[/quote]
,
,
is chosen on the side 
.
and 
,
and 
are drawn, respectively. The point 
is the circumcenter of 
.
and 
lies on the line 
.
be a point in the plane of triangle 
such that the segments 
,
,
are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to 
is acute.
and 
be two distinct rays not lying on the same line, and let 
be a circle with center 
that is tangent to ray 
and ray 
.
be a point on segment 
.
be the intersection of lines 
and 
be the intersection of line 
is tangent to 
lie on the plane such that 
is a parallelogram. 
is the orthocenter of 
intersects 
at 
and 
intersects 
at 
The circle with diameter 
intersects the circumcircle of 
again at 
Prove that: The circumcircle of 
passes through the reflection point of 
,
denote the incenter and 
.
perpendicular to 
intersects the tangent from 
.
and 
.
.
such that for any positive integer number 
and any arrangement of the numbers from 1 to 
on a circle, there exists a pair of adjacent numbers whose ratio (when dividing the larger number by the smaller one) is less than 
from the nonnegative integers to the integers satisfying 
and ![\[ f(mn) = f(m) + f(n) + f(m)f(n) \]](http://latex.artofproblemsolving.com/e/f/f/eff10a32cffaec742403ebfc46ff09bc8ef7b7c5.png)
for all nonnegative integers 
and 
.
![\[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]](http://latex.artofproblemsolving.com/e/7/f/e7fb8c20a43535b5aaed5ab254f1ef043263c62b.png)
true?
be the binary representation of 
.
,
,
,
equals 
.
such that![\[
4k^3 + 4k + 1
\]](http://latex.artofproblemsolving.com/f/3/5/f355e456cd47ecfdb13597d52fec3d848b67c594.png)
numbers, none of which appears more than 
times. Prove that they can be divided into 
groups of 
be a cyclic quadrilateral. 
be the projections of 
,
is a kite.
.
and 
,
,
.
such moves the blackboard will contain a single number. What is its maximum possible value? (Reminder that 
is the maximum integer not exceeding 
.
Y by Davi-8191, Kezer, rkm0959, Tawan, brokendiamond, HolyMath, megarnie, Adventure10, Mango247, lian_the_noob12
Find the smallest constant 
for which the following statement holds: among any five positive real numbers 
(not necessarily distinct), one can always choose distinct subscripts 
such that
![\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]](//latex.artofproblemsolving.com/a/e/e/aeeab8e846634c7922e9312ef139739ad3fbe6eb.png)
for which the following statement holds: among any five positive real numbers 
(not necessarily distinct), one can always choose distinct subscripts 
such that![\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]](http://latex.artofproblemsolving.com/a/e/e/aeeab8e846634c7922e9312ef139739ad3fbe6eb.png)
This post has been edited 1 time. Last edited by mathwizard888, Jul 19, 2017, 6:09 PM
Reason: match official wording
Reason: match official wording
.
,
,
,
for arbitrarily small 
,![\[ a/c, \quad b/d, \quad c/e, \quad a/d, \quad b/e \]](http://latex.artofproblemsolving.com/9/d/8/9d81467c30cd5d70d2d33fceabfd4623c77a9199.png)
and arrange them in a pentagon. These fractions are less than 
;
,
(check this). Thus given five distinct numbers 
.
such that![\[ \left| \frac{p}{q} - \frac {r}{s} \right| \le C. \]](http://latex.artofproblemsolving.com/4/c/2/4c246b97bb9cfac7f04529ea86208cc386f5e744.png)
.
with an arbitrarily large 
.
to eventually prove 
.
are both larger or both smaller than 
,
.
.
but a repeated use of the Lemma on those numbers give us 
.![\[ \left| \frac{a_4}{a_5} -\frac{a_1}{a_3} \right| \leq \frac12, \]](http://latex.artofproblemsolving.com/1/b/d/1bd8c035dabd8cccb8566a986b7b0ab983e5e847.png)
a contradiction.
can be done in the same way, just reverse the inequality signs. Thus, we are done.
