【A】Define the function
http://data.artofproblemsolving.com/images/latex/2/d/c/2dc523310d2460f724c9174d535ce437dbb7e942.gif by
http://data.artofproblemsolving.com/images/latex/e/6/0/e60b4c9b9e3cbec8db5a078d1618d5585d0b9768.gif The condition
http://data.artofproblemsolving.com/images/latex/6/9/9/69938bc91b5fc5036208964513e57b5c0a5e3f3e.gif
simplifies to
http://data.artofproblemsolving.com/images/latex/d/b/2/db294f7843327a0969f6542786affc1b872d3f2e.gif
and the condition
http://data.artofproblemsolving.com/images/latex/b/8/4/b84fd8e72981daf27ab1c45a4e1e90a02cb8cfe1.gif
simplifies to
http://data.artofproblemsolving.com/images/latex/e/f/8/ef8d362e2e4a73fa8ff8659deb6d4df50ee4edee.gif
Thus, we want to show that there is a unique positive integer
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
so that
http://data.artofproblemsolving.com/images/latex/3/7/9/379b5557ed7cefca53730db8e59367f4162cb40c.gif
We claim that
http://data.artofproblemsolving.com/images/latex/4/a/0/4a0a19218e082a343a1b17e5333409af9d98f0f5.gif
is strictly decreasing over the positive integers: we have
http://data.artofproblemsolving.com/images/latex/4/a/3/4a3082ed08300f2c075cdece961fa5d0a08d70c4.gif since
http://data.artofproblemsolving.com/images/latex/b/2/f/b2f54cb5489a95edf2816226fd49c06674db27bd.gif
for all
http://data.artofproblemsolving.com/images/latex/1/3/f/13fbd79c3d390e5d6585a21e11ff5ec1970cff0c.gif
. Thus,
http://data.artofproblemsolving.com/images/latex/4/a/0/4a0a19218e082a343a1b17e5333409af9d98f0f5.gif
is strictly decreasing, as claimed. Further, since
http://data.artofproblemsolving.com/images/latex/3/6/2/36218153edff84ca52b3b10d692d46a6d904718b.gif
is always an integer, we must have
http://data.artofproblemsolving.com/images/latex/7/6/6/7669494af09d495ff894355cff53e4dcee6e3aea.gif
for all
http://data.artofproblemsolving.com/images/latex/1/3/f/13fbd79c3d390e5d6585a21e11ff5ec1970cff0c.gif
.
Note that
http://data.artofproblemsolving.com/images/latex/c/b/3/cb3254a0ac885edce73cfd8e65dc9210c126dc49.gif
. Then, we have
http://data.artofproblemsolving.com/images/latex/f/3/8/f380871f3918e39575a1fe9571c001d49444b5ad.gif
,
http://data.artofproblemsolving.com/images/latex/0/4/b/04b7602167f78e828498fedae2b88acbcc499fdb.gif
, and by a straightforward induction argument,
http://data.artofproblemsolving.com/images/latex/9/6/d/96db34cbd4e8b6314a1c98d11e9407b7b3c1d399.gif
for all
http://data.artofproblemsolving.com/images/latex/a/3/5/a3591077c9082a64de086ad3230600a3722c0feb.gif
Then
http://data.artofproblemsolving.com/images/latex/a/1/b/a1b443673658c961459c9ee029318ff089ea1b0c.gif
so there is a positive integer
http://data.artofproblemsolving.com/images/latex/6/b/0/6b0d31c0d563223024da45691584643ac78c96e8.gif
so that
http://data.artofproblemsolving.com/images/latex/a/b/f/abff5ea1069a76eec75cf58e033d0e9d2f1b2bea.gif
Let
http://data.artofproblemsolving.com/images/latex/d/e/7/de77ab004b23d17751b294b7e14ed6a4509a249b.gif
be the least positive integer
http://data.artofproblemsolving.com/images/latex/6/b/0/6b0d31c0d563223024da45691584643ac78c96e8.gif
for which
http://data.artofproblemsolving.com/images/latex/4/3/6/4365141fc078ae40db41ec793f7eff395916435f.gif
. Since
http://data.artofproblemsolving.com/images/latex/4/a/0/4a0a19218e082a343a1b17e5333409af9d98f0f5.gif
is strictly decreasing,
http://data.artofproblemsolving.com/images/latex/4/e/1/4e1016767dca23771bb1dc7696fcc732998a2518.gif
if and only if
http://data.artofproblemsolving.com/images/latex/6/f/9/6f99b774e90a7085d1932af3d3215df3ed9dfb8e.gif
. Then, we must have
http://data.artofproblemsolving.com/images/latex/3/3/e/33e6edec8bcbcc12adee2ce1ad33442f243d8311.gif
, so
http://data.artofproblemsolving.com/images/latex/1/e/d/1ed9c5ac6a578400db4c6e6dcd2902c5121e31e6.gif
Thus,
http://data.artofproblemsolving.com/images/latex/5/8/5/585435f29acd1aa4c3301ef99a7c32fcde91e0dd.gif
is an integer satisfying the given conditions. Further, there
cannot be another integer
http://data.artofproblemsolving.com/images/latex/b/5/c/b5cc4e61197fff0813ad65c2d30cf6e661738fd7.gif satisfying the given
conditions. If there was such an
http://data.artofproblemsolving.com/images/latex/2/6/2/26203c06e0ca4d4ce2aa69366021fa7d28db3ceb.gif
, then
http://data.artofproblemsolving.com/images/latex/f/a/9/fa9af3a03e2e3b1addb380a2fe92035ef120737a.gif
, so
http://data.artofproblemsolving.com/images/latex/a/e/0/ae08039e82296408bf8df07126f8d51a841ac18f.gif
. However, then
http://data.artofproblemsolving.com/images/latex/9/e/6/9e62eb64130598756b669e487801054fc3aa91a9.gif
so
http://data.artofproblemsolving.com/images/latex/0/c/0/0c0c784a9f4069606bd471e36f5ba423fda196e1.gif
, a contradiction. Thus there is a unique positive
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
satisfying the given conditions.
http://data.artofproblemsolving.com/images/latex/7/d/1/7d166ea04691d4aa72d47a3183a5e5d1efeec0cf.gif
【B】 Observe that the second inequality for
a given
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
is equivalent to the opposite of the first inequality for
http://data.artofproblemsolving.com/images/latex/d/b/2/db2a943efe93404e43f6ecbec79e0a4fe81b1649.gif
. For
http://data.artofproblemsolving.com/images/latex/9/2/e/92ee913214330816c21ed1490df01843b5cb2075.gif
the first inequality is satisfied, and if the second isn't,
then, by what has been just said, the first inequality is satisfied
for
http://data.artofproblemsolving.com/images/latex/9/a/3/9a3240ad86b2b1d076825686aaf98565beeaac04.gif
. If no
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
as wanted exists, then, by continuing in this manner one sees
that for any
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
, the first inequality must always be satisfied. However,
http://data.artofproblemsolving.com/images/latex/b/7/0/b70de23dea27f1d784c764520af2fa79eee61c9c.gif
which is larger than
http://data.artofproblemsolving.com/images/latex/5/0/c/50cee995a2bc63b380f245be9141953ec97a4436.gif
for
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
large enough. Hence there must exist at least one
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
as wanted. For this
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
we have, in particular (from the second inequality)
http://data.artofproblemsolving.com/images/latex/9/a/7/9a7402f94b8cdd65b764f4f65df84439e61a64b0.gif
Were there an index
http://data.artofproblemsolving.com/images/latex/4/3/7/437860c8131692ea51fe11932271de36af65c093.gif for which the first
inequality held again, that is,
http://data.artofproblemsolving.com/images/latex/b/2/9/b290702ba0f471708f944fa96c7ae9c1926ea37c.gif
then one can apply the previous inequality for the first
http://data.artofproblemsolving.com/images/latex/d/b/2/db2a943efe93404e43f6ecbec79e0a4fe81b1649.gif
terms of the right hand side and notice that each of the
remaining ones is smaller than
http://data.artofproblemsolving.com/images/latex/d/5/c/d5c812e96c2f1f0b670f4c8cadd143cea5ae1cde.gif
to obtain a contradiction.
