MathDB

1

a^m has last digit 6 and the sum of the other digits is 3?

n

such that the number

2^n

has last digit

6

and the sum of the other digits is

2

? b) Are there natural numbers

a

and

m\ge 3

such that the number

a^m

has last digit

6

and the sum of the other digits is 3?

6

1

Geometry with inscribed triangle areas proportional to radii

Circles ω1 and ω2 have no common point. Where is outerior tangents a and b, interior tangent c. Lines a, b and c touches circle ω1 respectively on points A1, B1 and C1, and circle ω2 – respectively on points A2, B2 and C2. Prove that triangles A1B1C1 and A2B2C2 area ratio is the same as ratio of ω1 and ω2 radii.

1

Easy geometry

Circle touches parallelogram‘s

ABCD

borders

AB, BC

and

CD

respectively at points

K, L

and

M

. Perpendicular is drawn from vertex

C

to

AB

. Prove, that the line

KL

divides this perpendicular into two equal parts (with the same length).

5

1

Lithuanian TST 2014 #5

Given real numbers

x

and

y

. Let

s_{1}=x+y, s_{2}=x^2+y^2, s_{3}=x^3+y^3, s_{4}=x^4+y^4

and

t=xy

. a) Prove, that number

t

is rational, if

s_{2}, s_{3}

and

s_{4}

are rational numbers. b) Prove, that number

s_{1}

is rational, if

s_{2}, s_{3}

and

s_{4}

are rational numbers. c) Can number

s_{1}

be irrational, if

s_{2}

and

s_{3}

are rational numbers?

3

1

Lithuanian TST 2014 #3

Given such positive real numbers

a, b

and

c

, that the system of equations:

\{\begin{matrix}a^2x+b^2y+c^2z=1&&\\xy+yz+zx=1&&\end{matrix}

has exactly one solution of real numbers

(x, y, z)

. Prove, that there is a triangle, which borders lengths are equal to

a, b

and

c

.

2

1

Lithuanian TST 2014 #2

Finite set

A

has such property: every six its distinct elements’ sum isn’t divisible by

6

. Does there exist such set

A

consisting of

11

distinct natural numbers?

2014 Lithuania Team Selection Test

Subcontests

a^m has last digit 6 and the sum of the other digits is 3?

Geometry with inscribed triangle areas proportional to radii

Easy geometry

Lithuanian TST 2014 #5

Lithuanian TST 2014 #3

Lithuanian TST 2014 #2