MathDB

Find all n

Source: Saudi Arabia BMO TST Day I Problem 5

8/3/2014

Find all positive integers

n

such that

3^n+4^n+\cdots+(n+2)^n=(n+3)^n.

inductioninequalitiesalgebra unsolvedalgebra

Prove that lines PQ,BC, and MT are concurrent

Source: Saudi Arabia BMO TST Day III Problem 5

8/3/2014

Let

ABC

be a triangle. Circle

\Omega

passes through points

B

and

C

. Circle

\omega

is tangent internally to

\Omega

and also to sides

AB

and

AC

at

T,~ P,

and

Q

, respectively. Let

M

be midpoint of arc

\widehat{BC}

(containing T) of

\Omega

. Prove that lines

P Q,~ BC,

and

MT

are concurrent.

geometry unsolvedgeometry

Forming an n by n array of numbers with conditions

Source: Saudi Arabia BMO TST Day II Problem 5

8/3/2014

Let

n > 3

be an odd positive integer not divisible by

3

. Determine if it is possible to form an

n \times n

array of numbers such that

[*] (a) the set of the numbers in each row is a permutation of

0, 1, \dots , n - 1

; the set of the numbers in each column is a permutation of

0, 1, \dots , n-1

;
[*] (b) the board is totally non-symmetric: for

1 \le i < j \le n

and

1 \le i' < j' \le n

, if

(i, j) \neq (i', j')

then

(a_{i,j} , a_{j,i}) \neq (a_{i',j'} , a_{j',i'})

where

a_{i,j}

denotes the entry in the

i^\text{th}

row and

j^\text{th}

column.

combinatorics unsolvedcombinatorics

5