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Difference between revisions of "Mock USAMO by probability1.01 dropped problems"

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== Problem 1 ==
 
== Problem 1 ==
 +
Let <math>n>1</math> be a fixed positive integer, and let <math>a_1,a_2,\ldots,a_n</math> be distinct positive integers. We define <math>S_k=a_1^k+a_2^k+\cdots+a_n^k</math>. Prove that there are no distinct positive integers <math>p,q,r</math> for which <math>S_p,S_q,S_r</math> is a geometric sequence.
  
−
 
+
''Reason: The result is somewhat interesting, but no clever or surprising steps are used to solve the problem.''
 
[[Mock USAMO by probability1.01 dropped problems/Problem 1|Solution]]
 
[[Mock USAMO by probability1.01 dropped problems/Problem 1|Solution]]
  
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at <math>D</math>, <math>E</math>, and <math>F</math> respectively. Let <math>P</math> be a point on <math>AD</math> on the opposite
 
at <math>D</math>, <math>E</math>, and <math>F</math> respectively. Let <math>P</math> be a point on <math>AD</math> on the opposite
 
side of <math>EF</math> from <math>D</math>. If <math>EP</math> and <math>AB</math> meet at <math>M</math>, and <math>FP</math> and <math>AC</math> meet
 
side of <math>EF</math> from <math>D</math>. If <math>EP</math> and <math>AB</math> meet at <math>M</math>, and <math>FP</math> and <math>AC</math> meet
−
at <math>N</math>, prove that <math>MN, EF, and BC</math> concur.
+
at <math>N</math>, prove that <math>MN</math>, <math>EF</math>, and <math>BC</math> concur.
−
''Reason: The whole incircle business seemed rather artificial. Besides, it wasn’t that difficult.''
+
 
 +
''Reason: The whole incircle business seemed rather artificial. Besides, it wasn't that difficult.''
  
 
[[Image:Mock_usamo.png]]
 
[[Image:Mock_usamo.png]]
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== Problem 3 ==
 
== Problem 3 ==
 +
In triangle <math>ABC</math>, let <math>P</math> be an interior point. Suppose the circumcircles of <math>APB</math> and <math>APC</math> intersect <math>BC</math> again at <math>M</math> and <math>N</math> respectively. Prove that <math>PB\cdot AC=PC\cdot AB</math> iff <math>\angle BAM= \angle CAN</math>.
  
 +
''Reason: This is really easy with inversion. It's also quite hard without ideas from inversion (try to find a way!). Too bad, it was a pretty nice problem otherwise.''
  
 
[[Mock USAMO by probability1.01 dropped problems/Problem 3|Solution]]
 
[[Mock USAMO by probability1.01 dropped problems/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
 +
Let <math>ABCD</math> be a cyclic quadrilateral. Prove that <cmath>\frac{|AB-CD|}{AB+CD}+\frac{|AD-BC|}{AD+BC}=\frac{|AC-BD|}{AC+BD}.</cmath>
  
 +
''Reason: This problem was interesting but too simple.''
  
 
[[Mock USAMO by probability1.01 dropped problems/Problem 4|Solution]]
 
[[Mock USAMO by probability1.01 dropped problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
 +
Let a sequence <math>\{a_n\}</math> be defined by <math>a_1=1</math> and <math>a_{n+1}=2a_n+\sqrt{3a_n^2-2}</math>. Prove that all numbers in the sequence are integers.
  
 +
''Reason: This was actually a pretty good problem, but it was vying for the number 1 or number 4 spot with lots of other problems. Plus, bubala made this one.''
  
 
[[Mock USAMO by probability1.01 dropped problems/Problem 5|Solution]]
 
[[Mock USAMO by probability1.01 dropped problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
 +
In the game of ''Laser Gun'', two players move along the <math>x</math>-axis, and a mirror lies along the segment connecting <math>(0,1)</math> and <math>(2006,1)</math>. A number of opaque computer-controlled tiles of width 1 unit can slide back and forth along the mirror. Each player tries to shoot a laser at himself by reflecting it off of the mirror, thus scoring a point. The computer moves its opaque tiles to try to block the shots. If the players each move at twice the speed of each tile, then what is the minimum number of tiles needed to ensure that neither player can ever score a point?
  
 +
''Reason: The wording is way too confusing, and the whole shooting yourself thing didn’t do Laser Gun justice. However, if you do actually understand this problem, it's rather interesting.''
  
 
[[Mock USAMO by probability1.01 dropped problems/Problem 6|Solution]]
 
[[Mock USAMO by probability1.01 dropped problems/Problem 6|Solution]]

Latest revision as of 01:46, 16 May 2009

Problem 1

Let $n>1$ be a fixed positive integer, and let $a_1,a_2,\ldots,a_n$ be distinct positive integers. We define $S_k=a_1^k+a_2^k+\cdots+a_n^k$. Prove that there are no distinct positive integers $p,q,r$ for which $S_p,S_q,S_r$ is a geometric sequence.

Reason: The result is somewhat interesting, but no clever or surprising steps are used to solve the problem. Solution

Problem 2

In triangle $ABC$, $AB \not= AC$, let the incircle touch $BC$, $CA$, and $AB$ at $D$, $E$, and $F$ respectively. Let $P$ be a point on $AD$ on the opposite side of $EF$ from $D$. If $EP$ and $AB$ meet at $M$, and $FP$ and $AC$ meet at $N$, prove that $MN$, $EF$, and $BC$ concur.

Reason: The whole incircle business seemed rather artificial. Besides, it wasn't that difficult.

Mock usamo.png

Solution

Problem 3

In triangle $ABC$, let $P$ be an interior point. Suppose the circumcircles of $APB$ and $APC$ intersect $BC$ again at $M$ and $N$ respectively. Prove that $PB\cdot AC=PC\cdot AB$ iff $\angle BAM= \angle CAN$.

Reason: This is really easy with inversion. It's also quite hard without ideas from inversion (try to find a way!). Too bad, it was a pretty nice problem otherwise.

Solution

Problem 4

Let $ABCD$ be a cyclic quadrilateral. Prove that \[\frac{|AB-CD|}{AB+CD}+\frac{|AD-BC|}{AD+BC}=\frac{|AC-BD|}{AC+BD}.\]

Reason: This problem was interesting but too simple.

Solution

Problem 5

Let a sequence $\{a_n\}$ be defined by $a_1=1$ and $a_{n+1}=2a_n+\sqrt{3a_n^2-2}$. Prove that all numbers in the sequence are integers.

Reason: This was actually a pretty good problem, but it was vying for the number 1 or number 4 spot with lots of other problems. Plus, bubala made this one.

Solution

Problem 6

In the game of Laser Gun, two players move along the $x$-axis, and a mirror lies along the segment connecting $(0,1)$ and $(2006,1)$. A number of opaque computer-controlled tiles of width 1 unit can slide back and forth along the mirror. Each player tries to shoot a laser at himself by reflecting it off of the mirror, thus scoring a point. The computer moves its opaque tiles to try to block the shots. If the players each move at twice the speed of each tile, then what is the minimum number of tiles needed to ensure that neither player can ever score a point?

Reason: The wording is way too confusing, and the whole shooting yourself thing didn’t do Laser Gun justice. However, if you do actually understand this problem, it's rather interesting.

Solution

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