Question 1
Let $a,b$ be positive integers, with $b\gt 2$. Is $2^a+1$ divisible by $2^b-1$?
Question bank VMO 1983
Exam papers
6 questions
Question 2
(a) Prove that $\sqrt2(\sin t+\cos t)\ge2\sqrt[4]{\sin2t}$ for every $0\le t\le\pi/2$.
(b) Find $y$ such that $1+\dfrac{2\cot2y}{\cot y}\ge\dfrac{\tan2y}{\tan y}$ “for every $y\in(0,\pi)$.”
(b) Find $y$ such that $1+\dfrac{2\cot2y}{\cot y}\ge\dfrac{\tan2y}{\tan y}$ “for every $y\in(0,\pi)$.”
Question 3
Let $ABC$ be a triangle and $M$ any point inside it. Drop perpendiculars $MA_1,MB_1,MC_1$ to the sides $BC,CA,AB$, respectively. Determine the locus of $M$ for which the area of triangle $A_1B_1C_1$ equals a given number $k$. Discuss.
Question 4
Can an integer be represented as either of the following sums?
(a) $\dfrac1{a_1}+\dfrac1{a_2}+\cdots+\dfrac1{a_6}$.
(b) $\dfrac1{a_1}+\dfrac1{a_2}+\cdots+\dfrac1{a_9}$.
Here $a_1,a_2,\ldots,a_9$ are distinct positive integers.
(a) $\dfrac1{a_1}+\dfrac1{a_2}+\cdots+\dfrac1{a_6}$.
(b) $\dfrac1{a_1}+\dfrac1{a_2}+\cdots+\dfrac1{a_9}$.
Here $a_1,a_2,\ldots,a_9$ are distinct positive integers.
Question 5
Compare
$$S_n=\sum_{k=1}^n\frac{k}{(2n-2k+1)(2n-k+1)},\qquad T_n=\sum_{k=1}^n\frac1k.$$
$$S_n=\sum_{k=1}^n\frac{k}{(2n-2k+1)(2n-k+1)},\qquad T_n=\sum_{k=1}^n\frac1k.$$
Question 6
In a tetrahedron the opposite edges are equal in pairs, with lengths $a,b,c$, where $a\lt b\lt c$. A plane cuts the tetrahedron in a quadrilateral. Find the positions of the plane for which the section has least perimeter. Under this condition, find the locus of the centroids of the quadrilateral sections.