MathDB

2

Part of 2006 All-Russian Olympiad

Problems(3)

Large a, b, c, d satisfying 1/a + 1/b + 1/c + 1/d = 1/(abcd)

Source: All-Russian Olympiad 2006 finals, problem 9.2

5/7/2006
Show that there exist four integers aa, bb, cc, dd whose absolute values are all >1000000>1000000 and which satisfy 1a+1b+1c+1d=1abcd\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=\frac{1}{abcd}.
inductionnumber theory proposednumber theory
If (a-1)^3 + a^3 + (a+1)^3 is a cube, then 4 | a

Source: All-Russian Olympiad 2006 finals, problem 10.2

5/7/2006
If an integer a>1a > 1 is given such that (a−1)3+a3+(a+1)3\left(a-1\right)^3+a^3+\left(a+1\right)^3 is the cube of an integer, then show that 4∣a4\mid a.
geometry3D geometrymodular arithmeticnumber theoryrelatively primenumber theory proposed
Sum and product of purely periodic decimal fractions

Source: All-Russian Olympiad 2006 finals, problem 11.2

5/6/2006
The sum and the product of two purely periodic decimal fractions aa and bb are purely periodic decimal fractions of period length TT. Show that the lengths of the periods of the fractions aa and bb are not greater than TT. Note. A purely periodic decimal fraction is a periodic decimal fraction without a non-periodic starting part.
number theory proposednumber theory