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 | + | |
+ | ''Reason: The whole incircle business seemed rather artificial. Besides, it wasn't that difficult.'' | ||
+ | |||
[[Image:Mock_usamo.png]] | [[Image:Mock_usamo.png]] | ||
+ | |||
[[Mock USAMO by probability1.01 dropped problems/Problem 2|Solution]] | [[Mock USAMO by probability1.01 dropped problems/Problem 2|Solution]] | ||
== 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.