Secondary Special Camp 2011: Problem Sets
Posted: Fri Apr 22, 2011 10:29 am
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.
viewtopic.php?f=25&t=913
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.
viewtopic.php?f=25&t=914
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.
viewtopic.php?f=25&t=915
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 $BC$.
viewtopic.php?f=25&t=916
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)
viewtopic.php?f=26&t=917
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 $n^2$.
viewtopic.php?f=26&t=918
Problem 3: Prove that $y^2 = x^3 + 7$ has no integer solutions.
viewtopic.php?f=26&t=919
Problem 4: Determine all the positive integers $n \geq 3$, such that $2^{2000}$ is divisible by \[
1+\binom{n}{1}+\binom{n}{2}+\binom{n}{3}\]
viewtopic.php?f=26&t=920
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.
viewtopic.php?f=25&t=913
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.
viewtopic.php?f=25&t=914
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.
viewtopic.php?f=25&t=915
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 $BC$.
viewtopic.php?f=25&t=916
Secondary Special Camp 2011 Geometry.pdf- (34.88KiB)Downloaded 307 times
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)
viewtopic.php?f=26&t=917
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 $n^2$.
viewtopic.php?f=26&t=918
Problem 3: Prove that $y^2 = x^3 + 7$ has no integer solutions.
viewtopic.php?f=26&t=919
Problem 4: Determine all the positive integers $n \geq 3$, such that $2^{2000}$ is divisible by \[
1+\binom{n}{1}+\binom{n}{2}+\binom{n}{3}\]
viewtopic.php?f=26&t=920
- Secondary Special Camp 2011 NT.pdf
- (41.75KiB)Downloaded 270 times