1989 APMO Problems

Problem 1

Let $x_1, x_2, x_3, \dots , x_n$ be positive real numbers, and let

$S=x_1+x_2+x_3+\cdots +x_n$ .

Prove that

$(1+x_1)(1+x_2)(1+x_3)\cdots (1+x_n)\leq 1+S+\dfrac{S^2}{2!}+\dfrac{S^3}{3!}+\cdots +\dfrac{S^n}{n!}$ .

Solution

Problem 2

Prove that the equation

$6(6a^2+3b^2+c^2)=5n^2$

has no solutions in integers except $a=b=c=n=0$ .

Solution

Problem 3

Let $A_1,A_2,A_3$ be three points in the plane, and for convenience, let $A_4=A_1$ , $A_5=A_2$ . For $n=1, 2,$ and $3$ , suppose that $B_n$ is the midpoint of $A_nA_{n+1}$ , and suppose that $C_n$ is the midpoint of $A_nB_n$ . Suppose that $A_nC_{n+1}$ and $B_nA_{n+2}$ meet at $D_n$ , and that $A_nB_{n+1}$ and $C_nA_{n+2}$ meet at $E_n$ . Calculate the ratio of the area of triangle $D_1D_2D_3$ to the area of triangle $E_1E_2E_3$ .

Solution

Problem 4

Let $S$ be a set consisting of $m$ pairs $(a,b)$ of positive integers with the property that $1\leq a<b\leq n$ . Show that there are at least

$4m\cdot \dfrac{\left(m-\dfrac{n^2}{4}\right)}{3n}$

triples $(a,b,c)$ such that $(a,b)$ , $(a,c)$ , and $(b,c)$ belong to $S$ .

Solution

Problem 5

Determine all functions $f$ from the reals to the reals for which

$(1)$ $f(x)$ is strictly increasing,

$(2)$ $f(x)+g(x)=2x$ for all real $x$ ,

where $g(x)$ is the composition inverse function to $f(x)$ . (Note: $f$ and $g$ are said to be composition inverses if $f(g(x))=x$ and $g(f(x))=x$ for all real $x$ .)

Solution

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1989 APMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

See Also