Difference between revisions of "2007 USA TST Problems"
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Circles <math>\omega_{1}</math> and <math>\omega_{2}</math> intersect at <math>P</math> and <math>Q</math>. <math>AC</math> and <math>BD</math> are chords of <math>\omega_{1}</math> and <math>\omega_{2}</math>, respectively, such that <math>P</math> is on segment <math>AB</math> and on ray <math>CD</math>. Lines <math>AC</math> and <math>BD</math> intersect at <math>X</math>. Let the line through <math>P</math> parallel to <math>AC</math> intersect <math>\omega_{2}</math> again at <math>Y</math>, and let the line through <math>P</math> parallel to <math>BD</math> intersect <math>\omega_{1}</math> again at <math>Z</math>. Prove <math>Q, X, Y, Z</math> are collinear. | Circles <math>\omega_{1}</math> and <math>\omega_{2}</math> intersect at <math>P</math> and <math>Q</math>. <math>AC</math> and <math>BD</math> are chords of <math>\omega_{1}</math> and <math>\omega_{2}</math>, respectively, such that <math>P</math> is on segment <math>AB</math> and on ray <math>CD</math>. Lines <math>AC</math> and <math>BD</math> intersect at <math>X</math>. Let the line through <math>P</math> parallel to <math>AC</math> intersect <math>\omega_{2}</math> again at <math>Y</math>, and let the line through <math>P</math> parallel to <math>BD</math> intersect <math>\omega_{1}</math> again at <math>Z</math>. Prove <math>Q, X, Y, Z</math> are collinear. | ||
+ | [[2007 USA TST Problems/Problem 1|Solution]] | ||
== Problem 2 == | == Problem 2 == | ||
Let <math>a_{1}\le a_{2}\le ...\le a_{n}</math>, <math>b_{1}\le b_{2}\le ... \le b_{n}</math> be two nonincreasing sequences of reals such that | Let <math>a_{1}\le a_{2}\le ...\le a_{n}</math>, <math>b_{1}\le b_{2}\le ... \le b_{n}</math> be two nonincreasing sequences of reals such that | ||
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For any real number <math>m</math>, the number of pairs <math>(i, j)</math> such that <math>a_{i}-a_{j}=m</math> is equal to the number of pairs <math>(k, l)</math> such that <math>b_{k}-b_{l}=m</math>. Prove that <math>a_{i}=b_{i}</math> for <math>i=1, 2, ..., n</math>. | For any real number <math>m</math>, the number of pairs <math>(i, j)</math> such that <math>a_{i}-a_{j}=m</math> is equal to the number of pairs <math>(k, l)</math> such that <math>b_{k}-b_{l}=m</math>. Prove that <math>a_{i}=b_{i}</math> for <math>i=1, 2, ..., n</math>. | ||
+ | [[2007 USA TST Problems/Problem 2|Solution]] | ||
== Problem 3 == | == Problem 3 == | ||
For some <math>\theta \in (0, \frac{\pi}{2})</math>, <math>\displaystyle \cos{\theta}</math> is irrational. If, for some positive integer <math>k</math>, <math>\displaystyle \cos{(k\theta)}</math> and <math>\displaystyle \cos{([k+1]\theta)}</math> are both rational, then show <math>\theta=\frac{\pi}{6}</math>. | For some <math>\theta \in (0, \frac{\pi}{2})</math>, <math>\displaystyle \cos{\theta}</math> is irrational. If, for some positive integer <math>k</math>, <math>\displaystyle \cos{(k\theta)}</math> and <math>\displaystyle \cos{([k+1]\theta)}</math> are both rational, then show <math>\theta=\frac{\pi}{6}</math>. | ||
− | + | [[2007 USA TST Problems/Problem 3|Solution]] | |
== Problem 4 == | == Problem 4 == | ||
Are there two positive integers <math>(a, b)</math> such that, for each positive integer <math>n</math>, <math>b^{n}-n</math> is not divisible by <math>a</math>? | Are there two positive integers <math>(a, b)</math> such that, for each positive integer <math>n</math>, <math>b^{n}-n</math> is not divisible by <math>a</math>? | ||
