MathDB

3

Part of 1998 Iran MO (3rd Round)

Problems(4)

Show that all indices are zero except four

Source: Iran Third Round MO 1998, Exam 1, P3

7/1/2012
Let A,BA,B be two matrices with positive integer entries such that sum of entries of a row in AA is equal to sum of entries of the same row in BB and sum of entries of a column in AA is equal to sum of entries of the same column in BB. Show that there exists a sequence of matrices A1,A2,A3,⋯ ,AnA_1,A_2,A_3,\cdots , A_n such that all entries of the matrix AiA_i are positive integers and in the sequence A=A0,A1,A2,A3,⋯ ,An=B,A=A_0,A_1,A_2,A_3,\cdots , A_n=B, for each index ii, there exist indexes k,j,m,nk,j,m,n such that \begin{array}{*{20}{c}} \\ {{A_{i + 1}} - {A_{i}} = } \end{array}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}}     \ \ j& \ \ \ {k} \end{array}} \\ {\begin{array}{*{20}{c}} m \\ n \end{array}\left( {\begin{array}{*{20}{c}} { + 1}&{ - 1} \\ { - 1}&{ + 1} \end{array}} \right)} \end{array} \ \text{or} \ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}}     \ \ j& \ \ \ {k} \end{array}} \\ {\begin{array}{*{20}{c}} m \\ n \end{array}\left( {\begin{array}{*{20}{c}} { - 1}&{ + 1} \\ { + 1}&{ - 1} \end{array}} \right)} \end{array}. That is, all indices of Ai+1−Ai{A_{i + 1}} - {A_{i}} are zero, except the indices (m,j),(m,k),(n,j)(m,j), (m,k), (n,j), and (n,k)(n,k).
linear algebramatrixlinear algebra unsolved
maximum possible number of points with integer coordinates

Source: Iran Third Round MO 1998, Exam 2, P3

10/31/2010
Let n(r)n(r) be the maximum possible number of points with integer coordinates on a circle with radius rr in Cartesian plane. Prove that n(r)<63πr23.n(r) < 6\sqrt[3]{3 \pi r^2}.
analytic geometrygeometry unsolvedgeometry
Red and Green Points

Source: Iran Third Round MO 1998, Exam 4, P3

10/31/2010
Let ABCABC be a given triangle. Consider any painting of points of the plane in red and green. Show that there exist either two red points on the distance 11, or three green points forming a triangle congruent to triangle ABCABC.
combinatorics proposedcombinatorics
Functional Equation

Source: Iran Third Round MO 1998, Exam 3, P3

10/31/2010
Find all functions f:R→Rf : \mathbb R \to \mathbb R such that for all x,y,x, y, f(f(x)+y)=f(x2−y)+4f(x)y.f(f(x) + y) = f(x^2 - y) + 4f(x)y.
algebrafunctional equation