MathDB

calculating the summation

Source: Indonesia IMO 2007 TST, Stage 2, Test 1, Problem 4

11/15/2009

Let

n

and

k

be positive integers. Please, find an explicit formula for

\sum y_1y_2 \dots y_k,

where the summation runs through all k\minus{}tuples positive integers

(y_1,y_2,\dots,y_k)

satisfying y_1\plus{}y_2\plus{}\dots\plus{}y_k\equal{}n.

combinatorics proposedcombinatorics

single representation problem

Source: 2007 Indonesia TST stage 2 test 2 p4

12/14/2020

Given a collection of sets

X = \{A_1, A_2, ..., A_n\}

. A set

\{a_1, a_2, ..., a_n\}

is called a single representation of

X

if

a_i \in A_i

for all i. Let

|S| = mn

,

S = A_1\cup A_2 \cup ... \cup A_n = B_1 \cup B_2 \cup ... \cup B_n

with

|A_i| = |B_i| = m

for all

i

. Prove that

S = C_1 \cup C_2 \cup ... \cup C_n

where for every

i, C_i

is a single represenation for

\{A_j\}_{j=1}^n

and

\{B_j\}_{j=1}^n

.

combinatoricsrepresentationset theorySubsets

points and segments

Source: Indonesia IMO 2007 TST, Stage 2, Test 2, Problem 4

11/15/2009

Let

X

be a set of

k

vertexes on a plane such that no three of them are collinear. Let

P

be the family of all

{k \choose 2}

segments that connect each pair of points. Determine

\tau(P)

.

combinatorics proposedcombinatorics

squares and the family

Source: Indonesia IMO 2007 TST, Stage 2, Test 4, Problem 4

11/15/2009

Let

S

be a finite family of squares on a plane such that every point on that plane is contained in at most

k

squares in

S

. Prove that

P

can be divided into 4(k\minus{}1)\plus{}1 sub-family such that in each sub-family, each pair of squares are disjoint.

combinatorics proposedcombinatorics

finding solution (n,p)

Source: Indonesia IMO 2007 TST, Stage 2, Test 5, Problem 4

11/15/2009

Determine all pairs

(n,p)

of positive integers, where

p

is prime, such that 3^p\minus{}np\equal{}n\plus{}p.

number theorygreatest common divisornumber theory proposed

4

Problems(5)