【C】 We rewrite the condition as
http://data.artofproblemsolving.com/images/latex/e/8/d/e8d41872349078144c43d42a50edd6d52cb9b9d4.gif
Note that for all
http://data.artofproblemsolving.com/images/latex/1/1/8/118e1b944d113cca9330c359103c3a486227c560.gif
http://data.artofproblemsolving.com/images/latex/b/3/7/b3742407fb6402a9eda0a05b1895b61c20a58aee.gif
which implies
http://data.artofproblemsolving.com/images/latex/6/e/3/6e3edcb27e7b849a0a18cdd3f8b5e7ecc3577d24.gif
is monotonically increasing. Obviously
http://data.artofproblemsolving.com/images/latex/b/9/3/b938563f5b4b79c1d55c8b4823f99fa7eefe3a28.gif
is unbounded from above. At
http://data.artofproblemsolving.com/images/latex/1/a/1/1a12c30d6442639dc2387067315b545adbe33725.gif
it equals zero. Now let
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
be the largest integer such that
http://data.artofproblemsolving.com/images/latex/e/d/5/ed51256bfafb799da92437947a099ff58c0be03b.gif
We have that
http://data.artofproblemsolving.com/images/latex/8/5/4/8548d1e14aaef9b4ac26e2cc614fc1b69d70503b.gif
by our hypothesis, so this is our desired
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
which is clearly unique.
http://data.artofproblemsolving.com/images/latex/4/a/4/4a4e9e431da45a27bc880a8a1ca44d8b1b9bc143.gif
【D】 Suppose that for every
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
we have
http://data.artofproblemsolving.com/images/latex/b/d/3/bd38f8b347078ce9db5ee275ac7b4a806fd52b12.gif
for
http://data.artofproblemsolving.com/images/latex/0/1/e/01ea4b6bd17ee603696dd6e63b08b3ba75b78dce.gif
,
http://data.artofproblemsolving.com/images/latex/0/e/a/0ea78ddc5c71aae7688441df156886d7a91b2431.gif
. For
http://data.artofproblemsolving.com/images/latex/2/0/9/2091fb295870e9f79b6d8a10d0f6046b091e6fe5.gif
,
http://data.artofproblemsolving.com/images/latex/e/0/6/e06115b36bf2a3c3e36436ce69dc9e02e9a3a832.gif
so
http://data.artofproblemsolving.com/images/latex/b/4/b/b4b672830dcdd40d4e86db103b1d4b68461db38d.gif
and by induction
http://data.artofproblemsolving.com/images/latex/d/f/2/df2d98e957b97020b4c8136e880a8d70d6364d4a.gif
which it is a contradiction because we have an infinite
sequence of positive integer and bounded !
Let
http://data.artofproblemsolving.com/images/latex/6/5/6/6563c0d3e2385f5b150eaca65199dad1b7e3e0a9.gif
be an integer such that
http://data.artofproblemsolving.com/images/latex/e/6/0/e602925399dfbff3ce97f706c3330126b4437d37.gif
we have
http://data.artofproblemsolving.com/images/latex/3/5/7/35760d439719ecbe322a9d6220461878e4f5a1b8.gif
if
http://data.artofproblemsolving.com/images/latex/b/f/1/bf1e8af3f317606e8ea5e2a67a9632fd43afb3f2.gif
we get the existence, if not then
http://data.artofproblemsolving.com/images/latex/f/a/9/fa94339921f28f05bd72a8d256cb66b0a550ee42.gif
and again if
http://data.artofproblemsolving.com/images/latex/b/6/2/b624737573db3cc1081ba3a1a638d11973ad99c5.gif
we have done and if not we have
http://data.artofproblemsolving.com/images/latex/e/c/c/ecc2ccc6a418d0b2dd4bcde17637dbc185f9ba89.gif
and we continue until reaching the integer
http://data.artofproblemsolving.com/images/latex/6/5/6/6563c0d3e2385f5b150eaca65199dad1b7e3e0a9.gif
for which we have
http://data.artofproblemsolving.com/images/latex/8/c/8/8c8c9b9b3642ad30e69d2b64d1d6aab6c438e4b6.gif
The uniqueness is obvious because if
http://data.artofproblemsolving.com/images/latex/3/b/0/3b0d30f311d25a5bec23f7b7b440a5cde3bf70bc.gif
then
http://data.artofproblemsolving.com/images/latex/a/4/1/a41135ac835f56554503e2d75d20853811ae85cf.gif
and since the sequence is strictly increasing we show easily
that for any
http://data.artofproblemsolving.com/images/latex/7/b/7/7b7feded015f1c08fe0de4c2c295fa2e7b609ad4.gif
we have
http://data.artofproblemsolving.com/images/latex/6/f/0/6f01878a2573eb9b15fc1b030d3796b70824ac30.gif
so the property can't be satisfied for
http://data.artofproblemsolving.com/images/latex/9/0/a/90a0b1caf6aa8f6fed868c5a72559350c7344d87.gif
【E】Key Lemma:If
http://data.artofproblemsolving.com/images/latex/1/0/3/10302ffae4978a299eae5ce526fdf24a5cae1797.gif
is a cumulative
sequence with
http://data.artofproblemsolving.com/images/latex/3/d/8/3d86fb9b40d184e1d87ececdec1872ba684718c7.gif
and
http://data.artofproblemsolving.com/images/latex/9/c/a/9ca7e210f1a757fc7e35a5d9d4c8095e0ec7ae13.gif
for all
http://data.artofproblemsolving.com/images/latex/0/4/2/042dc4512fa3d391c5170cf3aa61e6a638f84342.gif
, then for every positive integer
http://data.artofproblemsolving.com/images/latex/1/3/f/13fbd79c3d390e5d6585a21e11ff5ec1970cff0c.gif
, there is a unique positive integer
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
so that,
http://data.artofproblemsolving.com/images/latex/4/b/e/4be1e57595a4cf68b5cb09fb8395c9401a6f866c.gif
.
Proof:
Almost obvious. But if you still need one, take
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
to be the smallest positive integer so that
http://data.artofproblemsolving.com/images/latex/3/9/6/3965908e7af3a0c374df93e9887fce8a6268147d.gif
. Then, we must have
http://data.artofproblemsolving.com/images/latex/4/3/b/43bcac7947041452457c53034b279125f5b03828.gif
. If not, it would contradict the minimality of
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
.
