MathDB

India proudly presents an ISL sequence

Source: IMO Shortlist 1993, India 1

3/15/2006

Define a sequence

\langle f(n)\rangle^{\infty}_{n=1}

of positive integers by

f(1) = 1

and

f(n) = \begin{cases} f(n-1) - n & \text{ if } f(n-1) > n;\\ f(n-1) + n & \text{ if } f(n-1) \leq n, \end{cases}

for

n \geq 2.

Let

S = \{n \in \mathbb{N} \;\mid\; f(n) = 1993\}.

(i) Prove that

S

is an infinite set. (ii) Find the least positive integer in

S.

(iii) If all the elements of

S

are written in ascending order as

n_1 < n_2 < n_3 < \ldots ,

show that

\lim_{i\rightarrow\infty} \frac{n_{i+1}}{n_i} = 3.

limitalgebraSequencerecurrence relationIMO Shortlist

Partition of the positive rationals into disjoint subsets

Source: IMO Shortlist 1993, India 4

3/15/2006

a) Show that the set

\mathbb{Q}^{ + }

of all positive rationals can be partitioned into three disjoint subsets.

A,B,C

satisfying the following conditions:

BA = B; \& B^2 = C; \& BC = A;

where

HK

stands for the set

\{hk: h \in H, k \in K\}

for any two subsets

H, K

of

\mathbb{Q}^{ + }

and

H^2

stands for

HH.

b) Show that all positive rational cubes are in

A

for such a partition of

\mathbb{Q}^{ + }.

c) Find such a partition

\mathbb{Q}^{ + } = A \cup B \cup C

with the property that for no positive integer

n \leq 34,

both

n

and

n + 1

are in

A,

that is,

\text{min} \{n \in \mathbb{N}: n \in A, n + 1 \in A \} > 34.

modular arithmeticnumber theoryrelatively primepartitionalgebraIMO Shortlist

Incenter I is the mid point of DE [mixtilinear incircle]

Source: IMO Shortlist 1993, Spain 1

3/29/2005

Let

ABC

be a triangle, and

I

its incenter. Consider a circle which lies inside the circumcircle of triangle

ABC

and touches it, and which also touches the sides

CA

and

BC

of triangle

ABC

at the points

D

and

E

, respectively. Show that the point

I

is the midpoint of the segment

DE

.

geometryincentercircumcircleratioIMO Shortlist

1

Problems(3)