MathDB

IMO ShortList 2001, geometry problem 5

Source: IMO ShortList 2001, geometry problem 5

9/30/2004

Let

ABC

be an acute triangle. Let

DAC,EAB

, and

FBC

be isosceles triangles exterior to

ABC

, with

DA=DC, EA=EB

, and

FB=FC

, such that \angle ADC = 2\angle BAC, \angle BEA= 2 \angle ABC, \angle CFB = 2 \angle ACB. Let

D'

be the intersection of lines

DB

and

EF

, let

E'

be the intersection of

EC

and

DF

, and let

F'

be the intersection of

FA

and

DE

. Find, with proof, the value of the sum

\frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}.

geometrycircumcircletrigonometryTriangleIMO Shortlist

IMO ShortList 2001, algebra problem 5

Source: IMO ShortList 2001, algebra problem 5

9/30/2004

Find all positive integers

a_1, a_2, \ldots, a_n

such that

\frac{99}{100} = \frac{a_0}{a_1} + \frac{a_1}{a_2} + \cdots + \frac{a_{n-1}}{a_n},

where

a_0 = 1

and

(a_{k+1}-1)a_{k-1} \geq a_k^2(a_k - 1)

for

k = 1,2,\ldots,n-1

.

inequalitiesalgebrarecurrence relationequationInteger sequenceIMO Shortlistimo shortlist 2001

IMO ShortList 2001, combinatorics problem 5

Source: IMO ShortList 2001, combinatorics problem 5

9/30/2004

Find all finite sequences

(x_0, x_1, \ldots,x_n)

such that for every

j

,

0 \leq j \leq n

,

x_j

equals the number of times

j

appears in the sequence.

combinatoricscountingInteger sequenceIMO Shortlist

5