Now, all we have to do is find an appropriate
http://data.artofproblemsolving.com/images/latex/f/e/1/fe1c3530c376057216084eea9e84d19d8bc8dd36.gif
and
http://data.artofproblemsolving.com/images/latex/c/5/7/c57e5f50b9251810a39299d4b8f96a8a168d43c2.gif
. Take,
http://data.artofproblemsolving.com/images/latex/d/1/6/d168c1d0eb95e45b728db5e4762367c687db7e0e.gif
, which is clearly increasing. We need,
http://data.artofproblemsolving.com/images/latex/5/a/1/5a1933ac91c07062493dfd5baa1c55d9ef8cde9c.gif
and
http://data.artofproblemsolving.com/images/latex/6/a/b/6ab13207a5ba854f383128d433cb6e3b29edca52.gif
. Clearly this want us to consider the sequence
http://data.artofproblemsolving.com/images/latex/1/3/6/136131c83eaba324a0f38c161fc498f1ee1616eb.gif
. It can be written as
http://data.artofproblemsolving.com/images/latex/a/d/b/adbcba58b602bbdb79b717cc8033d164f8eb52cf.gif
where
http://data.artofproblemsolving.com/images/latex/9/4/b/94bb2c84baedace52bd346620ab65541e714c6c3.gif
. Obviously,
http://data.artofproblemsolving.com/images/latex/b/b/8/bb864f872c043ea92a4101a1b37133a52da7c6ce.gif
for all
http://data.artofproblemsolving.com/images/latex/0/4/2/042dc4512fa3d391c5170cf3aa61e6a638f84342.gif
. Thus,
http://data.artofproblemsolving.com/images/latex/2/7/8/278c20e95705afbd9ff550df305b7c84ad7767fe.gif
is a cumulative sequence, and therefore, such a
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
always exists no matter what the choice of
http://data.artofproblemsolving.com/images/latex/4/a/5/4a5997da73aadd118038761e69d01e24586bf958.gif
is as long as it is a positive integer.
Remark: Something similar can not be said if it is replaced by
non-negative integers. Because then the sequence
http://data.artofproblemsolving.com/images/latex/c/5/7/c57e5f50b9251810a39299d4b8f96a8a168d43c2.gif
would not be strictly increasing. Also, the crucial idea seems
kind of trivial to me if someone has ever done "Binary Search" or
something like that in an algorithm course/in CSE,
whatever.【F】 Denote
http://data.artofproblemsolving.com/images/latex/0/9/f/09ffa4f620c801c74ed5e8d7ec86838958b90a56.gif
. Then
http://data.artofproblemsolving.com/images/latex/b/6/2/b62739e9a776fc80b7b711eea8f889d022a44d75.gif
,
http://data.artofproblemsolving.com/images/latex/5/4/d/54d608cbef1529c3089b9281aa6dcb16a2e65f1f.gif
increases and we are searching for
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
such that
http://data.artofproblemsolving.com/images/latex/9/6/1/961d6ac310113282ec6d965c2384ec2883674b64.gif
. It clearly exists and is unique.
Of course, the fact the sequence
http://data.artofproblemsolving.com/images/latex/b/d/5/bd5c118c105a9302cdbd987d761d10129d96d713.gif
has integer terms is irrelevant; all that matters is that the
sequence
http://data.artofproblemsolving.com/images/latex/5/5/c/55c7c43c212ed81a6f2d278646bc91848ee81fe2.gif
is unbounded.
Problem
2. Let
http://data.artofproblemsolving.com/images/latex/d/2/4/d24742f926d9941b8bbb1c164f09acecaf8c8933.gif
be an integer. Consider an
http://data.artofproblemsolving.com/images/latex/9/4/0/940c9f17d1a1df149e433fd817009f4f3c5e8019.gif
chessboard consisting of
http://data.artofproblemsolving.com/images/latex/a/0/a/a0a0ecc4fa470e45995967b80a24950e78459583.gif
unit squares. A configuration of
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
rooks on this board is peaceful if every row and every column
contains exactly one rook. Find the greatest positive integer
http://data.artofproblemsolving.com/images/latex/1/3/f/13fbd79c3d390e5d6585a21e11ff5ec1970cff0c.gif
such that, for each peaceful configuration of
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
rooks, there is a
http://data.artofproblemsolving.com/images/latex/6/9/3/693296daf9e6ef22f6007a932bd8a09b282058c0.gif
square which does not contain a rook on any of its
http://data.artofproblemsolving.com/images/latex/e/8/c/e8c715dc9932381137652eed513cefcebacb8f7e.gif
unit squares.
【A】 will show that
http://data.artofproblemsolving.com/images/latex/2/3/7/2375693b7acb1293303f087ca2cf34b26487c807.gif
.
First we show the upper bound. Take some
http://data.artofproblemsolving.com/images/latex/d/f/2/df2766e9cf7fb30d5859c9f48b8e2438f0a7262f.gif
square for
http://data.artofproblemsolving.com/images/latex/1/7/7/177b0b61fb9d9507836a0300e171947a77b88c78.gif
. The rooks will be in row
http://data.artofproblemsolving.com/images/latex/4/d/c/4dc7c9ec434ed06502767136789763ec11d2c4b7.gif
, column
http://data.artofproblemsolving.com/images/latex/8/d/b/8db15135d4b65e63026f05106dd421194a4cb870.gif
where
http://data.artofproblemsolving.com/images/latex/4/d/c/4dc7c9ec434ed06502767136789763ec11d2c4b7.gif
ranges from
http://data.artofproblemsolving.com/images/latex/b/6/5/b6589fc6ab0dc82cf12099d1c2d40ab994e8410c.gif
to
http://data.artofproblemsolving.com/images/latex/f/9/b/f9b4c2e012c899149774587525bb53a854342352.gif
and the row/column numbers are taken modulo
http://data.artofproblemsolving.com/images/latex/a/5/6/a56085464d79461b6efba083c6fca2f9e59a74e4.gif
. It's hard to visualize but I'm bad at drawing http://www.artofproblemsolving.com/Forum/images/smilies/sad.gif
No
http://data.artofproblemsolving.com/images/latex/7/c/3/7c3e404a0147b4dcc4cb152e14d56b2c1b813e63.gif
subsquares can be found in this arrangement, so
http://data.artofproblemsolving.com/images/latex/9/5/1/951d9c41bc67b19d3dcf246314fad6a75bb3eb53.gif
.
When
http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif
is not a perfect square, taking
http://data.artofproblemsolving.com/images/latex/a/5/6/a56085464d79461b6efba083c6fca2f9e59a74e4.gif
such that
http://data.artofproblemsolving.com/images/latex/6/a/0/6a0bd8a2a5e35366c6b4cbdde95c01395ae62fe4.gif
gives the same upper bound.
Now we show the lower bound. Let
http://data.artofproblemsolving.com/images/latex/7/3/0/730657b4c274d00e9bc696d6635b5874e1671795.gif
Consider some column of
http://data.artofproblemsolving.com/images/latex/5/1/6/516b9783fca517eecbd1d064da2d165310b19759.gif
adjacent
http://data.artofproblemsolving.com/images/latex/4/9/f/49f8f462abedc093d293c86b3ec1ad338f63ec32.gif
subsquares. Only
http://data.artofproblemsolving.com/images/latex/5/1/6/516b9783fca517eecbd1d064da2d165310b19759.gif
rooks can be placed in this column, so there either exists an
empty subsquare or every subsquare has a rook in it. In the former
case, we can find an empty
http://data.artofproblemsolving.com/images/latex/4/9/f/49f8f462abedc093d293c86b3ec1ad338f63ec32.gif
subsquare. In the latter case, since rooks cannot share rows
or columns, one rook must be in the corner of its
http://data.artofproblemsolving.com/images/latex/4/9/f/49f8f462abedc093d293c86b3ec1ad338f63ec32.gif
subsquare, so we can find an empty
http://data.artofproblemsolving.com/images/latex/3/2/8/3289c7d4ee043e54b1180c0a1050a29686766d2c.gif
subsquare. Therefore, a
http://data.artofproblemsolving.com/images/latex/3/2/8/3289c7d4ee043e54b1180c0a1050a29686766d2c.gif
subsquare can be found regardless of configuration and we have
the lower bound as well.【B】Let this maximum be
http://data.artofproblemsolving.com/images/latex/a/5/3/a53ffd447bcf898b6cc05d3a4f5cf05db89f6f08.gif
. It's obvious that
http://data.artofproblemsolving.com/images/latex/a/5/3/a53ffd447bcf898b6cc05d3a4f5cf05db89f6f08.gif
is nondecreasing.