; For 
consider the 5 fraction 
. Since they are in ![$(0,1]$](http://latex.artofproblemsolving.com/1/0/1/101678300d1aa11009468da8419aaf2c57b795bd.png)
so atleast 3 of them are in same side of 
is get easy when one notes the fractions at that order and notes that any 3 of them contains 4 elements of the given 5 tupple and thinks about PHP which is trivial ! 
works. For a construction, consider 
;
.
,
.
and 
must not lie on same side of 
.
lie on same side of 
,
where 
.
less than 
.
,
,
,
if 
and 
if 
.
,
,
,
. 
.
with an arbitrarily large 
denote the 
numbers. We claim the answer is 
with 
gives that 
,
such that for any 
distinct reals 
out of these five numbers, we have 
Note that 
,
If 
,
which is not according to our assumption that 
.
In a similar fashion, if 
,
contrary to the previous statement. Else 
which is again a contradiction to what we had discovered earlier. Thus, no such sequence exists, as desired. 
,
.
.
is greater or equal to 
.
and 
,
.
.
and 
.
and we are done.
In this way, all adjacent fractions have distinct subscripts.
.
,
.
.
,
then 
.
,
,
.
,
.
,
.
,
.
,
,
will lie on the same intervals between ![$(0,0.5]$](http://latex.artofproblemsolving.com/e/f/f/eff9da40e5de1e637511d74ff57b478c35177b5d.png)
or ![$(0.5,1]$](http://latex.artofproblemsolving.com/8/2/9/8290718d4222b60f4c9a1853d0da4b25a0ee5e25.png)
.
ways to choose a pair of numbers. We have to put each pair into one of the intervals. Let a pair be 
and let the sets of pairs in the given intervals be 
.![$\frac{a_1}{a_2} \in (0,0.5] \Rightarrow (a_1,a_2) \in A$](http://latex.artofproblemsolving.com/b/d/a/bda8a71582cb5918ffffc11ab064314657a13954.png)
.
.
in neither 
and 
,
is the best constant. We can't do any better because of the counterexample 1, 1, 1, 2, 
and consider only ratios 
with 
,
)
)
and 
cannot be both small, so at least one of them is large. But then either both 
and 
or 
and 
are both large, with which we are done.
.
,![\[a=\varepsilon,b=1,c=2-\varepsilon,d=2,e=2+\varepsilon,\]](http://latex.artofproblemsolving.com/d/e/2/de27bba1bc27890f0dbbdf8b8a2c76c55e303b03.png)
Then, all fractions get arbitrarily close to half-integers, so the minimum difference gets arbitrarily close to 
all of which are disjoint from the adjacent fraction. Since 5 is odd, there exists some series of 
,
,
for arbitrarily large 
.
such that 
for 
.
are between 
and 
.
.
.
and 
must be greater than 
.
and the 
)
,
,
for arbitrarily small 
,
.
we clearly also have 
so that 
.
which is a contradiction. 
is the smallest constant possible.
where 
or 
, then 
.
means distance between 
,
,
contains at least two elements lying on interval ![$(0, \frac{1}{2}]$](http://latex.artofproblemsolving.com/e/e/f/eef922650e59b11d60cd72d1876da39e59c29c8a.png)
or ![$[\frac{1}{2},1]$](http://latex.artofproblemsolving.com/1/e/5/1e5a8da9d6ee3dc023fc95487947dbec9acc2400.png)
.
,
by WLOG. Since 
,
with arbitrary 
we have 
To see that value 
suffices assume WLOG 
and consider five fractions 
in a cycle. They all don't exceed 
Two of these are adjacent in a cycle, and the absolute difference between them doesn't exceed 
,
,
or 
,
,
,
.
