Secondary Special Camp 2011
Exam 1: Geometry
Problem 1: Let $I$ be the incenter of triangle $ABC$. $O_1$ a circle passing through $B$ and tangent to the line $C I$ at $I$ and $O_2$ a circle passing through $C$ and tangent to the line $BI$ at $I$. Prove that $O_1,O_2$ and the circumcircle of $ABC$ pass through a single point.
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Problem 2:The incircle of triangle ABC touches the sides $BC,CA,AB$ at $A',B',C'$ respectively. Let the midpoint of the arc $AB$ of the circumcircle (not containing $C$) be $C''$, and define $A''$ and $B''$ similarly. Prove that the lines $A'A'',B'B'', C'C''$ are concurrent.
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Problem 3: Let $ABC$ be a triangle. Extend the side $BC$ past $C$, and let $D$ be the point on the extension such that $CD = AC$. Let $P$ be the second intersection of the circumcircle of $ACD$ with the circle with diameter $BC$. Let $BP$ and $AC$ meet at $E$, and let $CP$ and $AB$ meet at $F$. Prove that $D,E,F$ are collinear.
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Problem 4: Let $ABC$ be an acute triangle and $D, E, F$ the feet of its altitudes from $A, B, C$, respectively. The line through $D$ parallel to $EF$ meets line $AC$ and line $AB$ at $Q$ and $R$, respectively. Let $P$ be the intersection of line $BC$ and line $EF$.
Prove that the circumcircle of $PQR$ passes through the midpoint of $BC$.
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Exam 2: Number Theory
Problem 1: (a) Prove that $x^2+y^2+z^2=2007^{2011}$ has no integer solution. (2 points)
(b) Find all positive integers $d$ such that $d$ divides both $n^2+1$ and $(n + 1)^2 + 1$ for some integer $n$. (5 points)
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Problem 2: Let $n$ be a positive integer. Prove that the number of ordered pairs $(a, b)$ of relatively prime positive divisors of $n$ is equal to the number of divisors of $n^2$.
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Problem 3: Prove that $y^2 = x^3 + 7$ has no integer solutions.
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Problem 4: Determine all the positive integers $n \geq 3$, such that $2^{2000}$ is divisible by \[
1+\binom{n}{1}+\binom{n}{2}+\binom{n}{3}\]
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Secondary Special Camp 2011: Problem Sets
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Re: Secondary Special Camp 2011: Problem Sets
I have written all the problems and solutions of number theory here.
- Attachments
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- Solution to number theory problems..pdf
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One one thing is neutral in the universe, that is $0$.