For
http://data.artofproblemsolving.com/images/latex/3/6/c/36cd031c7c4525bd4d13d84e2c4f92332365a3e1.gif
, we have
http://data.artofproblemsolving.com/images/latex/6/1/a/61afa8fbf1ee491c9e42f480e5c3ac452c06bfee.gif
(I omit my counterexample for redundancy's sake). Now we show
that there exists a
http://data.artofproblemsolving.com/images/latex/3/2/8/3289c7d4ee043e54b1180c0a1050a29686766d2c.gif
square in the board. Assume for the sake of contradiction that
in each
http://data.artofproblemsolving.com/images/latex/3/2/8/3289c7d4ee043e54b1180c0a1050a29686766d2c.gif
square there is at least one rook. There are
http://data.artofproblemsolving.com/images/latex/8/8/7/88798dbd8c33866ce4ef45a7b7b4de78a0d448c7.gif
such squares, counting each rook at most
http://data.artofproblemsolving.com/images/latex/e/e/d/eedecb931ce025b2865ada7326580ddef2dfcdbd.gif times. Therefore, there
must be at least
http://data.artofproblemsolving.com/images/latex/8/9/f/89f374b948762a3fd5f71b628c9433d7808462db.gif
rooks, a contradiction. Therefore,
http://data.artofproblemsolving.com/images/latex/7/c/5/7c54db3ff97a4dbe677c7d2a60ecef597d740081.gif
For
http://data.artofproblemsolving.com/images/latex/9/a/e/9ae2cb70365777315ab0415d7d72effa3925e4ae.gif
, we may use the same logic. Since
http://data.artofproblemsolving.com/images/latex/6/b/2/6b227ff02e78a20413f7c8160aca14d7f0296a67.gif , we have
http://data.artofproblemsolving.com/images/latex/0/6/2/06255ca8ff8e05528a6c652e8fd4db6aef1fef31.gif
; we assert that
http://data.artofproblemsolving.com/images/latex/6/7/a/67a70641c5ad7970eff0a49b7c2434746f29ed65.gif
. Again, assume for the sake of contradiction that in each
http://data.artofproblemsolving.com/images/latex/4/9/f/49f8f462abedc093d293c86b3ec1ad338f63ec32.gif
square there is at least one rook. Then there are
http://data.artofproblemsolving.com/images/latex/1/2/6/126f070ed8c34bdf77317b398ce61c2c2aa29a84.gif
disjoint sets of squares and
http://data.artofproblemsolving.com/images/latex/4/4/4/444e6f19eeb1cfaf12908b3dcbd57d63e253ae4f.gif
rooks, so the pigeonhole principle tells us that there exists
an unoccupied square. Therefore,
http://data.artofproblemsolving.com/images/latex/6/7/a/67a70641c5ad7970eff0a49b7c2434746f29ed65.gif
.
Since
http://data.artofproblemsolving.com/images/latex/9/e/3/9e3bb2105d0d97023cffd8b2be32dc5b6313277d.gif
and
http://data.artofproblemsolving.com/images/latex/a/5/3/a53ffd447bcf898b6cc05d3a4f5cf05db89f6f08.gif
is nondecreasing, we have
http://data.artofproblemsolving.com/images/latex/1/4/3/143e7f21c4468d658ac9b418b51a6fa425552d37.gif
for all
http://data.artofproblemsolving.com/images/latex/f/c/2/fc294d4a2fc1b41f0508e343cc7a75cb3e8a7073.gif
. Therefore,
http://data.artofproblemsolving.com/images/latex/6/8/6/686ec6b7be4e10443de3cb6e4eaa1e1d028e7d7b.gif
.
【C】 Let
http://data.artofproblemsolving.com/images/latex/2/e/c/2ecc87653db45f13422e225cc29c38624e620f9f.gif
where
http://data.artofproblemsolving.com/images/latex/0/7/b/07bffd90f1fdc12a402d852a05ff2bb44efa3661.gif
.
We will prove this
http://data.artofproblemsolving.com/images/latex/1/3/f/13fbd79c3d390e5d6585a21e11ff5ec1970cff0c.gif
will be that one, we want.
Look to a rook in the uppermost row.
Select
http://data.artofproblemsolving.com/images/latex/1/3/f/13fbd79c3d390e5d6585a21e11ff5ec1970cff0c.gif
consecutive columns which contain the previous rook in the
first row.
Now we can divide these columns in
http://data.artofproblemsolving.com/images/latex/1/3/f/13fbd79c3d390e5d6585a21e11ff5ec1970cff0c.gif
http://data.artofproblemsolving.com/images/latex/b/7/7/b7792b0bcf384910f5dbfdeabe91b979d923aa55.gif
squares and a
http://data.artofproblemsolving.com/images/latex/e/2/6/e266c049ed10040664f24d67ba3ed80e2cf4e84c.gif
block above.
Because there are only
http://data.artofproblemsolving.com/images/latex/4/1/3/4136a771b69c092ff42aa6115afbc9160e5353c9.gif
rooks yet to place in the
http://data.artofproblemsolving.com/images/latex/1/3/f/13fbd79c3d390e5d6585a21e11ff5ec1970cff0c.gif
http://data.artofproblemsolving.com/images/latex/b/7/7/b7792b0bcf384910f5dbfdeabe91b979d923aa55.gif
squares, there is such a square empty.
Hence our searched
http://data.artofproblemsolving.com/images/latex/1/3/f/13fbd79c3d390e5d6585a21e11ff5ec1970cff0c.gif
is at least this value.
Now we prove there can always be a peaceful configuration with
equality:
We uses the coordinates of the latices from
http://data.artofproblemsolving.com/images/latex/f/5/9/f5978307681424a1310665bf2621e14da106ca14.gif
to
http://data.artofproblemsolving.com/images/latex/d/3/8/d388aa53c12baac90d86da6ac1421241e45d9829.gif
Place a rook in the origin and in each next column we place a rook
http://data.artofproblemsolving.com/images/latex/8/d/2/8d2d1975719162e2210a96c3bd5bebfdf42caa9b.gif
higher than the column before until we can't do it
anymore.
Now, in the next column we place a rook in row with index
http://data.artofproblemsolving.com/images/latex/3/5/6/356a192b7913b04c54574d18c28d46e6395428ab.gif
(http://data.artofproblemsolving.com/images/latex/4/5/f/45f0241f56d9823eb2d24a228d7ffe62c5fdcdc2.gif)
and again going to the right by placing a rook
http://data.artofproblemsolving.com/images/latex/1/3/f/13fbd79c3d390e5d6585a21e11ff5ec1970cff0c.gif
higher each time.
Each time we have to finish as we can't go
http://data.artofproblemsolving.com/images/latex/1/3/f/13fbd79c3d390e5d6585a21e11ff5ec1970cff0c.gif
higher, we place it in the smallest row yet attainable and
continue the same process.
It is easy to see we made a peaceful configuration.
Assume there is a
http://data.artofproblemsolving.com/images/latex/6/7/3/67318a3f19b3b8c509f154d74aa3cf60b111f8fd.gif
square without a rook.
Looking to the full
http://data.artofproblemsolving.com/images/latex/8/d/2/8d2d1975719162e2210a96c3bd5bebfdf42caa9b.gif
columns ( a
http://data.artofproblemsolving.com/images/latex/1/d/d/1dd1dd0edc60892d454a5ccbb8b911a8ffd89e4a.gif
block) of the square, we see except once the difference
between consecutive rooks is
http://data.artofproblemsolving.com/images/latex/8/d/2/8d2d1975719162e2210a96c3bd5bebfdf42caa9b.gif
and hence there isn't a gap of more than
http://data.artofproblemsolving.com/images/latex/8/d/2/8d2d1975719162e2210a96c3bd5bebfdf42caa9b.gif
.
