MathDB

2

Part of 2004 India IMO Training Camp

Problems(7)

USAMO 2003 Problem 1

Source:

9/27/2005
Prove that for every positive integer nn there exists an nn-digit number divisible by 5n5^n all of whose digits are odd.
AMCUSA(J)MOUSAMOinductionnumber theory
Triples

Source: Indian IMOTC 2004 Practice Test 2 Problem 2

9/23/2005
Find all triples (x,y,n)(x,y,n) of positive integers such that (x+y)(1+xy)=2n (x+y)(1+xy) = 2^{n}
number theory unsolvednumber theory
Determine all integers s.t.

Source: Indian IMOTC 2004 Day 1 Problem 2

9/23/2005
Determine all integers aa such that ak+1a^k + 1 is divisible by 1232112321 for some kk
modular arithmeticnumber theory unsolvednumber theory
A power eqn

Source: Indian IMOTC 2004 Day 2 Problem 2

9/23/2005
Show that the only solutions of te equation pk+1=qm p^{k} + 1 = q^{m} , in positive integers k,q,m>1k,q,m > 1 and prime pp are (i) (p,k,q,m)=(2,3,3,2)(p,k,q,m) = (2,3,3,2) (ii) k=1,q=2,k=1 , q=2,and pp is a prime of the form 2m−12^{m} -1, m>1∈Nm > 1 \in \mathbb{N}
trigonometrynumber theory unsolvednumber theory
Prove a function

Source: Indian IMOTC 2004 Day 3 Problem 2

9/23/2005
Define a function g:N↦Ng: \mathbb{N} \mapsto \mathbb{N} by the following rule: (a) gg is nondecrasing (b) for each nn, g(n)g(n) i sthe number of times nn appears in the range of gg, Prove that g(1)=1g(1) = 1 and g(n+1)=1+g(n+1−g(g(n)))g(n+1) = 1 + g( n +1 - g(g(n))) for all n∈Nn \in \mathbb{N}
functionalgebra unsolvedalgebra
Square-free residues

Source: Bulgarian TST's 2004 --- Problem 2.

5/27/2004
Find all primes p≥3p \geq 3 with the following property: for any prime q<pq<p, the number p−⌊pq⌋q p - \Big\lfloor \frac{p}{q} \Big\rfloor q is squarefree (i.e. is not divisible by the square of a prime).
floor functionnumber theory solvednumber theory
A polynomial ineq

Source: Indian IMOTC 2004 Day 5 Problem 2

9/23/2005
Let P(x)=x4+ax3+bx2+cx+dP(x) = x^4 + ax^3 + bx^2 + cx + d and Q(x)=x2+px+qQ(x) = x^2 + px + qbe two real polynomials. Suppose that there exista an interval (r,s)(r,s) of length greater than 22 SUCH THAT BOTH P(x)P(x) AND Q(x)Q(x) ARE nEGATIVE FOR X∈(r,s)X \in (r,s) and both are positive for x>sx > s and x<rx<r. Show that there is a real x0x_0 such that P(x0)<Q(x0)P(x_0) < Q(x_0)
algebrapolynomialalgebra unsolved