MathDB

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Part of 2009 Ukraine National Mathematical Olympiad

Problems(7)

Solve the equation n^3 - 2 = k! - [UKRMO 2009 Grade 8]

Source:

1/20/2011
Find all positive integer solutions of equation n3−2=k!.n^3 - 2 = k! .
modular arithmetic
(1+a^2)(1+b^2)(1+c^2) is perfect square-[UKRMO 2009 Grade 8]

Source:

1/21/2011
Let a,b,ca, b, c be integers satisfying ab+bc+ca=1.ab + bc + ca = 1. Prove that (1+a2)(1+b2)(1+c2)(1+ a^2 )(1+ b^2 )(1+ c^2 ) is a perfect square.
Build the set of points - [UKRMO 2009 Grade 9]

Source:

1/23/2011
Build the set of points (x,y)( x, y ) on coordinate plane, that satisfies equality: 1−x2+1−y2=2−x2−y2. \sqrt{1-x^2}+\sqrt{1-y^2}=2-x^2-y^2.
analytic geometryalgebra proposedalgebra
Find all possible values of abc - [UKRMO 2009 Grade 9]

Source:

1/23/2011
Pairwise distinct real numbers a,b,ca, b, c satisfies the equality a+1b=b+1c=c+1a.a +\frac 1b =b + \frac 1c =c+\frac 1a. Find all possible values of abc.abc .
Compare number of prime divisors - [UKRMO 2009 Grade 10]

Source:

1/23/2011
Compare the number of distinct prime divisors of 2002⋅2012⋅...⋅9002200^2 \cdot 201^2 \cdot ... \cdot 900^2 and (2002−1)(2012−1)⋅...⋅(9002−1).(200^2 -1)(201^2 -1)\cdot ... \cdot (900^2 -1) .
Solve the system of equations - [UKRMO 2009 Grade 11]

Source:

1/23/2011
Solve the system of equations {x3=2y3+y−2 y3=2z3+z−2 z3=2x3+x−2\{\begin{array}{cc}x^3=2y^3+y-2\\ \text{ } \\ y^3=2z^3+z-2 \\ \text{ } \\ z^3 = 2x^3 +x -2\end{array}
functionalgebrasystem of equations
System of equations have one soln - [UKRMO 2009 Grade 11]

Source:

1/23/2011
Find all possible real values of aa for which the system of equations {x+y+z=0 xy+yz+azx=0\{\begin{array}{cc}x +y +z=0\\\text{ } \\ xy+yz+azx=0\end{array} has exactly one solution.
algebrasystem of equationsalgebra unsolved