As there is a rook in the uppermost and lowermost
http://data.artofproblemsolving.com/images/latex/6/7/3/67318a3f19b3b8c509f154d74aa3cf60b111f8fd.gif
square, the assumption leads to a contradiction.
【D】
If
http://data.artofproblemsolving.com/images/latex/f/2/7/f279a788564ce1ad584e918e9a99a563ecd57109.gif
, then
http://data.artofproblemsolving.com/images/latex/b/6/4/b6450c209d91e79366a7a7e929000430655dccb8.gif
. In other words,
http://data.artofproblemsolving.com/images/latex/f/2/7/f27fd044505445d3836fa8c8753166922dd5a78a.gif
.
If
http://data.artofproblemsolving.com/images/latex/6/f/b/6fbb8b750a939d799f4fd6c0eb80fc22618b570a.gif
, then without loss of generality there is no rook in right
lower corner. We take lower row, right column and
http://data.artofproblemsolving.com/images/latex/a/5/6/a56085464d79461b6efba083c6fca2f9e59a74e4.gif
squares
http://data.artofproblemsolving.com/images/latex/7/c/3/7c3e404a0147b4dcc4cb152e14d56b2c1b813e63.gif
disjoint from them and from each other. Totally
http://data.artofproblemsolving.com/images/latex/f/f/c/ffcd1e783ed372aa9f53c3d7ed11e185c37cfdd5.gif
sets, by pigeonhole principle one of them does not contain a
rook, and it is a square. If
http://data.artofproblemsolving.com/images/latex/0/9/b/09b315600370cb316d951e338539deb41a2e1def.gif
, remove last row and last column, add a rook if necessary and
reduce the problem to
http://data.artofproblemsolving.com/images/latex/8/1/f/81f998c5713fded10a77db292e4114ec87997543.gif
.
If
http://data.artofproblemsolving.com/images/latex/5/d/2/5d281061132a79b374a8d593a54012a33cdd917b.gif
, then enumerate rows and columns from 0 to
http://data.artofproblemsolving.com/images/latex/f/9/b/f9b4c2e012c899149774587525bb53a854342352.gif
and put rooks with coordinates
http://data.artofproblemsolving.com/images/latex/2/4/d/24d8c9170023590db7c59a67d4ca0ed8e3c657fc.gif
for
http://data.artofproblemsolving.com/images/latex/d/7/6/d76317eae15905ec2b74614ff490b59d81c598f8.gif
. Straightforward check shows that there is no empty
http://data.artofproblemsolving.com/images/latex/7/c/3/7c3e404a0147b4dcc4cb152e14d56b2c1b813e63.gif
square. Example for
http://data.artofproblemsolving.com/images/latex/8/1/f/81f998c5713fded10a77db292e4114ec87997543.gif
without empty
http://data.artofproblemsolving.com/images/latex/7/c/3/7c3e404a0147b4dcc4cb152e14d56b2c1b813e63.gif
square is obtained from the example for
http://data.artofproblemsolving.com/images/latex/7/c/3/7c3e404a0147b4dcc4cb152e14d56b2c1b813e63.gif
as above: remove last row and last column and add rook if
necessary.
【E】For
http://data.artofproblemsolving.com/images/latex/b/6/3/b635f8e9d8fdf43db5b890591712803e078e4862.gif
the largest such square has side length
http://data.artofproblemsolving.com/images/latex/4/2/e/42e0b5e544d4438ccb1d048d1a64e2ae6a322da1.gif
for the following configuration:
Place rooks on the following squares:
http://data.artofproblemsolving.com/images/latex/e/5/1/e51ed1267ed6f082c369175a751d43b97662745f.gif
,
http://data.artofproblemsolving.com/images/latex/4/0/c/40c9087c5fea6e4972af02ea9af125108be7727c.gif
,
http://data.artofproblemsolving.com/images/latex/8/c/5/8c5270acfc9bc78dec68b0926f5e065bc9a74fd1.gif
,
http://data.artofproblemsolving.com/images/latex/9/0/8/908c2f8e0e2aa96db646337f6a92f5762f919565.gif
,
http://data.artofproblemsolving.com/images/latex/9/a/2/9a20be6cea2335a226dffcd4879777a0337196ad.gif
http://data.artofproblemsolving.com/images/latex/4/d/4/4d4821fcaf651de38a1170368491de763ac56944.gif
,
http://data.artofproblemsolving.com/images/latex/2/3/a/23ab2d21ec0123e477898de291c5dd98b3cda20f.gif
,
http://data.artofproblemsolving.com/images/latex/9/0/8/908c2f8e0e2aa96db646337f6a92f5762f919565.gif
,
http://data.artofproblemsolving.com/images/latex/a/7/9/a79414fe259b11521ca7411a54602fba2012143b.gif
http://data.artofproblemsolving.com/images/latex/9/0/8/908c2f8e0e2aa96db646337f6a92f5762f919565.gif
http://data.artofproblemsolving.com/images/latex/9/0/8/908c2f8e0e2aa96db646337f6a92f5762f919565.gif
http://data.artofproblemsolving.com/images/latex/f/1/d/f1d0d4a1eebbc2ad9535e20ef125e14b5be623b2.gif
,
http://data.artofproblemsolving.com/images/latex/0/b/b/0bb1366d1bf30f7fe757314a3af48d8fd2bee14e.gif
,
http://data.artofproblemsolving.com/images/latex/9/0/8/908c2f8e0e2aa96db646337f6a92f5762f919565.gif
http://data.artofproblemsolving.com/images/latex/3/c/a/3ca25a44f0fb484e0954a33ca18c64439741c1f5.gif
Problem 3.
Convex quadrilateral
http://data.artofproblemsolving.com/images/latex/f/b/2/fb2f85c88567f3c8ce9b799c7c54642d0c7b41f6.gif
has
http://data.artofproblemsolving.com/images/latex/1/b/0/1b0392e3fe78cbd53157002b31c24313797ba254.gif
. Point
http://data.artofproblemsolving.com/images/latex/7/c/f/7cf184f4c67ad58283ecb19349720b0cae756829.gif
is the foot of the perpendicular from
http://data.artofproblemsolving.com/images/latex/6/d/c/6dcd4ce23d88e2ee9568ba546c007c63d9131c1b.gif
to
http://data.artofproblemsolving.com/images/latex/0/f/4/0f4d56d1e20778bf2e1052ecb3219509238fb660.gif
. Points
http://data.artofproblemsolving.com/images/latex/0/2/a/02aa629c8b16cd17a44f3a0efec2feed43937642.gif
and
http://data.artofproblemsolving.com/images/latex/c/2/c/c2c53d66948214258a26ca9ca845d7ac0c17f8e7.gif
lie on sides
http://data.artofproblemsolving.com/images/latex/0/6/d/06d945942aa26a61be18c3e22bf19bbca8dd2b5d.gif
and
http://data.artofproblemsolving.com/images/latex/6/d/9/6d95c1847219c633950f8f1ceca9761315abfc19.gif
, respectively, such that
http://data.artofproblemsolving.com/images/latex/7/c/f/7cf184f4c67ad58283ecb19349720b0cae756829.gif
lies inside triangle
http://data.artofproblemsolving.com/images/latex/0/c/b/0cbe2ce8c8c9f0fbbdd9f1c35a60825b52d7f90a.gif
and
http://data.artofproblemsolving.com/images/latex/a/9/6/a967f73c8375be9c5197e1439e75dadf1f7a3135.gif
Prove that line
http://data.artofproblemsolving.com/images/latex/0/f/4/0f4d56d1e20778bf2e1052ecb3219509238fb660.gif
is tangent to the circumcircle of triangle
http://data.artofproblemsolving.com/images/latex/4/3/a/43af71bc25adf3286d1a277400c81c406b814271.gif
.
