VN-TST 1998

Let $f(x)$ be a real function such that for each positive real $c$ there exist a polynomial $P(x)$ (maybe dependent on $c$ ) such that $| f(x) - P(x)| \leq c \cdot x^{1998}$ for all real $x$ . Prove that $f$ is a real polynomial.

In the plane we are given the circles $\Gamma$ and $\Delta$ tangent to each other and $\Gamma$ contains $\Delta$ . The radius of $\Gamma$ is $R$ and of $\Delta$ is $\frac{R}{2}$ . Prove that for each positive integer $n \geq 3$ , the equation: $$(p(1) - p(n))^2 = (n-1)^2 \cdot (2 \cdot (p(1) + p(n)) - (n-1)^2 - 8)$$ is the necessary and sufficient condition for $n$ to exist $n$ distinct circles $\Upsilon_1, \Upsilon_2, \ldots, \Upsilon_n$ such that all these circles are tangent to $\Gamma$ and $\Delta$ and $\Upsilon_i$ is tangent to $\Upsilon_{i+1}$ , and $\Upsilon_1$ has radius $\frac{R}{p(1)}$ and $\Upsilon_n$ has radius $\frac{R}{p(n)}$ .

Let $p(1), p(2), \ldots, p(k)$ be all primes smaller than $m$ , prove that

$$\sum^{k}_{i=1} \frac{1}{p(i)} + \frac{1}{p(i)^2} > ln(ln(m)).$$

Find all integer polynomials $P(x)$ , the highest coefficent is 1 such that: there exist infinitely irrational numbers $a$ such that $p(a)$ is a positive integer.

Let $d$ be a positive divisor of $5 + 1998^{1998}$ . Prove that $d = 2 \cdot x^2 + 2 \cdot x \cdot y + 3 \cdot y^2$ , where $x, y$ are integers if and only if $d$ is congruent to 3 or 7 $\pmod{20}$ .

In a conference there are $n \geq 10$ people. It is known that:

I. Each person knows at least $\left[\frac{n+2}{3}\right]$ other people. II. For each pair of person $A$ and $B$ who don't know each other, there exist some people $A(1), A(2), \ldots, A(k)$ such that $A$ knows $A(1)$ , $A(i)$ knows $A(i+1)$ and $A(k)$ knows $B$ . III. There doesn't exist a Hamilton path.

Prove that: We can divide those people into 2 groups: $A$ group has a Hamilton cycle, and the other contains of people who don't know each other.