[/quote]
,
satisfying the following equations: 
be positive integers such that 
is a square. Prove that 
is not prime.
be the midpoint of the side 
of triangle 
.
intersects the side 
.
intersect the lines 
and 
in 
and 
respectively. If the midpoint of 
is 
,
.
be then 
-
my question is dose,
My reservations about this is that 
and 
so 
.
is a set and 
is a set, for each 
that 
we can have, for 
we can have,
for some 
for which 
is a perfect square.
,
lie on a line in this order and satisfy 
.
and 
,
is acute, and let 
be its orthocentre. Points 
lie on lines 
and 
,
and 
and 
and 
be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of 
.
chords, one of which has length 2025. Find the minimum possible value of 
is strictly increasing : 
that satisfies 
for all real 
satisfying the equation![\[ 4x^2 - 3y^2 = 1 \]](http://latex.artofproblemsolving.com/d/2/8/d284c2c3a7d7fbe12a7988a4534df16e7adc8947.png)
such that 
.
and then factor the terms with x and then take log base 2 for both sides? I am not sure how to do it, so could you please help?
Y by Exponent11, MathDolphin95, jkim0656, EthanNg6, H1u2g4o4, tofubear, Yiyj1, RainbowJessa, aidan0626, MathNinja9, PikaIsChu12, CheckeredPenguin10, corgi61, wimpykid
I hit my 1234th post! (I think I missed it, I'm kinda late, 
)
But here's a puzzle for you all! Try to create the numbers 1 through 25 using the numbers 1, 2, 3, and 4! You are only allowed to use addition, subtraction, multiplication, division, and parenthesis. If you're post #1, try to make 1. If you're post #2, try to make 2. If you're post #3, try to make 3, and so on. If you're a post after 25, then I guess you can try to make numbers greater than 25 but you can use factorials, square roots, and that stuff. Have fun!
1:
)But here's a puzzle for you all! Try to create the numbers 1 through 25 using the numbers 1, 2, 3, and 4! You are only allowed to use addition, subtraction, multiplication, division, and parenthesis. If you're post #1, try to make 1. If you're post #2, try to make 2. If you're post #3, try to make 3, and so on. If you're a post after 25, then I guess you can try to make numbers greater than 25 but you can use factorials, square roots, and that stuff. Have fun!
1:
