if
,
,
?
,
,
in terms of 
.
.
to z because we already know the value of 
.
(in the equation 
)
.
.
and cross multiply.
.
.
be real numbers that satisfy![\[
p^2+q^2=r^2+s^2
\]](http://latex.artofproblemsolving.com/5/1/b/51b19eb831481a9181fee897d22e7f71015425c3.png)
![\[
p^2+s^2-ps=q^2+r^2+qr
\]](http://latex.artofproblemsolving.com/3/7/c/37cc3145b42eb86b7019f13f1014ab0908753d28.png)
Hence find the value of 
![$f \in C([0,1];[0,1])$](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?
,
is 
.
with a radius of 
.![\begin{align*}
(a+bi)\cdot(a-bi) +[(a+bi)+(a-bi)] =a^2+b^2+2a.
\end{align*}](http://latex.artofproblemsolving.com/4/0/9/409b0aed187f4f2cbc8e3f204f8b5fac9bcb44b0.png)
and 
.
with radius 
red balls and 
of the remaining balls from bag without replacement, and then setting them aside. A draw is called a match if the two balls have the same color. Compute the expected number of draws until either a match occurs or the bag is empty.
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} ] } $](http://latex.artofproblemsolving.com/a/9/f/a9f640a2ba0f2c58a1a7085e843e82ae41c72951.png)
is continous on 
![$ ] \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 
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 
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 
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
,
,
,
and 
could be chosen to satisfy 
and 
.
be a set of 
be the set of numbers that occur as averages of two distinct
,
Y by TLH, Adventure10
Let ABCD be a rectangle with 
. The diagonals 
and
intersect at 
. A line passing through and perpendicular to 
meets 
and 
at 
and 
respectively. Let 
be a point on side 
such that 
. Show that 
.
. The diagonals 
and
intersect at 
. A line passing through and perpendicular to 
meets 
and 
at 
and 
respectively. Let 
be a point on side 
such that 
. Show that 
.This post has been edited 1 time. Last edited by parmenides51, Jan 31, 2020, 4:43 AM
cyclic implies 
cyclic, done!
