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Romanian Math Olympiad 1997 - 11th grade
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Problem \# 1

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Consider p a positive integer and a square
n x n matrix $ A = (a_{ij}) $ with positive entries. Prove that for
any permutation $\phi$ of \{1, 2, ... n\} there exists a function f: \{1, 2, ...
n\} $\longrightarrow$ \{0,1\} such that the matrix $B = (b_{ij})$ has the
determinant not divisible by p where B is obtained by replacing the
elements $a_{i\phi(i)}$ in the matrix A with $a_{i\phi(i)} + f(i)$.

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Problem \# 2
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Consider A an invertible square matrix
with integer odd entries with odd dimension. Prove that it is not possible
that all the minors of the elements of an arbitrary row to have the same
absolute value.

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Problem \# 3
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Take F the set of differentiable real functions
f such that f(x) $\geq$ f(x + sin x) for all real numbers x. Prove that
F contains non-constant functions and if f is in F then f' is zero on an
infinite subset of R.

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Problem \# 4
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If f and g are two continuous bijective
real functions such that f(g$^{-1}$(x)) + g(f$^{-1}$(x))
= 2x for all real numbers x and there is a real number a such that f(a)
= g(a) then f = g.

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\it 12th grade\rm
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Problem \# 1
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Take a complex non-rational number a such
that the set $A$ = \{m + na $\mid m$ and $n$ are integers\} is a ring with
unity
having exactly four invertible elements. Prove that $A$ is isomorphic with
$Z[i]$.

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Problem \# 2
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If f: [-1 , 1] $\longrightarrow R$ is a continous 
real function prove that
$2\cdot(\int_{-1}^1{f^2(x)}\,dx)\geq(\int_{-1}^1{f(x)}\,dx)^2
+ 3(\int_{-1}^1{xf(x)}\,dx)^2$. When does the equality occur?

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Problem \# 3
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If $K$ is a finite field and $f$ is an irreducible
polynomial in $K[X]$ of degree n then the polynomial $g - 1$ is divisible
by $f$ where $g$ is the product of all non-constant polynomials in $K[X]$
of degree at most n-1.

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Problem \# 4

Consider $f_n$ a sequence of non-negative
functions defined on the closed interval [0, 1] such that $f_0$
is continuous and $f_{n+1}(x) = \int_0^x { 1 \over {1+f_n(t) } }\,dt$.
Prove that the sequence $f_n$ converges
pointwisely to a function $f$ and find that function.


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First Test for IMO - Suceava - March 1997

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Problem \# 1
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There are given a line d in the plane and
three circles of centers A, B, C all tangent to d and any two of them are
externally tangent. Prove that ABC has an obtuse angle and find the maximum
value of that angle.

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Problem \# 2
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Find the number of all systems of 9 positive
distinct integers such that any positive integer less of equal to 500 can
be written as sum of some distinct numbers of the considered system.

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Problem \# 3
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Consider a set $M$ of $n$ points in the plane
$(n \geq 3)$ and a function $f: M \longrightarrow R$ such that
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a) not all points of $M$ lie on a circle
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b) every 3 points of $M$ are not on the same line
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c) for every circle $C$ passing through at
least three points of $M$, $\sum_{P\in M\cap C} {f(P)} = 0$
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Prove that $f$ is the null function.
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Problem \# 4
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Let ABC be a triangle inscribed in the
circle O and D an arbitrary point on the side
BC of ABC. Consider the circles K and L
tangent to O and to (AD), K also being tangent
to the segment (BD) and L to the segment (DC).
Prove that K and L
are tangent iff AD bisects the angle BAC.

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