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IMO ShortList 2002, number theory problem 3

Source: IMO ShortList 2002, number theory problem 3

9/28/2004

Let

p_1,p_2,\ldots,p_n

be distinct primes greater than

3

. Show that

2^{p_1p_2\cdots p_n}+1

has at least

4^n

divisors.

algebranumber theoryprimesDivisorsIMO Shortlist

IMO ShortList 2002, algebra problem 3

Source: IMO ShortList 2002, algebra problem 3

9/28/2004

Let

P

be a cubic polynomial given by

P(x)=ax^3+bx^2+cx+d

, where

a,b,c,d

are integers and

a\ne0

. Suppose that

xP(x)=yP(y)

for infinitely many pairs

x,y

of integers with

x\ne y

. Prove that the equation

P(x)=0

has an integer root.

algebrapolynomialIMO Shortlistequationinfinitely many solutions

IMO ShortList 2002, combinatorics problem 3

Source: IMO ShortList 2002, combinatorics problem 3

9/28/2004

Let

n

be a positive integer. A sequence of

n

positive integers (not necessarily distinct) is called full if it satisfies the following condition: for each positive integer

k\geq2

, if the number

k

appears in the sequence then so does the number

k-1

, and moreover the first occurrence of

k-1

comes before the last occurrence of

k

. For each

n

, how many full sequences are there ?

combinatoricsIMO Shortlistcountingbijection

3