MathDB

1

2017 | a_ib_i- a_jb_j i for permutations a_i, a_j of 1-2016

a_1, a_2, ... , a_{2016}

and

b_1, b_2, ... , b_{2016}

each contain all natural numbers from

1

to

2016

exactly once each (in other words, they are both permutations of the numbers

1, 2, ..., 2016

). Prove that different indices

i

and

j

can be found such that

a_ib_i- a_jb_j

is divisible by

2017

.

15

1

exactly four stable endings of a number of length n

Let's call the number string

D = d_{n-1}d_{n-2}...d_0

a stable ending of a number , if for any natural number

m

that ends in

D

, any of its natural powers

m^k

also ends in

D

. Prove that for every natural number

n

there are exactly four stable endings of a number of length

n

.
[hide=original wording]Ciparu virkni

D = d_{n-1}d_{n-2}...d_0

sauksim par stabilu skaitļa nobeigumu, ja jebkuram naturālam skaitlim m, kas beidzas ar D, arī jebkura tā naturāla pakāpe

m^k

beidzas ar D. Pierādīt, ka katram naturālam n ir tieši četri stabili skaitļa nobeigumi, kuru garums ir n.

14

1

ab + bc + ac = (a + b -c)(b + c - a)(c + a - b) for gcd(a,b,c)=1

Can you find three natural numbers

a, b, c

whose greatest common divisor is

1

and which satisfy the equality

ab + bc + ac = (a + b -c)(b + c - a)(c + a - b) ?

13

1

\sqrt{1 + 1/n^2 + 1/(n+1)^2} is rational

Prove that the number

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

is rational for all natural

n

.

5

1

max area of magic octagon, sides follow lines of chessboard

A magic octagon is an octagon whose sides follow the lines of the checkerboard's checkers and the side lengths are

1, 2, 3, 4, 5, 6, 7, 8

(in any order). What is the largest possible area of the magic octagon?
[hide=original wording]Burvju astoņstūris ar astoņstūris, kura malas iet pa rūtiņu lapas rūtiņu līnijām un malu garumi ir 1, 2,3, 4, 5, 6, 7, 8 (jebkādā secībā). Kāds ir lielākais iespējamais burvju astoņstūra laukums?

8

1

2017 chess players in chess tournament, trios wanted

2017

chess players participated in the chess tournament, each of them played exactly one chess game with each other. Let's call a trio of chess players

A, B, C

a principled one, if

A

defeated

B

,

B

defeated

C

, and

C

defeated

A

. What is the largest possible number of threes of principled chess players?

7

1

all six-digit natural numbers from 100000 to 999999 in a row no spaces

All six-digit natural numbers from

100000

to

999999

are written on the page in ascending order without spaces. What is the largest value of

k

for which the same

k

-digit number can be found in at least two different places in this string?

6

1

naturals in 13x13 grid, sum of numbers at 8 squares divisible by 8

A natural number is written in each box of the

13 \times 13

grid area. Prove that you can choose

2

rows and

4

columns such that the sum of the numbers written at their

8

intersections is divisible by

8

.

4

1

P(x) = 2x^3-30x^2+cx , for consecutive integers it takes consecutive values

The values of the polynomial

P(x) = 2x^3-30x^2+cx

for any three consecutive integers are also three consecutive integers. Find these values.

3

1

f(x+y)+f(xy)=f(x)f(y)+1 in Z

Find all functions

f (x) : Z \to Z

defined on integers, take integer values, and for all

x,y \in Z

satisfy

f(x+y)+f(xy)=f(x)f(y)+1

2

1

(x+y)(2- sin(x+y)/ 4 sin^2(x+y)=xy/(x+y)

Find all pairs of real numbers

(x, y)

that satisfy the equation

\frac{(x+y)(2-\sin(x+y))}{4\sin^2(x+y)}=\frac{xy}{x+y}

1

\sqrt{1/(3x+1)}+\sqrt{ (x/x+3)} >=1

Prove that for all real

x > 0

holds the inequality

\sqrt{\frac{1}{3x+1}}+\sqrt{\frac{x}{x+3}}\ge 1.

For what values of

x

does the equality hold?

