AoPS Academy
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
be a right angled triangle with 
.Let 
be the midpoint of 
, and 
be an arbitrary point on 
. The reflection of 
over 
intersects lines 
and at 
and 
, respectively. The circumcircles of 
and intersect again at 
. Prove that the center of the circumcircle of 
lies on 
.
define 
if 
or 
, and 
where 
are all prime factors of . Find all polynomials 
with nonnegative integer coefficients such that 
divides 
for every nonnegative integer .
satisfy the equation 
. Prove that there exist positive integers 
such that
![$f \in C([0,1];[0,1])$](http://latex.artofproblemsolving.com/4/7/5/475518b7db0b787dd90b93f602c7b912c663fba3.png)
bijective, 
and ![$(a_k) \in [0,1]^\mathbb N$](http://latex.artofproblemsolving.com/a/d/8/ad8a80d847c60ee8e916a1f9b08b7b03ec17c814.png)
with 
converge.
converge?
such that![\[
x f(2f(y) - x) + y^2 f(2x - f(y)) = \frac{f(x)^2}{x} + f(y f(y))
\]](http://latex.artofproblemsolving.com/f/9/e/f9e1ccf18bc744433e70a733aa8a2424befc8497.png)
with 
.
be real numbers. Define 
and 
.
and hence find the limit.
![$ f(x)= \frac {x - \sqrt{2}}{ [x \sqrt{2} ] } $](http://latex.artofproblemsolving.com/a/9/f/a9f640a2ba0f2c58a1a7085e843e82ae41c72951.png)
is continous on 
on 
on ![$ ] \frac {1}{ \sqrt{2} } ; 2 \sqrt{2} [ $](http://latex.artofproblemsolving.com/d/0/4/d04f852d2494e9cd545a1ff20462e7283595c9ce.png)
is differentiable and 
Show that for some 
![$f:[0,1]\to\mathbb R$](http://latex.artofproblemsolving.com/e/a/0/ea02b990406104357c11912a8bf57bea7eeb519d.png)
is differentiable with 
If ![$|f'(x)|\le f(x)\;\forall\;x\in[0,1],$](http://latex.artofproblemsolving.com/b/9/d/b9d6088657cd85b37190b98abac1d94a18bb5916.png)
then show that 
be the unit circle in the complex plane. Let 
be the map given by 
We define 
and 
for 
The smallest positive integer such that 
is called period of 
Determine the total number of points 
of period 
denote the set of natural numbers, and let 
be nine distinct tuples in 
Show that there are 
distinct elements in the set 
whose product is a perfect cube.
and let 
be positive integers such that 
Prove that 
and determine when equality holds.
, be normed spaces with a Banach space. Suppose 
is a family of continuous linear maps. Prove that the set![\[
X = \left\{ x \in E \mid \sup_{n\geq 1}|| A_n(x)||< +\infty \right\}
\]](http://latex.artofproblemsolving.com/e/c/d/ecd2cd83e90604ce128e8efb5e5b0748ad354ee3.png)
is either of first category in or is equal to the whole space .
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
, there exists some constant 
, for any 
, among the sequence 
and 
could be chosen to satisfy 
and 
.
be a set of distinct real numbers. Let 
be the set of numbers that occur as averages of two distinct. For a given 
, what is the smallest possible number of elements in ?
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that Prove that
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that Prove that Prove that
Equality holds iff 
or 
and any cyclic permutations.
be non-negative real numbers such that Prove that
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that Prove that![\[\sum a^2+8\sum ab\ge 3\sum a\sum\sqrt{ab}.\]](http://latex.artofproblemsolving.com/2/b/a/2bac168124825dbc152f3085a0d3723fa34caf47.png)
Changing 
it becomes a fourth degree inequality, so is linear in 
. Thus it remains to check only the cases 
and 
which is easy.
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
be positive real variables with 
Prove that
Equality holds iff 
be non-negative real variables with 
Prove thatEquality holds iff or where ![\[\frac{12a(a+2b)}{4ab+bc+ca} \geqslant \frac{32a^3+3(33b+56c)a^2+3(26b^2+102bc+13c^2)a-4(4b+c)(b-2c)^2}{11[ab(a+b)+bc(b+c)+ca(c+a)]+51abc}.\]](http://latex.artofproblemsolving.com/a/b/0/ab0f41dc6c9cf3e143885b3e7e7079048d3841c3.png)
be three non-negative real numbers with Prove that
When does equality hold?
be three non-negative real numbers with Prove that
When does equality hold?
be non-negative real variables with Prove that
Equality holds iff or 
where 
and 
.
,
and 
be non-negative numbers such that 
.
and set 
The inequality becomes
ưhere
which ends the proofs 