Question 1
Find all integer solutions of
$$x^3-y^3=2xy+8.$$
$$x^3-y^3=2xy+8.$$
Question bank VMO 1985
Exam papers
6 questions
Question 2
Let $M$ consist of all real-valued functions $f$ defined on the integers such that:
(a) For all integers $x,y$, $f(x)f(y)=f(x+y)+f(x-y)$.
(b) $f(0)\ne0$.
Find all $f\in M$ satisfying $f(1)=5/2$.
(a) For all integers $x,y$, $f(x)f(y)=f(x+y)+f(x-y)$.
(b) $f(0)\ne0$.
Find all $f\in M$ satisfying $f(1)=5/2$.
Question 3
A rectangular box of dimensions $a,b,c$ is cut by a plane passing through the intersection of its space diagonals and perpendicular to one of those diagonals. Find the area of the section.
Question 4
Write $\gcd(a,b)$ for the greatest common divisor of natural numbers $a,b$. Prove that, for natural numbers $a,b,m$, a necessary and sufficient condition for the existence of a natural number $n$ such that $(a^n-1)b$ is divisible by $m$ is
$$\gcd(ab,m)=\gcd(b,m).$$
$$\gcd(ab,m)=\gcd(b,m).$$
Question 5
Find all real values of $a$ for which
$$16x^4-ax^3+(2a+17)x^2-ax+16=0$$
has four distinct roots forming a geometric progression.
$$16x^4-ax^3+(2a+17)x^2-ax+16=0$$
has four distinct roots forming a geometric progression.
Question 6
A triangular pyramid $O.ABC$ has base $ABC$ of area $S$. Find its volume, knowing that each altitude drawn respectively from $A,B,C$ is not smaller than the arithmetic mean of the two lateral edges belonging to the face opposite that vertex.