AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be real numbers satisfying 
and 
Prove that 
Let be real numbers satisfying 
and 
Prove that 
be a positive integer. Call a rearrangement 
of 
nice if for every 
, we have that 
is not divisible by 
. is odd, prove that there is no nice arrangement of .
is even, find a nice arrangement of .
are non-negative real numbers and 
Prove that
be real numbers satisfying![\[
x_1 + x_2 + \cdots + x_6 = 6,
\]](http://latex.artofproblemsolving.com/6/2/2/622a329112ce9613a5c389b4c9078cd43c1c3f05.png)
![\[
x_1^2 + x_2^2 + \cdots + x_6^2 = 18.
\]](http://latex.artofproblemsolving.com/8/f/a/8fa3f74ba7c3587576116a7bdbb4e9f88d389133.png)
Find the maximum possible value of the product![\[
x_1 x_2 x_3 x_4 x_5 x_6.
\]](http://latex.artofproblemsolving.com/3/b/4/3b48315e2775b4ac3913d7c9b02dbc81daed8d15.png)
+1 w such that, for each positive divisor 
of , there are integers 
and 
such that 
.
. For the triangle that has the minimum area while satisfying the condition, find the lengths of the other two sides.
be a triangle with circumcenter 
and let the angle bisector of 
intersect 
. The point 
is such that 
and 
. Lines 
and 
intersect at
. Show that the circle centered at and passing through 
is tangent to segment.
be reals such that 
.Prove that
Where 
is divisible by 
.(where 
)Prove that the number 
is divisible by 
for which the inequality ![\[ (a^2+3)(b^2+3)(c^2+3)(d^2+3) + k(a-1)(b-1)(c-1)(d-1) \ge 0 \]](http://latex.artofproblemsolving.com/4/d/b/4db42ebabc7309bba9629a9a7125be43abf113f0.png)
holds for all real numbers 
, 
, 
, and . be an acute triangle (
) with circumcircle 
. Point 
is chosen arbitrarily on arc 
, not containing 
. Points and 
lie on lines 
and respectively, such that 
- parallelogram. Point is chosen on arc , not containing , such that 
. Line 
intersects 
at 
, and line 
intersects 
at 
. Line 
intersects , and 
at points , 
and 
respectively.
.
: 
such :
be a general quadrangle which does not form an orthocentric system. Let , 
, , be the Euler-Poncelet point (
), isogonal center (
), midray homothetic center (
), inscribed square axes crosspoint (
) of , respectively.
be the orthocenter of 
, and define 
, 
, 
cyclically.
(
).
(
).
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
lie on the same conic, here denoted by 
.
be the nine-point conic of . and have 4 distinct real intersection points, and let 
, 
, 
be the intersections, other than , of and . with respect to 
passes through and , and show that the center of and the orthocenter of coincide with .
.
for 
means that 
.
. Then, we wish to prove that 
. By AM-GM, 
so it suffices to prove that 
, which is true because 
.
, we just need to prove that 
implies that 
. Expanding gives 
as desired.
, the assertion is true. We will now prove that the assertion is true for .
such that 
, 
, 
don't form the side lengths of a triangle, then 
and 
by AM-GM so 
as desired.
. By AM-GM and our claim, 
as desired.
are equal to , , 
where , are positive reals and 
is nonnegative. By Cauchy-Schwarz, we have
Now
by AM-GM. To reach contradiction it is now sufficient to show 
which is true. We are done.
. Then ![\[t_1 + t_2 + t_3 = \frac25 t_1 + \frac35 t_1 + (t_2 + t_3) \geq \frac25 t_1 + \frac85(t_2 + t_3)\]](http://latex.artofproblemsolving.com/0/a/c/0acd6d3ef414f23102f5794365916ac04db3c1c9.png)
and ![\[\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} \geq \frac{1}{t_1} + \frac{4}{t_2 + t_3}\]](http://latex.artofproblemsolving.com/a/7/2/a7235fbb9869141c281b54ebbd1937a89bad445e.png)
by Cauchy-Schwarz, so we have ![\[(t_1 + t_2 + \dots + t_n)\left(\frac{1}{t_1} + \frac{1}{t_2} + \dots + \frac{1}{t_n}\right)\]](http://latex.artofproblemsolving.com/7/6/6/76643350b294b5322309ee8358dca57f236afcb5.png)
![\[ \geq \left(\frac25 t_1 + \frac85(t_2 + t_3) + t_4 + \dots + t_n\right)\left(\frac{1}{t_1} + \frac{4}{t_2 + t_3} + \frac{1}{t_4} + \dots + \frac{1}{t_n}\right)\]](http://latex.artofproblemsolving.com/5/6/b/56be74ad5093a6bdb0b8c517fedd46f1846faee2.png)
![\[ \overset{\text{C-S}}{\geq} \left(\sqrt{\frac25} + \sqrt{\frac{32}{5}} + (n - 3)\right)^2 = \left(n + \sqrt{10} - 3\right)^2.\]](http://latex.artofproblemsolving.com/7/c/0/7c0c0364f68030796905b430e213165b3fd94381.png)
But 
is strictly increasing, so 
, a contradiction. 