12

1

collinear wanted, circumcircle related

A diameter

AK

is drawn for the circumscribed circle

\omega

of an acute-angled triangle

ABC

, an arbitrary point

M

is chosen on the segment

BC

, the straight line

AM

intersects

\omega

at point

Q

. The foot of the perpendicular drawn from

M

on

AK

is

D

, the tangent drawn to the circle

\omega

through the point

Q

, intersects the straight line

MD

at

P

. A point

L

(different from

Q

) is chosen on

\omega

such that

PL

is tangent to

\omega

. Prove that points

L

,

M

and

K

lie on the same line.

11

1

perimeter of PBC <= perimeter of ABC, PL=AL on angle bisector AL

On the extension of the angle bisector

AL

of the triangle

ABC

, a point

P

is placed such that

P L = AL

. Prove that the perimeter of triangle

PBC

does not exceed the perimeter of triangle

ABC

.

10

1

<AST = 90^o wanted, incircle and altitude related

In an obtuse triangle

ABC

, for which

AC < AB

, the radius of the inscribed circle is

R

, the midpoint of its arc

BC

(which does not contain

A

) is

S

. A point

T

is placed on the extension of altitude

AD

such that

D

is between

A

and

T

and

AT = 2R

. Prove that

\angle AST = 90^o

.

9

1

AC// DI , OD _|_IB if AC=BC

In an isosceles triangle

ABC

in which

AC = BC

and

\angle ABC < 60^o

,

I

and

O

are the centers of the inscribed and circumscribed circles, respectively. For the triangle

BIO

, the circumscribed circle intersects the side

BC

again at

D

. Prove that: i) lines

AC

and

DI

are parallel, ii) lines

OD

and

IB

are perpendicular.

2017 Latvia Baltic Way TST

Subcontests

2017 | a_ib_i- a_jb_j i for permutations a_i, a_j of 1-2016

exactly four stable endings of a number of length n

ab + bc + ac = (a + b -c)(b + c - a)(c + a - b) for gcd(a,b,c)=1

\sqrt{1 + 1/n^2 + 1/(n+1)^2} is rational

max area of magic octagon, sides follow lines of chessboard

2017 chess players in chess tournament, trios wanted

all six-digit natural numbers from 100000 to 999999 in a row no spaces

naturals in 13x13 grid, sum of numbers at 8 squares divisible by 8

P(x) = 2x^3-30x^2+cx , for consecutive integers it takes consecutive values

f(x+y)+f(xy)=f(x)f(y)+1 in Z

(x+y)(2- sin(x+y)/ 4 sin^2(x+y)=xy/(x+y)

\sqrt{1/(3x+1)}+\sqrt{ (x/x+3)} >=1

collinear wanted, circumcircle related

perimeter of PBC <= perimeter of ABC, PL=AL on angle bisector AL

<AST = 90^o wanted, incircle and altitude related

AC// DI , OD _|_IB if AC=BC

2017 Latvia Baltic Way TST

Subcontests

2017 | a_ib_i- a_jb_j i for permutations a_i, a_j of 1-2016

exactly four stable endings of a number of length n

ab + bc + ac = (a + b -c)(b + c - a)(c + a - b) for gcd(a,b,c)=1

\sqrt{1 + 1/n^2 + 1/(n+1)^2} is rational

max area of magic octagon, sides follow lines of chessboard

2017 chess players in chess tournament, trios wanted

all six-digit natural numbers from 100000 to 999999 in a row no spaces

naturals in 13x13 grid, sum of numbers at 8 squares divisible by 8

P(x) = 2x^3-30x^2+cx , for consecutive integers it takes consecutive values

f(x+y)+f(xy)=f(x)f(y)+1 in Z

(x+y)(2- sin(x+y)/ 4 sin^2(x+y)=xy/(x+y)

\sqrt{1/(3x+1)}+\sqrt{ (x/x+3)} &gt;=1

collinear wanted, circumcircle related

perimeter of PBC &lt;= perimeter of ABC, PL=AL on angle bisector AL

&lt;AST = 90^o wanted, incircle and altitude related

AC// DI , OD _|_IB if AC=BC

\sqrt{1/(3x+1)}+\sqrt{ (x/x+3)} >=1

perimeter of PBC <= perimeter of ABC, PL=AL on angle bisector AL

<AST = 90^o wanted, incircle and altitude related