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a + bc is perfect square - - 1997 Romania NMO VII p2

Source:

8/13/2024

Let

a \ne 0

be a natural number. Prove that

a

is a perfect square if and only if for every

b \in N^*

there exists

c \in N^*

such that

a + bc

is a perfect square.

number theory

Nice but seems difficult

Source: Romania 1997

3/31/2006

I found this inequality in "Topics in Inequalities" (I 85) For all positive reals

x,y,z

with

xyz=1

prove:

\frac{x^9+y^9}{x^6+x^3y^3+y^6}+\frac{y^9+z^9}{y^6+y^3z^3+z^6}+\frac{z^9+x^9}{z^6+z^3x^3+x^6}\geq 2

inequalitiesinequalities proposed

very very beautiful

Source: Romania 1997

9/10/2005

Suppse that

n\geq 3

be an inetegr number and

x\in \mathbb{R}

s.t.

x,x^2

and

x^n

have a same decimal representation. Prove that

x

is an integer number.

number theory proposednumber theory

Integral inequlity

Source: Romanian national olympiad

2/20/2006

Prove that:

\int_{-1}^1f^2(x)dx\ge \frac 1 2 (\int_{-1}^1f(x)dx)^2 +\frac 3 2(\int_{-1}^1xf(x)dx)^2

Please give a proof without using even and odd functions. (the oficial proof uses those and seems to be un-natural) :D

calculusintegrationLaTeXfunctionreal analysisreal analysis unsolved

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