MathDB

3

Part of 1997 Iran MO (3rd Round)

Problems(3)

Coloring the points with rational coordinates

Source: Iran Third Round MO 1997, Exam 2, P3

6/30/2012
Let dd be a real number such that d2=r2+s2d^2=r^2+s^2, where rr and ss are rational numbers. Prove that we can color all points of the plane with rational coordinates with two different colors such that the points with distance dd have different colors.
analytic geometrynumber theoryleast common multiplecombinatorics unsolvedcombinatorics
An interesting problem about weighing coins

Source: Iran Third Round MO 1997, Exam 1, P3

6/30/2012
There are 3030 bags and there are 100100 similar coins in each bag (coins in each bag are similar, coins of different bags can be different). The weight of each coin is an one digit number in grams. We have a digital scale which can weigh at most 999999 grams in each weighing. Using this scale, we want to find the weight of coins of each bag.
(a) Show that this operation is possible by 1010 times of weighing, and
(b) It's not possible by 99 times of weighing.
combinatorics unsolvedcombinatorics
Rational

Source: Iran Third Round MO 1997, Exam 3, P3

10/18/2005
Let S={x0,x1,…,xn}S = \{x_0, x_1,\dots , x_n\} be a finite set of numbers in the interval [0,1][0, 1] with x0=0x_0 = 0 and x1=1x_1 = 1. We consider pairwise distances between numbers in SS. If every distance that appears, except the distance 11, occurs at least twice, prove that all the xix_i are rational.
algebra proposedalgebralinear algebra