2006 VTRMC

Part of VTRMC

Subcontests

(7)

1

masses in spheres, cylinders

Three spheres each of unit radius have centers

P,Q,R

with the property that the center of each sphere lies on the surface of the other two spheres. Let

C

denote the cylinder with cross-section

PQR

(the triangular lamina with vertices

P,Q,R

) and axis perpendicular to

PQR

M

denote the space which is common to the three spheres and the cylinder

C

, and suppose the mass density of

M

at a given point is the distance of the point from

PQR

. Determine the mass of

M

1

triangle inside another, bisectors

In the diagram below,

BP

\angle ABC

CP

\angle BCA

PQ

is perpendicular to

BC

BQ\cdot QC=2PQ^2

AB+AC=3BC

. https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOC8zL2IwZjNjMDAxNWEwMTc1ZGNjMTkwZmZlZmJlMGRlOGRhYjk4NzczLnBuZw==&rn=VlRSTUMgMjAwNi5wbmc=

1

counting sequences of letters, Cs in a block

S(n)

denote the number of sequences of length

n

formed by the three letters

A,B,C

with the restriction that the

C

's (if any) all occur in a single block immediately following the first

B

(if any). For example

ABCCAA

AAABAA

ABCCCC

are counted in, but

ACACCB

CAAAAA

are not. Derive a simple formula for

S(n)

and use it to calculate

S(10)

1

no 1 in base 3 for n or n^2

Find, with proof, all positive integers

n

such that neither

n

n^2

1

when written in base

3

1

sequence, convergence of alternating sum

\{a_n\}

be a monotonically decreasing sequence of positive real numbers with limit

0

\{b_n\}

be a rearrangement of the sequence such that for every non-negative integer

m

b_{3m+1}

b_{3m+2}

b_{3m+3}

are a rearrangement of the terms

a_{3m+1}

a_{3m+2}

a_{3m+3}

. Prove or give a counterexample to the following statement: the series

\sum_{n=1}^\infty(-1)^nb_n

1

existence of DE solutions

We want to find functions

p(t)

q(t)

f(t)

p

q

are continuous functions on the open interval

(0,\pi)

f

is an infinitely differentiable nonzero function on the whole real line

(-\infty,\infty)

f(0)=f'(0)=f''(0)

y=\sin t

y=f(t)

are solutions of the differential equation

y''+p(t)y'+q(t)y=0

(0,\pi)

.
Is this possible? Either prove this is not possible, or show this is possible by providing an explicit example of such

f,p,q

1

VTRMC 2006 - Number 3.

Hey, This problem is from the VTRMC 2006. 3. Recall that the Fibonacci numbers

F(n)

are defined by F(0) \equal{} 0, F(1) \equal{} 1 and F(n) \equal{} F(n \minus{} 1) \plus{} F(n \minus{} 2) for

n \geq 2

. Determine the last digit of

F(2006)

(e.g. the last digit of 2006 is 6). As, I and a friend were working on this we noticed an interesting relationship when writing the Fibonacci numbers in "mod" notation. Consider the following, 01 = 1 mod 10 01 = 1 mod 10 02 = 2 mod 10 03 = 3 mod 10 05 = 5 mod 10 08 = 6 mod 10 13 = 3 mod 10 21 = 1 mod 10 34 = 4 mod 10 55 = 5 mod 10 89 = 9 mod 10 Now, consider that between the first appearance and second apperance of

5 mod 10

, there is a difference of five terms. Following from this we see that the third appearance of

5 mod 10

occurs at a difference 10 terms from the second appearance. Following this pattern we can create the following relationships. F(55) \equal{} F(05) \plus{} 5({2}^{2}) This is pretty much as far as we got, any ideas?