.![\[\frac{a_1}{a_2}, \frac{a_3}{a_4},\frac{a_1}{a_5},\frac{a_2}{a_3},\frac{a_4}{a_5}\]](http://latex.artofproblemsolving.com/3/a/1/3a1ba3e146f1274e0317bab41b76a447584206cd.png)
note that all of them are less than 
and have all different subscripts then we are done. Fortunately, we have by pigeonhole principle that there will be two cyclically adjacent fractions in our above list that will lie on the same side of 
where 
,
.
for some arbitrarily large positive integer ![\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| > \frac{1}{2}. \]](http://latex.artofproblemsolving.com/6/e/8/6e87299ebd6447c9f8d40c1bf4103981e47f8e24.png)
Therefore if 
and 
,
are on different sides of 
are all on a different side of 
are all on the same side of 
.
and 
have an element in common, they must both be on the same side of ![\[\left | \frac{a_2}{a_3} - \frac{a_4}{a_5} \right| \le \frac{1}{2} ,\]](http://latex.artofproblemsolving.com/7/3/2/732cd2f45773119e2104f5e5791965e548e9e3de.png)
absurd. Therefore 
![\[ \frac{a}{c} - \frac{b}{d}, \frac{b}{c} - \frac{a}{d}, \frac{c}{b} - \frac{a}{d} \]](http://latex.artofproblemsolving.com/d/1/2/d129a5ba678d09502d456db6bbf260ad97ce35cb.png)
Since 
,
is minimal.![\[ \frac{a}{c} - \frac{b}{d}, \frac{a}{c} - \frac{b}{e}, \frac{a}{d} - \frac{b}{e}, \frac{a}{d} - \frac{c}{e}, \frac{b}{d} - \frac{c}{e} \]](http://latex.artofproblemsolving.com/2/c/c/2ccdc815c6368192ff06fdc0d7a17372d1816e96.png)
Arrange these values in a circle, 
.
and 
.
and taking 
shows that 
Just Consider 
where 
and 
atleast 
of the fractions lie in same intervals. Among any 
with 
ensures that 
.
chosen from the 
where 
are distinct and chosen from 
implies that 
which basically proves the above claim.
,
. Similarly as 
and 
,we get 
and 
.
which is a clear contradiction to 
for some permutation 
of 
.
,
and 
.
from the four variable case. Namely, these five differences arise from the 
ways of choosing numbers from 
.
, 
, 
, 
and 
.
.
.
.
be positive real numbers, and let 
denote 
,
is a permutation of 
.
,
.
and 
such that 
for all permutations 
of 
and 
or 
and 
.
is positive. Suppose 
.
,
,
.
,
,
.
.
,
.
.
,
,
is monotonically increasing on 
.
.![\[\left |\frac{a_1}{a_2} - \frac{a_3}{a_4} \right| \ge \left |\frac{a_1}{a_3} - \frac{a_2}{a_4} \right| .\]](http://latex.artofproblemsolving.com/2/d/2/2d209267466fd1e89739610a7ed146116db3eb2d.png)
Multiply both sides by 
,![\[ \left |\frac{a_3}{a_2} \right| \ge \left |\frac{a_1a_4-a_2a_3}{a_1a_4-a_2a_3} \right| = 1,\]](http://latex.artofproblemsolving.com/f/0/3/f03450dc0d3348998a0b77850013802f364e7d28.png)
which is obviously true. Our steps are reversible, so this proves the claim.
.![\[\left |\frac{a_1}{a_4} - \frac{a_2}{a_3} \right| \ge \left |\frac{a_1}{a_3} - \frac{a_2}{a_4} \right| .\]](http://latex.artofproblemsolving.com/4/f/1/4f18fb590f6282c4aa5d2937b92fa8ea72dc54e4.png)
Multiply both sides by 
to obtain 
.
,
,
,
,
is the smallest possible value of 
.![\[C\ge \min \left\{ |d(1,3) - d(2,4)|, |d(1,3) - d(2,5)|, |d(1,4) - d(2,5)|, |d(1,4) - d(3,5)|, |d(2,4) - d(3,5)|\right\},\]](http://latex.artofproblemsolving.com/3/5/a/35afbfc9646e9f45b436c932b1175d7c81e28980.png)
so it suffices to show that the right hand side (which we will denote 
for some very small 
,
,
.
approaches 
,
all approach 
.
.
.![$(0, \tfrac{1}{2}]$](http://latex.artofproblemsolving.com/6/c/b/6cbadd7ca8a156157e6b5482e4ed8067aff98b88.png)
,![$(\tfrac{1}{2}, 1]$](http://latex.artofproblemsolving.com/3/1/f/31f8960811146d17d28ce793810a83762149ea04.png)
.
.
,
