MathDB

Problem 1

Part of 2003 Federal Math Competition of S&M

Problems(3)

2011 as sum of 10 4th powers

Source: Serbia 2003 1st Grade P1

5/12/2021
Find the number of solutions to the equationx14+x24+…+x104=2011x_1^4+x_2^4+\ldots+x_{10}^4=2011in the set of positive integers.
number theoryDiophantine equation
existence of triangle with sides √a,√b,√c

Source: Serbia 2003 2nd Grade P1

5/13/2021
Given a △ABC\triangle ABC with the edges a,ba,b and cc and the area SS:
(a) Prove that there exists △A1B1C1\triangle A_1B_1C_1 with the sides a,b\sqrt a,\sqrt b and c\sqrt c. (b) If S1S_1 is the area of △A1B1C1\triangle A_1B_1C_1, prove that S12≥S34S_1^2\ge\frac{S\sqrt3}4.
geometryTriangle
10^n|floor((5+√35)^(2n-1))

Source: Serbia 2003 3&4th Grade P1

5/13/2021
Prove that the number ⌊(5+35)2n−1⌋\left\lfloor\left(5+\sqrt{35}\right)^{2n-1}\right\rfloor is divisible by 10n10^n for each n∈Nn\in\mathbb N.
number theoryfloor functionDivisibilityrecursion