MathDB

1

a_{1998} > 63 if a_n =a_{n-1} + 1/ a_{n-1}

a_0

be a positive real number and consider the general term sequence

a_n

defined by

a_n =a_{n-1} + \frac{1}{a_{n-1}} \,\,\, n=1,2,3,...

Prove that

a_{1998} > 63

.

4

1

[\frac{3^{31}+2^{31}}{3^{29}+2^{29}}] +1 - Portugal OPM 1998 p4

What is the largest integer less than or equal to

\frac{3^{31}+2^{31}}{3^{29}+2^{29}} \,\,\, ?

3

1

2000 -element subset of 1-3000 where none of them is twice another

Could the set

\{1,2,3,...,3000\}

contain a subset of

2000

elements such that none of them is twice the size of another?

1

72 chickens sold for 679 escudos

A chicken breeder went to check what price per chick he had charged the previous year. He found an invoice, half erased, which read:

72

chickens sold for

*679*

escudos” (the digits of the units and tens of thousands were illegible). What price did each chick sell for last year?

2

1

PA^2+PB^2+...+PH^2 =?, regular octagon ABC...H 1998 Portugal p2

The regular octagon of the following figure is inscribed in a circle of radius

1

and

P

is a arbitrary point of this circle. Calculate the value of

PA^2 + PB^2 +...+ PH^2

. https://cdn.artofproblemsolving.com/attachments/4/c/85e8e48c45970556077ac09c843193959b0e5a.png

5

1

Real IMO Shortlist 2019 G2 - Archimedes Broken Chord

Let

F

be the midpoint of circle arc

AB

, and let

M

be a point on the arc such that

AM <MB

. The perpendicular drawn from point

F

on

AM

intersects

AM

at point

T

. Show that

T

bisects the broken line

AMB

, that is

AT =TM+MB

.
KöMaL Gy. 2404. (March 1987), Archimedes of Syracuse

1998 Portugal MO

Subcontests

a_{1998} > 63 if a_n =a_{n-1} + 1/ a_{n-1}

[\frac{3^{31}+2^{31}}{3^{29}+2^{29}}] +1 - Portugal OPM 1998 p4

2000 -element subset of 1-3000 where none of them is twice another

72 chickens sold for 679 escudos

PA^2+PB^2+...+PH^2 =?, regular octagon ABC...H 1998 Portugal p2

Real IMO Shortlist 2019 G2 - Archimedes Broken Chord

1998 Portugal MO

Subcontests

a_{1998} &gt; 63 if a_n =a_{n-1} + 1/ a_{n-1}

[\frac{3^{31}+2^{31}}{3^{29}+2^{29}}] +1 - Portugal OPM 1998 p4

2000 -element subset of 1-3000 where none of them is twice another

72 chickens sold for *679* escudos

PA^2+PB^2+...+PH^2 =?, regular octagon ABC...H 1998 Portugal p2

Real IMO Shortlist 2019 G2 - Archimedes Broken Chord

a_{1998} > 63 if a_n =a_{n-1} + 1/ a_{n-1}

72 chickens sold for 679 escudos