and 
? I originally tried using 
's, which was very close but broke slightly for 
. But if we use arbitrary coefficients and 
, it suffices to maximize the function ![\[f(x) = \sqrt x + 2\sqrt{2 - x},\]](http://latex.artofproblemsolving.com/c/3/7/c37b71145f5b1d1c15cfcfa44cbe556c1213ebbe.png)
which by C-S or similar methods is greatest when 
. This fits perfectly when .
. Assume to the contrary that there exists some tuple 
such that the inequality is satisfied and 
. We show that if 
we can form a similar tuple with 
elements. Of course, just remove 
, the decrease on the left is 
, the decrease on the right is 
by AM-GM, so the inequality is preserved. Now we can just reduce to the case, where we MUST have 
. Since 
are increasing for 
, if the inequality holds true for 
, it must hold for 
. Now plugging this value of gives 
, which is NEVER true by AM-GM, so the original inequality can never hold in the conditions described.
WLOG we can assume is the largest. Thus we need to show 
. If we rewrite as a quadratic function in respect to (considering and as parameters) we have:
We however know:
.WLOG we can assume 
. We know that 
. We will first show that 
. This follows from the fact:
From this it directly follows that 
, implying that 
are the sides of a triangle for 
. We are now left to show that 
from which the desired will follow (because 
are the smallest). For simplicity let 
. FTSOC let 
for some 
. We know that by Cauchy-Schwarz 
. We will now show that 
.
From here it follows that:
![\[\left(t_1 + t_2 + \dots + t_n\right)
\left( \frac{1}{t_1} + \frac{1}{t_2} + \dots + \frac{1}{t_n} \right) = n+\sum_{i, j \in \{1, 2, \dots, n\}, i\neq j} \frac{t_i}{t_j} +\frac{t_j}{t_i}\]](http://latex.artofproblemsolving.com/6/0/3/603a241e4653fd3df03efa30e822979ccf517f55.png)
Thus we get if for any 
, 
and that 
we get a contradiction,
that 
, thus I will
, we get that proving that is equivalent to proving![\[\frac{1}{a}+\frac{1}{b}-\frac{a+b}{c(c+k)}\geq 0\]](http://latex.artofproblemsolving.com/d/6/3/d6347eedbee72e919170ae49d15d63dc6872409c.png)
When and is a positive real. Thus the above is equivalent to:![\[\frac{1}{a}+\frac{1}{b}-\frac{1}{a+b}\geq 0\]](http://latex.artofproblemsolving.com/e/6/e/e6e5ffd6f245a148efe013b7f83b0e5e68252740.png)
![\[\frac{b+a}{ab}-\frac{1}{a+b}\]](http://latex.artofproblemsolving.com/4/9/7/497c7d2bc149121cfb147ceba6006cffeb0b1315.png)
The above is minimised when 
, thus its equivalent to:![\[\frac{2a}{a^2}-\frac{1}{2a}\geq 0\]](http://latex.artofproblemsolving.com/3/d/9/3d9bcb384a9e7159c50d6748733c5ecf1e01c2f4.png)
Which is clearly true.
, and FTSOC let 
. The condition then gives by Cauchy that 
where the summation is taken over all ordered pairs in . Now notice that 
has no integer solutions for 
, so we just need to prove that 
by AMGM. Also, 
by Cauchy so this reduces into 
which is obvious.
that if there exists at least one triple 
that are not the sides of a triangle, then 
For 
we must show that if 
cannot be the sides of a triangle, then 
WLOG, assume that 
then we have that 
Also, because our inequality is homogenous, assume that 
Letting 
it suffices to show that 
But by AM-GM, 
so it suffices to show that 
But, 
by 
applications of AM-GM at the end. Therefore, our base case is proven.
Now, assume that for 
our proposition holds. Consider the sequence 
of positive real numbers. Then if there is at least one triple, say WLOG 
which are not the sides of a triangle then we have by our inductive hypothesis that
Therefore, our induction is complete. QED which are not the sides of a triangle, and 
But by our induction above, we have that 
a contradiction. Therefore, all are the sides of a triangle. QED
such that 
Then, we have that:
Thus, all that is left is to prove that 
Expanding, we have that 
Now, differentiate w.r.t. 
We get 
We claim that 
However, this is clearly true because 
while 
Thus, 
is increasing as 
increases. Thus, we can assume that 
This clearly implies that 
Hence, we are done.
the equality hold for 
or 
or any
,![\[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\]](http://latex.artofproblemsolving.com/1/a/b/1ab410ebc17d9f6c38a3e7ae0f6efa955ffddfd3.png)
Show that 
.
