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{\centerline {\bf THE 1992 ASIAN PACIFIC MATHEMATICAL OLYMPIAD}}
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{\it Time allowed: 4 hours}
{\it NO calculators are to be used.}
{\it Each question is worth seven points.}
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{\bf Question 1}
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A triangle with sides $a$, $b$, and $c$ is given. Denote by $s$ the
semiperimeter, that is $s = (a + b + c)/2$. Construct a triangle with sides $s
- a$, $s - b$, and $s - c$. This process is repeated until a triangle can no
longer be constructed with the side lengths given.
For which original triangles can this process be repeated indefinitely?
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{\bf Question 2}
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In a circle $C$ with centre $O$ and radius $r$, let $C_1$, $C_2$ be two circles
with centres $O_1$, $O_2$ and radii $r_1$, $r_2$ respectively, so that each
circle $C_i$ is internally tangent to $C$ at $A_i$ and so that $C_1$, $C_2$ are
externally tangent to each other at $A$.
Prove that the three lines $OA$, $O_1 A_2$, and $O_2 A_1$ are concurrent.
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{\bf Question 3}
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Let $n$ be an integer such that $n > 3$. Suppose that we choose three numbers
from the set $\{1, 2, \ldots, n\}$. Using each of these three numbers only
once and using addition, multiplication, and parenthesis, let us form all
possible combinations.
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(a) Show that if we choose all three numbers greater than $n/2$, then the
values of these combinations are all distinct.
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(b) Let $p$ be a prime number such that $p \leq \sqrt{n}$. Show that the
number of ways of choosing three numbers so that the smallest one is $p$ and
the values of the combinations are not all distinct is precisely the number of
positive divisors of $p - 1$.
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{\bf Question 4}
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Determine all pairs $(h,s)$ of positive integers with the following property:
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If one draws $h$ horizontal lines and another $s$ lines which satisfy
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(i) they are not horizontal,
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(ii) no two of them are parallel,
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(iii) no three of the $h + s$ lines are concurrent,
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then the number of regions formed by these $h + s$ lines is 1992.
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{\bf Question 5}
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Find a sequence of maximal length consisting of non-zero integers in which the
sum of any seven consecutive terms is positive and that of any eleven
consecutive terms is negative.
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