【A】Is this a request for more detail? Notice
that
http://data.artofproblemsolving.com/images/latex/8/b/a/8ba3491c3d1f407f808868e7a3acbcd9af38d74d.gif
where
http://data.artofproblemsolving.com/images/latex/0/6/2/062b0d1ce40194c489554375471d3125d2bc1aa6.gif
, whence
http://data.artofproblemsolving.com/images/latex/9/a/d/9add0dc6112d9826e53bd29c7ce790a961253617.gif
Moreover,
http://data.artofproblemsolving.com/images/latex/c/b/7/cb7be8a577ccceee58d09427ad489c97f3662a6f.gif
Now, if we plug in the
http://data.artofproblemsolving.com/images/latex/1/1/f/11f6ad8ec52a2984abaafd7c3b516503785c2072.gif
in the right-hand side of the above, we obtain
http://data.artofproblemsolving.com/images/latex/b/f/8/bf84684df4a96764efb77acd0b62d00f640a8a60.gif
Pulling out a factor of
http://data.artofproblemsolving.com/images/latex/c/d/3/cd3d9b3f0f56dbc86f6c3763c5fcd548555345c6.gif
from the rightmost term, we get something that is symmetric in
http://data.artofproblemsolving.com/images/latex/e/9/d/e9d71f5ee7c92d6dc9e92ffdad17b8bd49418f98.gif
and
http://data.artofproblemsolving.com/images/latex/3/c/3/3c363836cf4e16666669a25da280a1865c2d2874.gif
, as required.【B】Problem. Convex quadrilateral
http://data.artofproblemsolving.com/images/latex/f/b/2/fb2f85c88567f3c8ce9b799c7c54642d0c7b41f6.gif
is cyclic. Point
http://data.artofproblemsolving.com/images/latex/5/1/1/511993d3c99719e38a6779073019dacd7178ddb9.gif
is a point inside triangle
http://data.artofproblemsolving.com/images/latex/1/a/5/1a55bbce46efeb4e65a53b9a31a2efffd48379ec.gif
such that
http://data.artofproblemsolving.com/images/latex/c/0/4/c04c2b7378e85dde89a50f0213dd8bc60accddbd.gif
. Points
http://data.artofproblemsolving.com/images/latex/0/2/a/02aa629c8b16cd17a44f3a0efec2feed43937642.gif
and
http://data.artofproblemsolving.com/images/latex/c/2/c/c2c53d66948214258a26ca9ca845d7ac0c17f8e7.gif
lie on sides
http://data.artofproblemsolving.com/images/latex/0/6/d/06d945942aa26a61be18c3e22bf19bbca8dd2b5d.gif
and
http://data.artofproblemsolving.com/images/latex/6/d/9/6d95c1847219c633950f8f1ceca9761315abfc19.gif
, respectively, such that
http://data.artofproblemsolving.com/images/latex/5/1/1/511993d3c99719e38a6779073019dacd7178ddb9.gif
lies inside triangle
http://data.artofproblemsolving.com/images/latex/0/c/b/0cbe2ce8c8c9f0fbbdd9f1c35a60825b52d7f90a.gif
and
http://data.artofproblemsolving.com/images/latex/0/8/8/088ea83a2e58bd93d7d09d29f6fb860b2ef05ca9.gif
. Prove that circumcenter of triangle
http://data.artofproblemsolving.com/images/latex/4/f/c/4fca696988e1671ac6d56b1ee95271001ca0db58.gif
lie on
http://data.artofproblemsolving.com/images/latex/b/c/5/bc5af774c819dc8b91c6b244583727dac105439a.gif
.
Solution (It is from idea
of leader).
Let
http://data.artofproblemsolving.com/images/latex/c/c/4/cc4200da2381358dd8b316a3b2eb7fd765d308af.gif
be reflection of
http://data.artofproblemsolving.com/images/latex/3/2/0/32096c2e0eff33d844ee6d675407ace18289357d.gif
through
http://data.artofproblemsolving.com/images/latex/0/7/8/0789b5b02a99468b86bed774a5bfdc42280543b5.gif
. We have
http://data.artofproblemsolving.com/images/latex/1/4/5/145cc7a3c18716b7a7062afe5c2d39dba7f253d0.gif
and
http://data.artofproblemsolving.com/images/latex/d/f/7/df76812aaa4e140cfe6459c2c8819481722a4026.gif
are cyclic with center
http://data.artofproblemsolving.com/images/latex/8/5/e/85e78aac7ba81d37cf4c8a81d9e6df3cf5b507ea.gif
. We must prove that perpendicular bisector of
http://data.artofproblemsolving.com/images/latex/2/0/6/20659b78c2d6d94a2e5e295072e647ecb0485452.gif
and
http://data.artofproblemsolving.com/images/latex/3/9/a/39a0f31b2a7446e0041c02281ac2fd664aeae8e6.gif
are concurrent. Apply Menelaus theorem for triangle
http://data.artofproblemsolving.com/images/latex/4/3/a/43a7c088e89befe04932bf3df61ea43292edb213.gif
we must prove that
http://data.artofproblemsolving.com/images/latex/0/2/6/0269bb23b5ffad97584d4cd90f19f802f54b2340.gif
or
http://data.artofproblemsolving.com/images/latex/5/7/a/57a2b0555ccf7d48f42768524870cf8899651f77.gif
Let
http://data.artofproblemsolving.com/images/latex/0/3/4/034f1965ccdbdf9e642feeb9858da5096b6d1a9a.gif
cuts perpendicular bisector of
http://data.artofproblemsolving.com/images/latex/3/9/a/39a0f31b2a7446e0041c02281ac2fd664aeae8e6.gif
at
http://data.artofproblemsolving.com/images/latex/c/3/1/c3156e00d3c2588c639e0d3cf6821258b05761c7.gif
and
http://data.artofproblemsolving.com/images/latex/0/3/4/034f1965ccdbdf9e642feeb9858da5096b6d1a9a.gif
cuts
http://data.artofproblemsolving.com/images/latex/b/1/f/b1fb3bec6fdb22e19a94fe4c6c4481ccba2ee9f0.gif
at
http://data.artofproblemsolving.com/images/latex/0/6/5/06576556d1ad802f247cad11ae748be47b70cd9c.gif
. We see
http://data.artofproblemsolving.com/images/latex/4/7/3/473c2c6c9716259178a2565a3ee58c24a9069abe.gif
From this we have
http://data.artofproblemsolving.com/images/latex/d/8/5/d851aefa460e6c99db1025aef516f1f591ea3a1b.gif
is tangent of
http://data.artofproblemsolving.com/images/latex/9/f/4/9f49d7213fefc16e18a7d68ad05d46c1b8514e36.gif
but
http://data.artofproblemsolving.com/images/latex/c/3/1/c3156e00d3c2588c639e0d3cf6821258b05761c7.gif
is circumcenter of triangle
http://data.artofproblemsolving.com/images/latex/1/b/f/1bfe152b965a4a3bcc493fe26b46fdf00509fa83.gif
so
http://data.artofproblemsolving.com/images/latex/c/f/7/cf74217c948d72a76cb96edaeee582babcb140e5.gif
is Apollonious circle for segment
http://data.artofproblemsolving.com/images/latex/0/3/4/034f1965ccdbdf9e642feeb9858da5096b6d1a9a.gif
. We are done.
