AoPS Academy
and 
be positive integers such that 
divides 
. Prove that 
.
where 
are nine distinct integers. Prove that there exists an integer 
such that for all integers 
the number 
is divisible by a prime number greater than 20.
be the lengths of the sides of a triangle. Prove that![\[\frac{1}{(a-4b)^2}+\frac{1}{(b-4c)^2}+\frac{1}{(c-4a)^2} \geqslant \frac{1}{ab+bc+ca}.\]](http://latex.artofproblemsolving.com/7/4/b/74b7aa5621bc52977d9acd448d74293dc27a8e00.png)
of points in the plane is balanced if, for any two different points 
and 
in , there is a point 
in such that 
. We say that is centre-free if for any three different points , and in , there is no points 
in such that 
.
, there exists a balanced set consisting of 
points. for which there exists a balanced centre-free set consisting of points.
be a triangle, 
is incenter of triangle . Draw 
perpendicular to 
at 
and 
perpendicular to 
at 
, 
. Prove that: Area of triangle 
be the solid square region with vertices at![\[
(0, 0), \left(\frac{1}{2}, \frac{1}{2}\right), (1, 0), \left(\frac{1}{2}, -\frac{1}{2}\right).
\]](http://latex.artofproblemsolving.com/a/9/8/a9898f9237f07eaef5d17a9ac689d80ca631863f.png)
consists of two copies of 
: one copy which is rotated 
counterclockwise around the origin and scaled by a factor of 
, and another copy which is also rotated counterclockwise around the origin and scaled by a factor of , and then translated by 
. and 
below.
be the limiting region of the sequence 
. can be written as 
for relatively prime positive integers and . Find 
.
shows that after each species divides and then each amoeba cell preys on exactly one bacterial cell, it turns out kill bacteria. Determine the ratio of the number of amoeba to the number of bacteria on the first day. is a positive integer. Let 
.
cm. The first nine balls are placed on the table so that cm and all the balls are arranged following the pattern of the arrangement of the balls made by Budi, calculate the height of the center of the topmost ball is measured from the table surface in the arrangement of the balls done by Bambang. whose sides are 
cm, 
cm, and 
cm. Find the maximum possible area of the rectangle can be made in the triangle .
people waiting in line to buy tickets to a show with the price of one ticket is 
Rp.. Known of them they only have 
Rp. in banknotes and the rest is only has a banknote of Rp. If the ticket seller initially only has Rp., what is the probability that the ticket seller have enough change to serve everyone according to their order in the queue?
with
. (It is implied that is written with a decimal representation.)
regions. Every pair of lines in the same group are parallel. Let 
and 
respectively be the number of lines in groups 1, 2, and 3. If no lines in group 3 go through the intersection of any two lines (in groups 1 and 2, of course), then the least number of lines required in order to have more than 2018 regions is ....
where 
and 
are squares with both planes being parallel. The length of the sides of and respectively are 
and 
, and the height of the frustum is 
. Points and respectively are intersections of the diagonals of and and the line 
is perpendicular to the plane . Construct the pyramids 
and 
and calculate the volume of the 3D figure which is the intersection of pyramids and .
is on the 3rd row, 4th column, and on the 6th diagonal. Meanwhile the position of the number 
is on the 3rd row, 5th column, and 7th diagonal.
based on its row, column, and diagonal.. is the set of 3-digit integers not containing the digit 
. Define a gadang number to be the element of whose digits are all distinct and the digits contained in such number are not prime, and (a gadang number leaves a remainder of 5 when divided by 7. If we pick an element of at random, what is the probability that the number we picked is a gadang number?
and satisfy 
dan 
. The number 
is a fraction in its simplest form.
. is (equal to some number) , determine all positive factors of . above (specifically in problem b), determine the probability of taking a factor who is a multiple of 4.
be given in the following graphs.
with 
for all 
where 
is the domain of 
..
so that 
. has side lengths 
cm and 
cm. All four of its vertices lie on a circle. Calculate the area of quadrilateral . and 
, with 
and 
. It is known that 
, where 
is a 3-digit number whose number in its tens place is 5. Determine the number/quantity of all possible values of .
(the original problem does not have an overline, which I fixed) is arranged from the set 
. Such number satisfies 
. Determine the quantity of different possible (such) numbers.
(i.e., the square on 
has as one of its sides and lies outside ). Show that the lines drawn from the vertices 
, 
, and 
to the centers of the opposite squares are concurrent.
. A circle O with center C has a radius of 2. Let P be a point on the circle O.
?
?, and 
are points on sides 
and 
, respectively, such that 
. If 
and 
, find the area of .
. Let the vertex on 
be , and Let the vertex 
. 
is a parallelogram. Let side of the 
rhombus be and let 
. Then we get 
Now draw a line passing through point , parallel to 
. Say that 
, (parallel line passing through ) intersects 
. Then it is easy to see 
. ===> 
We have to show it is also equal to 
. Let the line passing through also intersect at . Then 
. 
==>
.
, 
. Done!
,
,
,
,
.
.
side length of 
area of 
is a right angle, and 
,
.
is a right angle, 
,
.
and 
both contain a right angle and 
through AA postulate. Because they are similar, 
,
,
,
.
.
,
.
and 
are arranged so that 
lies on the segment 
.
.