AoPS Academy
such that 
is the square of an integer.
be an acute triangle with 
. Let 
be the centroid of triangle 
and let 
be the foot of the perpendicular from 
to side 
. The median 
intersects the circumcircle 
of triangle at a second point 
. Let 
be the point where intersects . The line 
intersects the circle at a point 
, such that lies between and . Prove that the points 
and lie on a circle.
written on the board. We apply the following rule: At each step, we will add all the numbers that are the sum of the two numbers on the board so that the sum does not appear on the board. For example, if the two initial numbers are 
; then the numbers on the board after step 1 are 
; after step 2 are 
; 
, prove that the number 2024 cannot appear on the board.
; 
, prove that the number 2024 can appear on the board.
(base-10) into base-, where is the smallest integer that has 8 factors. The answer will be the code to the safe.[/quote]
such that there exist positive integers 
and 
with 
satifying the condition that,
is an integer.
.
is a nonzero polynomial with integer coefficients. Prove that there exists infinitely many prime numbers 
such that for some natural number , 
.
be positive real numbers that fulfill![\[
\frac{x^2}{y^2}+\frac{4x^2-3xy-4y^2}{2xy-5y^2}=2
\]](http://latex.artofproblemsolving.com/6/c/c/6ccba85deb3e6aa205abd3bc8e9e9bb68f4df6fd.png)
Hence find the value of 
to 
on a different unit square of a 
square grid, such that consecutive integers are on adjacent squares, and is not adjacent to ? (Note that adjacent squares are squares that share a common side.)
be a prime and 
be positive integers such that 
and 
is prime. Prove that 
, 
, , and 
. The railway forms a complete loop, as shown below, and trains go in both directions. Suppose that a trip between two adjacent towns costs one ticket. Using exactly eight tickets, how many distinct ways are there of traveling from town and ending at town ? (You may pass through town in the middle).
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?
on the whiteboard. In each move, she selects two distinct numbers, and , erases them, and replaces them with 
. She repeats this process until only one number, 
, remains. What are all the possible values of ?
, and the sum of all numbers decreases by 
as well. This value is equal to 
, and it will always be equal to 
. We add to get the sum of all numbers when there is number, and the only possible value of is 
.
