MathDB

3

Part of 2008 China National Olympiad

Problems(2)

China Mathematics Olympiads (National Round) 2008 Problem 3

Source:

11/28/2010
Given a positive integer nn and x1≤x2≤…≤xn,y1≥y2≥…≥ynx_1 \leq x_2 \leq \ldots \leq x_n, y_1 \geq y_2 \geq \ldots \geq y_n, satisfying ∑i=1nixi=∑i=1niyi\displaystyle\sum_{i = 1}^{n} ix_i = \displaystyle\sum_{i = 1}^{n} iy_i Show that for any real number α\alpha, we have ∑i=1nxi[iα]≥∑i=1nyi[iα]\displaystyle\sum_{i =1}^{n} x_i[i\alpha] \geq \displaystyle\sum_{i =1}^{n} y_i[i\alpha]
Here [β][\beta] denotes the greastest integer not larger than β\beta.
inequalitiesalgebra unsolvedalgebra
China Mathematics Olympiads (National Round) 2008 Problem 6

Source:

11/28/2010
Find all triples (p,q,n)(p,q,n) that satisfy q^{n+2} \equiv 3^{n+2} (\mod p^n) ,  p^{n+2} \equiv 3^{n+2} (\mod q^n) where p,qp,q are odd primes and nn is an positive integer.
modular arithmeticalgebrapolynomialVietanumber theory unsolvednumber theory