MathDB

2

Part of 2010 District Olympiad

Problems(6)

n(n + 1)(n + 2) mod (n - 1) - 2010 Romania District VII p2

Source:

9/1/2024
Let nn be an integer, n≥2n \ge 2. Find the remainder of the division of the number n(n+1)(n+2)n(n + 1)(n + 2) by n−1n - 1.
number theoryremainder
(x + y)^2 / (x^3 + xy^2- x^2y -y^3) not an integer

Source: 2010 Romania District VII p2

9/1/2024
Let x,yx, y be distinct positive integers. Show that the number (x+y)2x3+xy2−x2y−y3\frac{(x + y)^2}{x^3 + xy^2- x^2y -y^3} is not an integer.
number theoryalgebraInteger
Romania District Olympiad 2010

Source: Grade IX

3/13/2010
Consider the sequence (xn)n≥0 (x_n)_{n\ge 0} where x_n\equal{}2^{n}\minus{}1\ ,\ n\in \mathbb{N}. Determine all the natural numbers p p for which: s_p\equal{}x_0\plus{}x_1\plus{}x_2\plus{}...\plus{}x_p is a power with natural exponent of 2 2.
algebra proposedalgebra
Romania District Olympiad 2010

Source: Grade X

3/13/2010
Consider two real numbers a∈[−2,∞) , r∈[0,∞) a\in [ - 2,\infty)\ ,\ r\in [0,\infty) and the natural number n≥1 n\ge 1. Show that: r2n+arn+1≥(1−r)2n r^{2n} + ar^n + 1\ge (1 - r)^{2n}
inequalities proposedinequalities
Romanian District Olympiad

Source: Grade XI

3/17/2010
Consider the matrix A,B∈lM3(C) A,B\in \mathcal l{M}_3(\mathbb{C}) with A=−tA A=-^tA and B=tB B=^tB. Prove that if the polinomial function defined by f(x)=det⁡(A+xB) f(x)=\det(A+xB) has a multiple root, then det⁡(A+B)=det⁡B \det(A+B)=\det B.
linear algebramatrixalgebrapolynomialfunctionlinear algebra unsolved
Romanian District Olympiad

Source: Grade XII

3/17/2010
Let G G be a group such that if a,b∈G a,b\in \mathbb{G} and a^2b\equal{}ba^2, then ab\equal{}ba. i)If G G has 2n 2^n elements, prove that G G is abelian. ii) Give an example of a non-abelian group with G G's property from the enounce.
abstract algebragroup theorysuperior algebrasuperior algebra unsolved