Let $ABC$ be a triangle with $BC=a,CA=b,AB=c$. Choose six distinct points $A_1,A_2,B_1,B_2,C_1,C_2$, none equal to $A,B,C$, such that $A_1,A_2$ lie on line $BC$, $B_1,B_2$ on line $CA$, and $C_1,C_2$ on line $AB$. Define real numbers $\alpha,\beta,\gamma$ by
$$\overrightarrow{A_1A_2}=\frac\alpha a\overrightarrow{BC},\quad \overrightarrow{B_1B_2}=\frac\beta b\overrightarrow{CA},\quad \overrightarrow{C_1C_2}=\frac\gamma c\overrightarrow{AB}.$$
Consider the circumcircles of $AB_1C_1,AB_2C_2,BC_1A_1,BC_2A_2,CA_1B_1,CA_2B_2$. Let $d_A,d_B,d_C$ be respectively the radical axes of the pair through $A$, the pair through $B$, and the pair through $C$. Prove that these three axes concur if and only if $a\alpha+b\beta+c\gamma\ne0$.
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Find every integer $k$ for which there are infinitely many integers $n\ge3$ such that $$P_n(x)=x^{n+1}+kx^n-870x^2+1945x+1995$$ is a product of two integer-coefficient polynomials, each of degree at least one.
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In space, $n\ge2$ points are given, no four coplanar, together with $\tfrac12(n^2-3n+4)$ segments whose endpoints are among those points. At least one segment has the following property: after removing it but retaining its endpoints, there are two distinct points which cannot be joined by any polygonal path made of the remaining segments. Find the largest number $k$ for which $k$ of the given segments form a closed polygonal path, each of whose vertices is incident with exactly two segments of that path.
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For each nonnegative integer $n$, let $f(n)$ be the largest nonnegative integer such that $2^{f(n)}$ divides $n+1$. A pair of nonnegative integers $(n,p)$ is called good if $2^{f(n)}>p$. Find every triple of nonnegative integers $(n,p,q)$ such that $(n,p)$, $(p,q)$ and $(n+p+q,n)$ are all good.
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Let $$f(x)=\frac{2x^3-3}{3(x^2-1)}.$$ 1. Prove that there exists a function $g$ continuous on $\mathbb R$ such that $f(g(x))=x$ and $g(x)>x$ for every real $x$. 2. Prove that there exists a real $a>1$ such that the sequence $a_0=a$, $a_{n+1}=f(a_n)$ for every integer $n\ge0$ is periodic with least positive period $1995$. The rational function has its natural domain $\mathbb R\setminus\{-1,1\}$.
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The integer sequence $\{a_n\}_{n=0}^{+\infty}$ is defined by $a_0=1$, $a_1=3$, and
$$a_{n+2}=\begin{cases}a_{n+1}+9a_n,&n\text{ even},\\ 9a_{n+1}+5a_n,&n\text{ odd}.\end{cases}$$
Prove that $a_{1995}^2+a_{1996}^2+a_{1997}^2+a_{1998}^2+a_{1999}^2+a_{2000}^2$ is divisible by $20$, and that no term $a_{2m+1}$ is a perfect square.
Let $ABC$ be a triangle with altitudes $AD,BE,CF$. Points $A',B',C'$ lie on $AD,BE,CF$, respectively, and satisfy
$$\frac{AA'}{AD}=\frac{BB'}{BE}=\frac{CC'}{CF}=k.$$
Determine all $k$ for which triangle $A'B'C'$ is similar to triangle $ABC$.
Let $ABCD$ be a tetrahedron. Let $A'$ be the circumcenter of the face opposite $A$, and define $B',C',D'$ similarly. Let $p_A$ be the plane through $A$ perpendicular to $C'D'$, $p_B$ the plane through $B$ perpendicular to $D'A'$, $p_C$ the plane through $C$ perpendicular to $A'B'$, and $p_D$ the plane through $D$ perpendicular to $B'C'$. Prove that these four planes have a common point. If this common point coincides with the circumcenter of tetrahedron $ABCD$, must the tetrahedron be regular? Explain.
Determine all polynomials $p(x)$ such that, for every $a\gt 1995$, the equation $p(x)=a$ has more than $1995$ real roots, all greater than $1995$ (a root of multiplicity greater than one is counted according to that multiplicity).
How many ways are there to color the vertices of a regular $2n$-gon with $n$ colors, so that each vertex receives one color and each color is used on two nonadjacent vertices? Two colorings are considered the same if one is obtained from the other by a rotation.