in New York , the Manhattan distance between two points

(a,b)

(c,d)

in the plane is defined to be

|a-c|+|b-d|

Suppose only two distinct Manhattan distance occur between all pairs of distinct points of some point set. What is the maximal number of points in such a set?

1

Combinatorial Game

Two players play the following game. At the outset there are two piles, containing

10,000

20,000

tokens,respectively . A move consists of removing any positive number of tokens from a single pile

or

x>0

tokens from one pile and

y>0

tokens from the other , where

x+y

is divisible by

2015

. The player who can not make a move loses. Which player has a winning strategy

1

Algebra .Inequality

n

be a positive integer and let

a_1,\cdots ,a_n

be real numbers satisfying

0\le a_i\le 1

i=1,\cdots ,n.

Prove the inequality

(1-{a_i}^n)(1-{a_2}^n)\cdots (1-{a_n}^n)\le (1-a_1a_2\cdots a_n)^n.

1

Combinatorics

n\geq 2

, an equilateral triangle is divided into

n^2

congruent smaller equilateral triangles. Detemine all ways in which real numbers can be assigned to the

\frac{(n+1)(n+2)}{2}

vertices so that three such numbers sum to zero whenever the three vertices form a triangle with edges parallel to the sides of the big triangle.

1

Concurrency. Geometry

In the non-isosceles triangle

ABC

an altitude from

A

BC

D

M

be the midpoint of

BC

N

be the reflection of

M

D

. The circumcirle of triangle

AMN

intersects the side

AB

P\ne A

AC

Q\ne A

AN,BQ

CP

are concurrent.

1

Algebra Polynomials

n>1

be an integer. Find all non-constant real polynomials

P(x)

satisfying , for any real

x

P(x)P(x^2)P(x^3)\cdots P(x^n)=P(x^{\frac{n(n+1)}{2}})

1

Polynomial

P(n)

the greatest prime divisor of

n

. Find all integers

n\geq 2

P(n)+\lfloor\sqrt{n}\rfloor=P(n+1)+\lfloor\sqrt{n+1}\rfloor

1

Triangle Sides

Three pairwairs distinct positive integers

a,b,c,

gcd(a,b,c)=1

a|(b-c)^2 ,b|(a-c)^2 , c|(a-b)^2

Prove that there doesnt exist a non-degenerate triangle with side lengths

a,b,c.

1

Geometry

D

be the footpoint of the altitude from

B

in the triangle

ABC

AB=1

. The incircle of triangle

BCD

coincides with the centroid of triangle

ABC

. Find the lengths of

AC

BC

1

Geometry

The diagonals of parallelogram

ABCD

E

. The bisectors of

\angle DAE

\angle EBC

F

ECFD

is a parellelogram . Determine the ratio

AB:AD

1

Functional Equation

Find all functions

f:\mathbb{R}\to\mathbb{R}

satisfying the equation

|x|f(y)+yf(x)=f(xy)+f(x^2)+f(f(y))

for all real numbers

x

y