MathDB

1

sum \frac{b_{n+1}}{b_1+b_2+...+b_n} > \frac{2013}{1013}

b_1,b_2,...

be a sequence of positive real numbers such that for each

n\ge 1

,

b_{n+1}^2 \ge \frac{b_1^2}{1^3}+\frac{b_2^2}{2^3}+...+\frac{b_n^2}{n^3}

Show that there is a positive integer

M

such that

\sum_{n=1}^M \frac{b_{n+1}}{b_1+b_2+...+b_n} > \frac{2013}{1013}

2

1

m^3-n^3=2mn +8 diophantine

Find all pairs of integers

(m,n)

such that

m^3-n^3=2mn +8

4

1

Finite Operations on Matrix

In the following

6\times 6

matrix, one can choose any

k\times k

submatrix, with

1<k\leq6

and add

1

to all its entries. Is it possible to perform the operation a finite number of times so that all the entries in the

6\times 6

matrix are multiples of

3

?

\begin{pmatrix} 2 & 0 & 1 & 0 & 2 & 0 \\ 0 & 2 & 0 & 1 & 2 & 0 \\ 1 & 0 & 2 & 0 & 2 & 0 \\ 0 & 1 & 0 & 2 & 2 & 0 \\ 1 & 1 & 1 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}

Note: A

p\times q

submatrix of a

m\times n

matrix (with

p\leq m

,

q\leq n

) is a

p\times q

matrix formed by taking a block of the entries of this size from the original matrix.

1

OLYMPIAD GEOMETRY, interesting

In the Triangle ABC AB>AC, the extension of the altitude AD with D lying inside BC intersects the circum-circle of the Triangle ABC at P. The circle through P and tangent to BC at D intersects the circum-circle of Triangle ABC at Q distinct from P with PQ=DQ. Prove that AD=BD-DC

5

1

SMO Senior Round 2 Qn 5 2013

Let

x,y

be distinct real numbers such that

\frac{x^n-y^n}{x-y}

is an integer for

4

consecutive positive integer

n

. Prove that

\frac{x^n-y^n}{x-y}

is an integer for all positive integers

n

.

2013 Singapore Senior Math Olympiad

Subcontests

sum \frac{b_{n+1}}{b_1+b_2+...+b_n} > \frac{2013}{1013}

m^3-n^3=2mn +8 diophantine

Finite Operations on Matrix

OLYMPIAD GEOMETRY, interesting

SMO Senior Round 2 Qn 5 2013

2013 Singapore Senior Math Olympiad

Subcontests

sum \frac{b_{n+1}}{b_1+b_2+...+b_n} &gt; \frac{2013}{1013}

m^3-n^3=2mn +8 diophantine

Finite Operations on Matrix

OLYMPIAD GEOMETRY, interesting

SMO Senior Round 2 Qn 5 2013

sum \frac{b_{n+1}}{b_1+b_2+...+b_n} > \frac{2013}{1013}