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

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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. | 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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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</math>, <math>EF</math>, and <math>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 | + | |

+ | ''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 02:46, 16 May 2009

## Problem 1

Let be a fixed positive integer, and let be distinct positive integers. We define . Prove that there are no distinct positive integers for which 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 , , let the incircle touch , , and at , , and respectively. Let be a point on on the opposite side of from . If and meet at , and and meet at , prove that , , and concur.

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

## Problem 3

In triangle , let be an interior point. Suppose the circumcircles of and intersect again at and respectively. Prove that iff .

*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.*

## Problem 4

Let be a cyclic quadrilateral. Prove that

*Reason: This problem was interesting but too simple.*

## Problem 5

Let a sequence be defined by and . 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.*

## Problem 6

In the game of *Laser Gun*, two players move along the -axis, and a mirror lies along the segment connecting and . 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.*