Question 1
Without solving $x^3-x+1=0$, find the sum of the eighth powers of its roots.
VMO 1975 — Division A
Question 2
Solve the equation
$$\frac{y^3+m^3}{(y+m)^3}+\frac{y^3+n^3}{(y+n)^3}+\frac{y^3+p^3}{(y+p)^3}-\frac32+\frac32\cdot\frac{y-m}{y+m}\cdot\frac{y-n}{y+n}\cdot\frac{y-p}{y+p}=0.$$
$$\frac{y^3+m^3}{(y+m)^3}+\frac{y^3+n^3}{(y+n)^3}+\frac{y^3+p^3}{(y+p)^3}-\frac32+\frac32\cdot\frac{y-m}{y+m}\cdot\frac{y-n}{y+n}\cdot\frac{y-p}{y+p}=0.$$
Question 3
Let $ABCD$ be a tetrahedron with $BA\perp AC$ and $DB\perp(BAC)$. Let $O$ be the midpoint of $AB$, and drop $OK\perp DC$. Prove that a necessary and sufficient condition for
$$\frac{V_{KOAC}}{V_{KOBD}}=\frac{AC}{BD}$$
is $2AC\cdot BD=AB^2$.
$$\frac{V_{KOAC}}{V_{KOBD}}=\frac{AC}{BD}$$
is $2AC\cdot BD=AB^2$.
Question 4
Given the arithmetic progression $-1,18,37,\ldots$, find the term of the progression that is written using only the digit 5.
Question 5
Prove that the sum of the local maximum and local minimum values of
$$y=\frac{\cot^3 x}{\cot3x}$$
on $(0,\pi/2)$ is rational.
$$y=\frac{\cot^3 x}{\cot3x}$$
on $(0,\pi/2)$ is rational.
Question 6
In space, let $\Delta$ be a fixed line and $A$ a fixed point outside $\Delta$. A variable line $d$ rotates about $A$. Let $MN$ be the common perpendicular of $d$ and $\Delta$, with $M\in d$ and $N\in\Delta$. Find the locus of $M$ and the locus of the midpoint of $MN$.