【C】Let
http://data.artofproblemsolving.com/images/latex/c/d/9/cd9093a2cbe9dac49cd03ca8f14ada3610144821.gif
and
http://data.artofproblemsolving.com/images/latex/3/d/b/3db654b22df4b35347f16117285c1ac03d82d21f.gif
be the centers of the circles. We need to prove that
http://data.artofproblemsolving.com/images/latex/1/4/5/145a3925a5ecef10c71a13322086ccbc997abd9b.gif
Let
http://data.artofproblemsolving.com/images/latex/b/5/1/b51a60734da64be0e618bacbea2865a8a7dcd669.gif
be he midpoint of
http://data.artofproblemsolving.com/images/latex/5/4/9/549368d07edbd45c470fc42eca1a6c826d9ffc77.gif
. Then
http://data.artofproblemsolving.com/images/latex/9/9/3/993358a6f6a763c5d8256eb1e279098f17981432.gif
. Thenquadrilaterals
http://data.artofproblemsolving.com/images/latex/8/a/9/8a91daa5a0aeda27ecce613aebbaeb99d0737b24.gif
and
http://data.artofproblemsolving.com/images/latex/f/0/7/f0722083757eb5fb0db13f3d1c4515cfe14c8122.gif
are inscribed. Since that
http://data.artofproblemsolving.com/images/latex/0/c/5/0c5b92848d536a2dd14d312df9ec6b6504b85d00.gif
. Also
http://data.artofproblemsolving.com/images/latex/4/e/7/4e7a5e6d51535374a60989f39743048a0c0cd776.gif
. As
http://data.artofproblemsolving.com/images/latex/c/8/4/c84eea47c06ae1c06614aee8a13145851c48eb92.gif
it is sufficient to prove that
http://data.artofproblemsolving.com/images/latex/1/6/6/166d57c6a3eeafa249c7029bec4f5afb400757e4.gif
or
http://data.artofproblemsolving.com/images/latex/e/f/6/ef633b04cbbf92eca691c45a987fa94177fb66f5.gif
. Finally
http://data.artofproblemsolving.com/images/latex/d/1/4/d14fa5c324f07ea33b90509b610feb848d7317d8.gif
(http://data.artofproblemsolving.com/images/latex/4/c/e/4cef0d391539bd1c77b3745a00d644c9eecf621a.gif
because
http://data.artofproblemsolving.com/images/latex/e/5/2/e52093914965db7cfddd6b1581744e2039315f26.gif
(http://data.artofproblemsolving.com/images/latex/1/5/6/156dcc9602b169ede5c94f820daf344d65b044f0.gif)
so
http://data.artofproblemsolving.com/images/latex/d/1/3/d131e9e3d4a9489e6b9194a246d0fe13522fd216.gif
where
http://data.artofproblemsolving.com/images/latex/e/0/1/e0184adedf913b076626646d3f52c3b49c39ad6d.gif
is the center of the circumcircle of
http://data.artofproblemsolving.com/images/latex/3/c/0/3c01bdbb26f358bab27f267924aa2c9a03fcfdb8.gif
.
What do you think about problem? As for me it was really
surprisingly to see geometry with angle equalities and without many
circles on the 3rd position. And i hope leaders wrote "http://data.artofproblemsolving.com/images/latex/7/c/f/7cf184f4c67ad58283ecb19349720b0cae756829.gif
lies inside triangle
http://data.artofproblemsolving.com/images/latex/9/9/3/993c8787a6ec77be0cb23f537cb6edcef37de759.gif
" in bold - i waste a lot of time finding what is
wrong
【D】Reflect
http://data.artofproblemsolving.com/images/latex/3/2/0/32096c2e0eff33d844ee6d675407ace18289357d.gif
about
http://data.artofproblemsolving.com/images/latex/a/e/4/ae4f281df5a5d0ff3cad6371f76d5c29b6d953ec.gif
and
http://data.artofproblemsolving.com/images/latex/5/0/c/50c9e8d5fc98727b4bbc93cf5d64a68db647f04f.gif
to get
http://data.artofproblemsolving.com/images/latex/1/7/9/179de509382b817db3c8c2a31a226c1f5aff8081.gif , then
http://data.artofproblemsolving.com/images/latex/9/c/9/9c90aeea05f9732116caea13478493ed9a1af754.gif
and
http://data.artofproblemsolving.com/images/latex/3/3/4/334152739372c4db25ccfcd855db671884794104.gif
are cyclic (this gets rid of the ugly angle condition and
makes the diagram better since now
http://data.artofproblemsolving.com/images/latex/6/d/c/6dcd4ce23d88e2ee9568ba546c007c63d9131c1b.gif
is the circumcenter of
http://data.artofproblemsolving.com/images/latex/b/8/1/b81e7fb1104e8be8a0c18902f437677f7a3e4c07.gif
). Furthermore,
http://data.artofproblemsolving.com/images/latex/5/c/b/5cb22470020bbecea3ede464798e91b130dd1f48.gif
and
http://data.artofproblemsolving.com/images/latex/0/7/d/07d29a5f14fd8b02a9888b08b1f5718a131666cf.gif
.
Now let
http://data.artofproblemsolving.com/images/latex/c/6/3/c63ae6dd4fc9f9dda66970e827d13f7c73fe841c.gif
be the midpoint of
http://data.artofproblemsolving.com/images/latex/0/2/4/0240bc2a44592b8a87c11750a73d170cccfed317.gif
and note that
http://data.artofproblemsolving.com/images/latex/a/1/5/a15abeba2fcbc3517b01bf91a03562b92581fe81.gif is a straight line and
perpendicular to
http://data.artofproblemsolving.com/images/latex/0/2/4/0240bc2a44592b8a87c11750a73d170cccfed317.gif
. Thus
http://data.artofproblemsolving.com/images/latex/9/9/3/99318eea8828621c8f1476e1a5b9e301f795a5fc.gif
. Simple angle chasing gives
http://data.artofproblemsolving.com/images/latex/5/d/6/5d65f96ff2ac3eb770204cf6d5d83db85143b295.gif
, and
http://data.artofproblemsolving.com/images/latex/d/1/7/d17bb082e04f08a5f47e5f92f26b8d0ba2b800de.gif
is the external angle bisector of
http://data.artofproblemsolving.com/images/latex/4/1/e/41e7239d6d5608e744e164b7094bf384d63a4c7b.gif
(same for the other side), useful information for
http://data.artofproblemsolving.com/images/latex/1/c/d/1cdca7c00f36a8baf99f3d025b214b7915fd8b78.gif
.
But now what? There is no way to find the other 2 angles of
http://data.artofproblemsolving.com/images/latex/1/c/d/1cdca7c00f36a8baf99f3d025b214b7915fd8b78.gif
(indeed, SHT). One very useful way to do this is to construct
another triangle where we can find the angles, and show they are
similar using length ratios. So you experiment around for a while,
and find that it is similar to the triangle formed by sticking the
isosceles triangles
http://data.artofproblemsolving.com/images/latex/3/f/7/3f793250c37887982c517b310579623e17877a22.gif
together. Now you are almost done, just need to formalise a
bit.
