Let $f$ be defined on $\mathbb R$. Suppose that for every real $c>0$ there is a polynomial $P_c$ with real coefficients such that
$$|f(x)-P_c(x)|\le cx^{1998}\qquad(x\in\mathbb R).$$
Prove that $f$ is a polynomial with real coefficients.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
In the plane a circle $(C)$ of radius $R$ contains and is internally tangent to a circle $(C')$ of radius $R/2$. Let $H$ be the family of circles inside $(C)$, outside $(C')$, and tangent to both. For each integer $n\ge3$ and positive real numbers $p_1,p_n$, prove that
$$(p_1-p_n)^2=(n-1)^2\bigl(2(p_1+p_n)-(n-1)^2-8\bigr)$$
is necessary and sufficient for there to be $n$ distinct circles $(C_1),\ldots,(C_n)$ in $H$ such that $(C_i)$ is externally tangent to $(C_{i-1})$ and $(C_{i+1})$ for $i=2,\ldots,n-1$, with radii $R/p_1$ and $R/p_n$ for $(C_1)$ and $(C_n)$ respectively.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
Let $m>3$ be an integer, and let $p_1,\ldots,p_k$ be all primes not exceeding $m$. Prove that
$$\sum_{i=1}^k\left(\frac1{p_i}+\frac1{p_i^2}\right)>\ln(\ln m).$$
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
Find all polynomials $P(x)$ with integer coefficients and leading coefficient 1 such that there are infinitely many irrational real numbers $\alpha$ for which $P(\alpha)$ is a positive integer.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
Let $d$ be a positive divisor of $5+1998^{1998}$. Prove that $d$ can be written as $d=2x^2+2xy+3y^2$ with integers $x,y$ if and only if its remainder upon division by 20 is 3 or 7.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
A conference has $n\ge10$ participants. Assume that:
1. Each person knows at least $\left\lfloor\frac{n+2}{3}\right\rfloor$ other participants.
2. Any two people $A,B$ who do not know each other are connected indirectly: there are $k\ge1$ people $A_1,\ldots,A_k$ such that $A$ knows $A_1$, $A_i$ knows $A_{i+1}$ for $i=1,\ldots,k-1$, and $A_k$ knows $B$.
3. It is impossible to place all $n$ people in a row so that every adjacent pair know each other.
Prove that the participants can be split into two groups: the first can sit around a round table so that every adjacent pair know each other, and in the second no two people know each other. (The square brackets printed in the source denote the integer part.)
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
Define the sequence $\{x_n\}$ by $x_1=a\ge1$ and
$$x_{n+1}=1+\ln\frac{x_n(x_n^2+3)}{1+3x_n^2}.$$
Prove that the sequence converges and determine its limit.
Let $O$ be the center of the circumsphere of tetrahedron $ABCD$. Let $A',B',C',D'$ lie on that sphere so that $AA',BB',CC',DD'$ are diameters. Let $A''$ be the centroid of triangle $BCD$, and define $B'',C'',D''$ analogously. Prove that the lines $A'A'',B'B'',C'C'',D'D''$ are concurrent. Suppose they meet at $X$. Prove that the line through $X$ and the midpoint of $AB$ is perpendicular to $CD$.
The sequence $a_0,a_1,a_2,\ldots$ is defined by $a_0=20$, $a_1=100$, and
$$a_{n+2}=4a_{n+1}+5a_n+20.$$
Find the smallest positive integer $m$ such that $a_m-a_0,a_{m+1}-a_1,a_{m+2}-a_2,\ldots$ are all divisible by $1998$.
Does there exist an infinite sequence of real numbers $\{x_n\}_{n=1}^{+\infty}$ such that $|x_n|\le0.666$ and
$$|x_m-x_n|\ge\frac1{n^2+n+m^2+m}$$
for all distinct positive integers $m,n$?
Find all positive integers $n$ for which there exists a polynomial $p(x)$ with real coefficients satisfying
$$p\left(x^{1998}+\frac1{x^{1998}}\right)=x^n+\frac1{x^n}.$$