MathDB

1

1 < a_n < 1 +1/3^n if a_n =(2a_{n-1} + 1)/(a_{n-1} + 2)

a_0, a_1, a_2, a_3,...

is defined by

a_0 = 2

and

a_n =\frac{2a_{n-1} + 1}{a_{n-1} + 2}

,

n = 1, 2, 3, ...

Prove that

1 < a_n < 1 + \frac{1}{3^n}

for all

n = 1, 2, 3, . .

4

1

no of distinct odd prime factors of n(n + 3) is a multiple of 3

Show that there are infinitely many positive integers

n

such that the number of distinct odd prime factors of

n(n + 3)

is a multiple of

3

.
(For instance,

180 = 2^2 \cdot 3^2 \cdot 5

has two distinct odd prime factors and

840 = 2^3 \cdot 3 \cdot 5 \cdot 7

has three.)

2

1

a deck of 52 cards is stacked in a pile facing down

A deck of

52

cards is stacked in a pile facing down. Tom takes the small pile consisting of the seven cards on the top of the deck, turns it around, and places it at the bottom of the deck. All cards are again in one pile, but not all of them face down, since the seven cards at the bottom now face up. Tom repeats this move until all cards face down again. In total, how many moves did Tom make?

3

1

AC // BM wanted, right triangle, projections, midpoints, circumcenter

Let

ABC

be a triangle with

\angle A = 90^o

and let

D

be an orthogonal projection of

A

onto

BC

. The midpoints of

AD

and

AC

are called

E

and

F

, respectively. Let

M

be the circumcentre of

\vartriangle BEF

. Prove that

AC\parallel BM

.

2017 Grand Duchy of Lithuania

Subcontests

1 < a_n < 1 +1/3^n if a_n =(2a_{n-1} + 1)/(a_{n-1} + 2)

no of distinct odd prime factors of n(n + 3) is a multiple of 3

a deck of 52 cards is stacked in a pile facing down

AC // BM wanted, right triangle, projections, midpoints, circumcenter

2017 Grand Duchy of Lithuania

Subcontests

1 &lt; a_n &lt; 1 +1/3^n if a_n =(2a_{n-1} + 1)/(a_{n-1} + 2)

no of distinct odd prime factors of n(n + 3) is a multiple of 3

a deck of 52 cards is stacked in a pile facing down

AC // BM wanted, right triangle, projections, midpoints, circumcenter

1 < a_n < 1 +1/3^n if a_n =(2a_{n-1} + 1)/(a_{n-1} + 2)