[/quote]
,
,
,
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
,
is an integer. Prove that there is a positive integer 
such that 
for all 
.
ornaments in 
different colours.
grid of real numbers. Determine the greatest possible number of cells 
in the grid such that the entry in 
,
has side lengths 
inside 
,
,
and denote them as 
,
,
.
?
.
be a point on side 
,
be a point on the extension of 
such that 
.
be the circumcenter of 
,
satisfying 
.
be a prime and suppose 
is prime. Prove that 
is prim
the orthocenter of an acute triangle 
.
,
,
,
respectively such that ![\[ OD + DH = OE + EH = OF + FH\]](http://latex.artofproblemsolving.com/a/8/7/a874af4cb802b9b341b055140a08e096aaf2112e.png)
and the lines 
,
,
are concurrent.
with 
;
,
,
.
very closely approximates 
,
is miniscule; then 
.![\[ (7m-2025)(7n-2025)=2025^2=3^8\cdot5^4, \]](http://latex.artofproblemsolving.com/6/e/1/6e1e8b8a2aa29a0c82efa08a9fe03aa1b4ca0b55.png)
and let 
,
.
,
.
and similarly for 
;
.
with 
,
,
.
,
,![\[ a\equiv5\pmod7\iff d-c\equiv1\pmod6, \]](http://latex.artofproblemsolving.com/c/2/f/c2fab77dab5397af66d7bc9d8a0c0d3cf00b8f80.png)
which gives the 
solutions 
.
and 
.
be the side length of 
and 
.
,
and![\[ \cos(30^\circ-\theta)=\frac{\sqrt3}{2}\cos\theta+\frac 12\sin\theta=\frac{20}{x}. \]](http://latex.artofproblemsolving.com/9/6/0/9609011bccd9bf73415675171e644dca5dcde8c8.png)
Substituting and solving gives![\begin{align*} & \frac{\sqrt3}{2}\cdot\frac{25}{x}+\frac 12\cdot\frac{\sqrt{x^2-625}}{x}=\frac{20}{x} \\
\implies & 25\sqrt3+\sqrt{x^2-625}=40 \\
\implies & x^2=(40-25\sqrt3)^2+625=100(41-20\sqrt3) \\
\implies & [APB]=\frac{x^2\sqrt3}{4}=25(41\sqrt3-60)\approx\boxed{275.35}. \end{align*}](http://latex.artofproblemsolving.com/4/d/0/4d051f434928dd977c2232e728f4688e1281aba0.png)
Y by nitride, MC_ADe
sprint p27 wrote:
What is the greatest value of 
such that 
divides 
.
such that 
divides 
.sprint p28 wrote:
Find the first positive digit after the decimal point in the rational number 
sprint p29 wrote:
How many order pairs of positive integers 
and such that 
and such that 
and finally, what you've all been waiting for:
target p8 wrote:
There exists two positive reals 
and 
such that 
and 
form an equilateral triangle. Find the area of this triangle.
and 
such that 
and 
form an equilateral triangle. Find the area of this triangle.apparently sprint p20 = target p8 in difficulty
and 
,
not 
because 
so the expansion of 
becomes 
Simplifying yields 
(which is positive) instead of 
(which is negative). However 
so nothing really changes
