IMO 1981

Official IMO 1981 problems  ·  6/6 solved, 1 verified.

Problem 1   solved · 13m46s · Solution →

$\displaystyle P$ is a point inside a given triangle $\displaystyle ABC$. $\displaystyle D, E, F$ are the feet of the perpendiculars from $\displaystyle P$ to the lines $\displaystyle BC, CA, AB$, respectively. Find all $\displaystyle P$ for which

$\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$

is least.

Problem 2   solved · 11m36s · Solution →

Let $\displaystyle 1 \le r \le n$ and consider all subsets of $\displaystyle r$ elements of the set ${ 1, 2, \ldots , n }$. Each of these subsets has a smallest member. Let $\displaystyle F(n,r)$ denote the arithmetic mean of these smallest numbers; prove that

$F(n,r) = \frac{n+1}{r+1}.$

Problem 3   solved · 8m08s · Solution →

Determine the maximum value of $\displaystyle m^2 + n^2$, where $\displaystyle m$ and $\displaystyle n$ are integers satisfying $m, n \in { 1,2, \ldots , 1981 }$ and $\displaystyle ( n^2 - mn - m^2 )^2 = 1$.

Problem 4   solved · 8m32s · Solution →

(a) For which values of $\displaystyle n>2$ is there a set of $\displaystyle n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $\displaystyle n-1$ numbers?

(b) For which values of $\displaystyle n>2$ is there exactly one set having the stated property?

Problem 5   solved · 3m47s · Solution →

Three congruent circles have a common point $\displaystyle O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle and the point $\displaystyle O$ are collinear.

Problem 6   ✓ verified · 7m05s · Solution →

The function $\displaystyle f(x,y)$ satisfies

(1) $\displaystyle f(0,y)=y+1,$

(2) $\displaystyle f(x+1,0)=f(x,1),$

(3) $\displaystyle f(x+1,y+1)=f(x,f(x+1,y)),$

for all non-negative integers $\displaystyle x,y$. Determine $\displaystyle f(4,1981)$.