MathDB

2

Part of 2008 Mongolia Team Selection Test

Problems(4)

Mongolian IMO TST 2008

Source: Day 1 Problem 2

5/12/2008
Let a,b,c,d a,b,c,d be the positive integers such that a>b>c>d a > b > c > d and (a \plus{} b \minus{} c \plus{} d) | (ac \plus{} bd) . Prove that if m m is arbitrary positive integer , n n is arbitrary odd positive integer, then a^n b^m \plus{} c^m d^n is composite number
number theory proposednumber theory
Mongolian TST 2008

Source: Day 2 Poblem 2

5/17/2008
Given positive integersm,n m,n such that m<n m < n. Integers 1,2,...,n2 1,2,...,n^2 are arranged in n×n n \times n board. In each row, m m largest number colored red. In each column m m largest number colored blue. Find the minimum number of cells such that colored both red and blue.
Gausscombinatorics proposedcombinatorics
Number of the inversion.

Source: Mongolian TST2008 day4 problem2

5/28/2008
Let a1,a2,...,an a_1,a_2,...,a_n is permutaion of 1,2,...,n 1,2,...,n. For this permutaion call the pair (ai,aj) (a_i,a_j) wrong pair if i<j i<j and ai>aj a_i >a_j.Let number of inversion is number of wrong pair of permutation a1,a2,a3,..,an a_1,a_2,a_3,..,a_n. Let n≥2 n \ge 2 is positive integer. Find the number of permutation of 1,2,..,n 1,2,..,n such that its number of inversion is divisible by n n.
combinatorics proposedcombinatorics
PQ is perpendicular to AC

Source: Mongolian TST 2008, day3 problem 2

5/23/2008
The quadrilateral ABCD ABCD inscribed in a circle wich has diameter BD BD. Let A′,B′ A',B' are symmetric to A,B A,B with respect to the line BD BD and AC AC respectively. If A'C \cap BD \equal{} P and AC\cap B'D \equal{} Q then prove that PQ⊥AC PQ \perp AC
geometrysymmetryincenterangle bisector