MathDB

1

Part of 1989 Romania Team Selection Test

Problems(4)

set of m x n matrices with entries in the set {0,1,2,3,4}

Source: Romania BMO TST 1989 p1

2/17/2020
Let MM denote the set of m×nm\times n matrices with entries in the set {0,1,2,3,4}\{0,1,2,3,4\} such that in each row and each column the sum of elements is divisible by 55. Find the cardinality of set MM.
combinatoricsSumnumber theory
1989 | a_n= n^6 +5n^4 -12n^2 -36

Source: Romania IMO TST 1989 1.1

2/17/2020
Let the sequence (ana_n) be defined by an=n6+5n4−12n2−36,n≥2a_n = n^6 +5n^4 -12n^2 -36, n \ge 2. (a) Prove that any prime number divides some term in this sequence. (b) Prove that there is a positive integer not dividing any term in the sequence. (c) Determine the least n≥2n \ge 2 for which 1989∣an1989 | a_n.
Sequencenumber theoryprime
f(f(x))-2 f(x)+x = 0 for all x \in N, f(1989) wanted

Source: Romania IMO TST 1989 2.1

2/17/2020
Let FF be the set of all functions f:N→Nf : N \to N which satisfy f(f(x))−2f(x)+x=0f(f(x))-2 f(x)+x = 0 for all x∈Nx \in N. Determine the set A={f(1989)∣f∈F}A =\{ f(1989) | f \in F\}.
functionalfunctional equationalgebra
Oldies but goldies

Source: Romanian selection test 1989, proposed by Gheorghe Eckstein

2/21/2005
Prove that 1+2+…+n<2\sqrt {1+\sqrt {2+\ldots +\sqrt {n}}}<2, ∀n≥1\forall n\ge 1.
inductioninequalitieslimitalgebra proposedalgebra