MathDB

1

a_m= 2^{2^{,,,{^{2}}}}}, A_{km} (x)=2^{2^{,,,^{x^{a_m}}}}} B_k(y)=4^{,,,^{4^y}}}

x,y,K,m \in N

, let us define:

a_m= \underset{k \, twos}{2^{2^{,,,{^{2}}}}}

,

A_{km} (x)= \underset{k \, twos}{ 2^{2^{,,,^{x^{a_m}}}}}

,

B_k(y)= \underset{m \, fours}{4^{4^{4^{,,,^{4^y}}}}}

,
Determine all pairs

(x,y)

of non-negative integers, dependent on

k>0

, such that

A_{km} (x)=B_k(y)

2

1

\Sigma^{b}_{k=a}\sqrt{k^2+3k+3} /(b-a+1), floor function sum

(i) For positive integers

a<b

, let

M(a,b)=\frac{\Sigma^{b}_{k=a}\sqrt{k^2+3k+3}}{b-a+1}

. Calculate

[M(a,b)]

(ii) Calculate

N(a,b)=\frac{\Sigma^{b}_{k=a}[\sqrt{k^2+3k+3}]}{b-a+1}

.

3

1

locus of P if A'BC, AB'C and ABC' are congruent to ABC

Let

P

be the point in the interior of

\vartriangle ABC

. Let

D

be the intersection of the lines

AP

and

BC

and let

A'

be the point such that

\overrightarrow{AD} = \overrightarrow{DA'}

. The points

B'

and

C'

are defined in the similar way. Determine all points

P

for which the triangles

A'BC, AB'C

, and

ABC'

are congruent to

\vartriangle ABC

.

6

1

Geometry Unleashed

The quadrilateral PQRS whose vertices are the midpoints of the sides AB, BC, CD, DA, respectively of a quadrilateral ABCD is called the midpoint quadrilateral of ABCD. Determine all circumscribed quadrilaterals whose mid-point quadrilaterals are squares. .

5

1

values of term in sequence

Let

n>1

and for

1 \leq k \leq n

let p_k \equal{} p_k(a_1, a_2, . . . , a_n) be the sum of the products of all possible combinations of k of the numbers

a_1,a_2,...,a_n

. Furthermore let P \equal{} P(a_1, a_2, . . . , a_n) be the sum of all

p_k

with odd values of

k

less than or equal to

n

. How many different values are taken by

a_j

if all the numbers

a_j (1 \leq j \leq n)

and

P

are prime?

4

1

numbers divisible by 2009

Let

a

be a positive integer. Consider the sequence

(a_n)

defined as a_0\equal{}a and a_n\equal{}a_{n\minus{}1}\plus{}40^{n!} for

n > 0

. Prove that the sequence

(a_n)

has infinitely many numbers divisible by

2009

.

2009 Federal Competition For Advanced Students, P2

Subcontests

a_m= 2^{2^{,,,{^{2}}}}}, A_{km} (x)=2^{2^{,,,^{x^{a_m}}}}} B_k(y)=4^{,,,^{4^y}}}

\Sigma^{b}_{k=a}\sqrt{k^2+3k+3} /(b-a+1), floor function sum

locus of P if A'BC, AB'C and ABC' are congruent to ABC

Geometry Unleashed

values of term in sequence

numbers divisible by 2009