.
,
,
.
are the three sides of some triangle for all 
.
terms is at least 
,
and the inequality will be violated.
.
when 
,
,
and 
verify 
then any three of them are sides of a triangle. I think I've managed to solve this problem and if my solution is correct, then it's a really hard and nice problem.
.
![\[ (t_1+ \cdots + t_n) ( \frac{1}{t_1} + \cdots + \frac{1}{t_n}) = n^2 + \sum_{i<j} (\sqrt{\frac{t_i}{t_j}} - \sqrt{\frac{t_j}{t_i}} )^2 \]](http://latex.artofproblemsolving.com/8/2/e/82e1a00585a2b5adbd6c59119f4bf55c41b04f5e.png)
then![\[{{ (\sqrt{a/b} - \sqrt{b/a})^2 + (\sqrt{a/c} - \sqrt{c/a})^2 =
\frac{ (a-b)^2}{ab} } + \frac{ (a-c)^2}{ac} }
\geq \frac{(a-b)c}{ab} + \frac{(a-c)b}{ac}
= c/b + b/c -(b+c)/a \geq 2 - 1 = 1 \]](http://latex.artofproblemsolving.com/3/c/1/3c1c7545ca6ebf41b70af49f67869215ec2039d5.png)
,
.
.
Thus, we want
if 
Since 
,
which is clearly a contradiction. Thus, for all 
,
and 
are side lengths of a triangle which implies that 
,
and 
are as well. Swapping the indices we get the conclusion for all 
,
,
and 
.
.
and do a second degree equation in 
.
,
so 
which is a contradiction from Cauchy.
Mainly for storage.
.
.
,
and 
.
.
.![$(s+u+v)(\frac{1}{s} + \frac{1}{u} + \frac{1}{v}) \ge (s+u+v)(\frac{4}{s+u} + \frac{1}{v}) = 5 + \frac{4v}{s+u} + \frac{s+u}{v} \ge 5 + 5 \cdot \sqrt[5]{\frac{v^3}{(s+u)^3}} \ge 10$](http://latex.artofproblemsolving.com/3/2/2/322de8dea08b6057f70ea16e1396449a2cc2af43.png)
when 
.
We rephrase the problem as such:
the number 
is written. A table is "good" if all of the numbers in it sum up to a number strictly less than 
.
for all 
.
written term by term:![\[
\begin{array}{c|ccccccc}
& \frac{1}{t_1}
& \frac{1}{t_2}
& \frac{1}{t_3}
& \frac{1}{t_4}
& \frac{1}{t_5}
& \cdots
& \frac{1}{t_n} \\
\hline
t_1 & & & & & & & \\
t_2 & & & & & & & \\
t_3 & & & & & & & \\
t_4 & & & & & & & \\
t_5 & & & & & & & \\
\vdots & & & & & & & \\
t_n & & & & & & & \\
\end{array}
\]](http://latex.artofproblemsolving.com/2/2/c/22c8c239fc3d7d4edd456b3ca64c7d274d2fa718.png)
Let us now start filling in the table with numbers - we immediately notice that the cells along the main diagonal all have the number 
inside of them, so we can immediately fill these ones up:![\[
\begin{array}{c|ccccccc}
& \frac{1}{t_1}
& \frac{1}{t_2}
& \frac{1}{t_3}
& \frac{1}{t_4}
& \frac{1}{t_5}
& \cdots
& \frac{1}{t_n} \\
\hline
t_1 &1 &a &b &c &d & &v \\
t_2 &a' &1 &g &h &i & &w \\
t_3 &b' &g' &1 &l &m & &x \\
t_4 &c' &h' &l' &1 &p & &y \\
t_5 &d' &i' &m' &p' &1 & &z \\
\vdots & & & & & &\ddots & \\
t_n &v' &w' &x' &y' &z' & &1 \\
\end{array}
\]](http://latex.artofproblemsolving.com/b/a/0/ba08e85c9ac36fc7a91ac7b4c59b0528aa25db6f.png)
Now we observe the following
etc.)
for all real 
.
with cell 
)
,
cells (two by two symmetric in regards to the main diagonal) with a sum of the numbers in them at least 
such that 
- assuming that such exists would invariably lead to a contradiction, thus completing the problem.
.
(note that if there exists a triple that does not form a triangle, then the triple 
also does not form one) and we have thus reduced the problem to proving that
Since the expression on the left is scalable, we can assume that 
,
and 
.
From here, we use the useful lemma 
(can be derived - pun intended - from the second derivative of the LHS function and proving that it is reached for 
) to get the inequality we need to prove to be simply
Since this function is increasing for 
(this can be verified by expressing the difference 
)
at 
,
cells with a sum of their numbers at least 
,