Construct
http://data.artofproblemsolving.com/images/latex/a/e/4/ae45432b4e79c594da3520e0a25f2169187e2beb.gif
such that
http://data.artofproblemsolving.com/images/latex/e/4/8/e48025dedae6e0455f4b606c36f5668ad0a380a2.gif
(by spiral similarities
http://data.artofproblemsolving.com/images/latex/2/d/1/2d1afa015c41673f05fbccd9bc77e37aa5016295.gif
). Now
http://data.artofproblemsolving.com/images/latex/8/f/7/8f7031202fc7e97c9524c63672f1f03ee09ce8bf.gif
and
http://data.artofproblemsolving.com/images/latex/2/e/7/2e74e6b582827b6b1881a3edd57d7060c38bf5ac.gif
(since
http://data.artofproblemsolving.com/images/latex/8/2/e/82e165118a5fd6886188016611857d877b917bf3.gif
and
http://data.artofproblemsolving.com/images/latex/d/a/c/dac291a67cb761af7501215e36e0cb2a45913435.gif
). So we also have
http://data.artofproblemsolving.com/images/latex/5/e/d/5ed6426bad9f6f856cb3103a3055fa94c79ddc54.gif
. Note that
http://data.artofproblemsolving.com/images/latex/8/8/b/88b3055abd23b88b870d365bb2bf7e9bd9c3c9fa.gif
, and the length ratio
http://data.artofproblemsolving.com/images/latex/8/c/f/8cfd7682b7cdba9eaeea53a4f5353ffa88736351.gif
. Thus
http://data.artofproblemsolving.com/images/latex/4/0/1/401665c998508ffc10c091c63e4ca69f6bff630d.gif
, and the rest is just angle chase.
【E】Here was a sketch of my solution. Not very elegant but quite
direct.
http://cache.artofproblemsolving.com/asyforum/b/e/9/be97582a72f860c67eaa3c4b8f095eb4600820e9.png
First by angle chasing one can show that
http://data.artofproblemsolving.com/images/latex/e/a/e/eaed73b0703abcd3df08f399d0b35688779d42ee.gif
, so the tangent to
http://data.artofproblemsolving.com/images/latex/2/d/7/2d724490b7f925f53a85751cc805ecbbdd133fa8.gif
at
http://data.artofproblemsolving.com/images/latex/c/2/c/c2c53d66948214258a26ca9ca845d7ac0c17f8e7.gif
is perpendicular to
http://data.artofproblemsolving.com/images/latex/6/d/9/6d95c1847219c633950f8f1ceca9761315abfc19.gif
. Thus the circumcenter
http://data.artofproblemsolving.com/images/latex/0/8/a/08a914cde05039694ef0194d9ee79ff9a79dde33.gif
of
http://data.artofproblemsolving.com/images/latex/8/8/4/884997c4033a2aaa13ae7497e4ff6e7d5046623a.gif
lies on
http://data.artofproblemsolving.com/images/latex/6/d/9/6d95c1847219c633950f8f1ceca9761315abfc19.gif
.
Let the perpendicular bisector of
http://data.artofproblemsolving.com/images/latex/a/4/8/a482167f2a1cb7f5dc21104d8f7590e325bccb8c.gif
meet
http://data.artofproblemsolving.com/images/latex/b/d/4/bd4b01946d29a04b84f9d1306289b7687d01b8c2.gif
at
http://data.artofproblemsolving.com/images/latex/5/1/1/511993d3c99719e38a6779073019dacd7178ddb9.gif
now. It suffices to show that
http://data.artofproblemsolving.com/images/latex/4/5/d/45dac411f8e0aaa52149525a53518cdfd6d254b4.gif
is symmetric in
http://data.artofproblemsolving.com/images/latex/c/e/1/ce1b9c42012ca27b7c437a28f105e8c24e5e82fc.gif
and
http://data.artofproblemsolving.com/images/latex/f/1/9/f19a7d58baaa0bca5faa6b0da708ee0594e9e21b.gif
, because then
http://data.artofproblemsolving.com/images/latex/5/1/1/511993d3c99719e38a6779073019dacd7178ddb9.gif
will be the circumcenter of
http://data.artofproblemsolving.com/images/latex/2/1/d/21d225a14443659864b2c64fd2eeedf32b04fe61.gif
. To do this, set
http://data.artofproblemsolving.com/images/latex/0/6/e/06efc93c79050e521270bddb23c063101768f66e.gif
and
http://data.artofproblemsolving.com/images/latex/2/9/b/29bfeaae60a9961789930cff79925204f5a5ff46.gif
. Use the Law of Cosines on
http://data.artofproblemsolving.com/images/latex/3/e/5/3e5d291442f25cbf9005ace1bd0cd4e86da14c8a.gif
and
http://data.artofproblemsolving.com/images/latex/f/b/9/fb9cb4b3d70b5d835fafd8cd2fd5bb847fe7d7a1.gif
, using variables
http://data.artofproblemsolving.com/images/latex/4/d/d/4dddd3d963a9bcc95e1e2c938082c5f91970aa29.gif
and
http://data.artofproblemsolving.com/images/latex/6/5/c/65ca70147bbfbb7a98c1c451a0c45541411b2153.gif
. We get that
http://data.artofproblemsolving.com/images/latex/6/0/0/600e7c279093a11922f1336855bc52b860f4785d.gif
By the Angle Bisector Theorem,
http://data.artofproblemsolving.com/images/latex/2/a/8/2a8c22b032f9c9a1996f75817ceec4a38844e960.gif
, so the rest is a ~15 minute computation
now.【A】
First of all let points
http://data.artofproblemsolving.com/images/latex/c/c/4/cc4200da2381358dd8b316a3b2eb7fd765d308af.gif
be symmetric points of
http://data.artofproblemsolving.com/images/latex/3/2/0/32096c2e0eff33d844ee6d675407ace18289357d.gif
wrt
http://data.artofproblemsolving.com/images/latex/f/c/b/fcba2db25b197842e5c8249ed6cd38e017591751.gif
. Now easily get
http://data.artofproblemsolving.com/images/latex/9/b/1/9b1b0d6ef0b28598967d7d5b9c602474236427f6.gif
cyclic. And let
http://data.artofproblemsolving.com/images/latex/2/9/c/29cdc2f7f9ea5f37004fbd3483b2b37eb6dc546d.gif
be the centers of these circles respectively. Now if you try
to prove by Menelaus theorem that the perp bisectors of
http://data.artofproblemsolving.com/images/latex/2/1/9/219d01092d6fa5b2dfe49031d34045b15339ee6d.gif
intersect on
http://data.artofproblemsolving.com/images/latex/b/d/4/bd4b01946d29a04b84f9d1306289b7687d01b8c2.gif
(which is equivalent to the given problem) you will reduce the
problem to
http://data.artofproblemsolving.com/images/latex/a/1/0/a108f2d7c7823d2374a4a71da18b8d2a21c65df4.gif
. Now this reduces to
http://data.artofproblemsolving.com/images/latex/d/6/a/d6a1dccd5cedd5f45557140659687d2105b7f50e.gif
. Now this can be proven if you prove that
http://data.artofproblemsolving.com/images/latex/e/7/6/e76d1eb7635932229fd1ba3cf2d57e2930acf3fb.gif
is the apollonious circle for segment
http://data.artofproblemsolving.com/images/latex/0/3/4/034f1965ccdbdf9e642feeb9858da5096b6d1a9a.gif
. If you intersect the perp bisector of
http://data.artofproblemsolving.com/images/latex/b/d/4/bd4b01946d29a04b84f9d1306289b7687d01b8c2.gif
with
http://data.artofproblemsolving.com/images/latex/0/3/4/034f1965ccdbdf9e642feeb9858da5096b6d1a9a.gif
this reduces to
http://data.artofproblemsolving.com/images/latex/d/a/d/dad904000d5485059b37e646dac138c57f5e5727.gif
now you can angle chase that
http://data.artofproblemsolving.com/images/latex/6/3/f/63f4bd12ff7eb173528a409c1f54fe6f3d9ccfaa.gif which is
trivial.