MathDB

1

Part of 2011 Moldova Team Selection Test

Problems(2)

Equation with real numbers

Source: Moldova TST 2011, day 1, problem 1

3/8/2011
Find all real numbers x,yx, y such that: y+3x+2=232+y2−49−16xy+3\sqrt{x+2}=\frac{23}2+y^2-\sqrt{49-16x}
calculusderivativeinequalitiesalgebra unsolvedalgebra
Sum of all elements of a set

Source: Moldova TST 2011, day 2, problem 1

3/8/2011
Natural numbers have been divided in groups as follow: (1),(2,4),(3,5,7),(6,8,10,12),(9,11,13,15,17),…(1), (2, 4), (3, 5, 7), (6, 8, 10, 12), (9, 11, 13, 15, 17), \ldots. Let SnS_n be the sum of the elements of the nnth group. Prove that S2n+12n+1−S2n2n\frac{S_{2n+1}}{2n+1}-\frac{S_{2n}}{2n} is even.
arithmetic sequencenumber theory unsolvednumber theory