+ | [[2007 USA TST Problems/Problem 4|Solution]] | ||
== Problem 5 == | == Problem 5 == | ||
Let the tangents at <math>B</math> and <math>C</math> to the circumcircle of <math>\triangle{ABC}</math> meet at <math>T</math>. Let the perpendicular to <math>AT</math> at <math>A</math> meet ray <math>BC</math> at <math>S</math>. Let <math>B_{1}, C_{1}</math> lie on <math>ST</math> such that <math>B_{1}T=C_{1}T=BT</math> and so that <math>T</math> lies between <math>S</math> and <math>B_{1}</math>. Prove that <math>\triangle{AB_{1}C_{1}}\sim \triangle{ABC}</math>. | Let the tangents at <math>B</math> and <math>C</math> to the circumcircle of <math>\triangle{ABC}</math> meet at <math>T</math>. Let the perpendicular to <math>AT</math> at <math>A</math> meet ray <math>BC</math> at <math>S</math>. Let <math>B_{1}, C_{1}</math> lie on <math>ST</math> such that <math>B_{1}T=C_{1}T=BT</math> and so that <math>T</math> lies between <math>S</math> and <math>B_{1}</math>. Prove that <math>\triangle{AB_{1}C_{1}}\sim \triangle{ABC}</math>. | ||
+ | [[2007 USA TST Problems/Problem 5|Solution]] | ||
== Problem 6 == | == Problem 6 == | ||
− | For any polynomial <math>P</math>, let <math>r(2i-1)</math> be the remainder mod <math>1024</math> from 0 to 1023, inclusive, of <math>P(2i-1)</math> for <math>i=1, 2, ..., 512</math>. Call the set <math>\{r(1), r(3), r(5), ..., r(1023)\}</math> the ''remainder sequence'' of <math>P</math>. Call a remaidner sequence '''complete''' if it is a permutation of <math>\{1, 3, 5, ..., 1023\}</math>. Show that the number of complete remainder sequences is at most <math>2^{35}</math>. | + | For any polynomial <math>P</math>, let <math>r(2i-1)</math> be the remainder mod <math>1024</math> from 0 to 1023, inclusive, of <math>P(2i-1)</math> for <math>i=1, 2, ..., 512</math>. Call the set <math>\{r(1),\ r(3),\ r(5),\ ...,\ r(1023)\}</math> the ''remainder sequence'' of <math>P</math>. Call a remaidner sequence '''complete''' if it is a permutation of <math>\{1, 3, 5, ..., 1023\}</math>. Show that the number of complete remainder sequences is at most <math>2^{35}</math>. |
+ | [[2007 USA TST Problems/Problem 6|Solution]] | ||
== See also == | == See also == | ||
*[[TST]] | *[[TST]] |
Latest revision as of 20:59, 24 May 2007
Problem 1
Circles and
intersect at
and
.
and
are chords of
and
, respectively, such that
is on segment
and on ray
. Lines
and
intersect at
. Let the line through
parallel to
intersect
again at
, and let the line through
parallel to
intersect
again at
. Prove
are collinear.
Problem 2
Let ,
be two nonincreasing sequences of reals such that
,
,
,
and
For any real number
, the number of pairs
such that
is equal to the number of pairs
such that
. Prove that
for
.
Problem 3
For some ,
is irrational. If, for some positive integer
,
and
are both rational, then show
.
Problem 4
Are there two positive integers such that, for each positive integer
,
is not divisible by
?
Problem 5
Let the tangents at and
to the circumcircle of
meet at
. Let the perpendicular to
at
meet ray
at
. Let
lie on
such that
and so that
lies between
and
. Prove that
.
Problem 6
For any polynomial , let
be the remainder mod
from 0 to 1023, inclusive, of
for
. Call the set
the remainder sequence of
. Call a remaidner sequence complete if it is a permutation of
. Show that the number of complete remainder sequences is at most
.