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Difference between revisions of "1989 APMO Problems"

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m (→‎Problem 5: Problem re-written in exact form as found on APMO website. (improved clarity))
 
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== Problem 5 ==
 
== Problem 5 ==
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Determine all functions <math>f</math> from the reals to the reals for which
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<math>f</math> is a strictly increasing real-valued function on reals. It has inverse <math>f^{-1}</math>. Find all possible <math>f</math> such that <math>f(x)+f'(x)=2x</math> for all <math>x</math>.
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<math>(1)</math> <math>f(x)</math> is strictly increasing,
 
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<math>(2)</math> <math>f(x)+g(x)=2x</math> for all real <math>x</math>,
 
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where <math>g(x)</math> is the composition inverse function to <math>f(x)</math>. (Note: <math>f</math> and <math>g</math> are said to be composition inverses if <math>f(g(x))=x</math> and <math>g(f(x))=x</math> for all real <math>x</math>.)
 
  
 
[[1989 APMO Problems/Problem 5|Solution]]
 
[[1989 APMO Problems/Problem 5|Solution]]

Latest revision as of 09:27, 11 April 2016

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

$f$ is a strictly increasing real-valued function on reals. It has inverse $f^{-1}$. Find all possible $f$ such that $f(x)+f'(x)=2x$ for all $x$.

Solution

See Also

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