MathDB

Problem 1

Part of 2001 Federal Math Competition of S&M

Problems(3)

SSSS and diagonals perpendicular

Source: Serbia & Montenegro 2001 1st Grade P1

6/1/2021
Let ABCDABCD and A1B1C1D1A_1B_1C_1D_1 be convex quadrangles in a plane, such that AB=A1B1AB=A_1B_1, BC=B1C1BC=B_1C_1, CD=C1D1CD=C_1D_1 and DA=D1A1DA=D_1A_1. Given that diagonals ACAC and BDBD are perpendicular to each other, prove that the same holds for diagonals A1C1A_1C_1 and B1D1B_1D_1.
geometry
3a=x^2+2y^2 implies a=b^2+2c^2

Source: Serbia & Montenegro 2001 2nd Grade P1

6/2/2021
Let S={x2+2y2∣x,y∈Z}S=\{x^2+2y^2\mid x,y\in\mathbb Z\}. If aa is an integer with the property that 3a3a belongs to SS, prove that then aa belongs to SS as well.
number theory
x^y + y = y^x + x

Source: unknown

4/12/2006
Solve in positive integers xy+y=yx+x x^y + y = y^x + x
calculusderivativenumber theory unsolvednumber theory