AoPS Academy
.
and 
, we have that 
and 
. Find the value of 
.
and 
. He notices that neither nor is a perfect square, but 
is a perfect square. What is the smallest possible value of 
?
whose digits sum to 
?
has 
and 
. The altitude drawn from 
intersects 
at 
, where 
. What is the area of ?
letters in the word for each positive integer is 
. Each letter of the word is chosen randomly and independently from the uppercase English alphabet. The probability of Archyuta’s program outputting “EMCC” can be written as 
for some positive integer 
. What is the greatest nonnegative integer such that 
divides ?
that can be expressed as the sum of seven consecutive whole numbers, as the sum of five consecutive whole numbers, and as the sum of three consecutive whole numbers?
with side length 
, square 
is inscribed in such that 
is on side and 
is on side 
such that 
and 
. If square PQRS inscribed in such that 
, find the side length of 
.
seconds to reach the top. Tired, he then walks at one-third of his running speed back down the escalator, taking 30 seconds to reach the bottom. Assuming his running speed and the escalator’s speed are constant, what is the ratio of his running speed to the escalator’s speed? bricks shaped like rectangular prisms each of dimension 
foot by foot by 
feet which he stores inside a feet by feet by feet hollow box. In how many ways can he fit his bricks into the box? (Rotations and reflections of a configuration are considered distinct.)
is a point lying inside rectangle . If 
, 
and 
, find 
in degrees.
be a positive integer. If of ’s divisors are prime and 
of ’s divisors are composite, how many of ’s divisors are perfect squares?
. Let 
be the sum of all distinct possible values of their absolute difference. Find the remainder when is divided by .
. The internal angle bisectors of each of its four vertices are drawn, bounding a square region with area 
. Find the perimeter of the rectangle.
be a right triangle, with a right angle at 
. The perpendicular bisector of hypotenuse 
splits the triangle into a smaller triangle and a quadrilateral. If the triangle has an area of 
and the quadrilateral has an area of 
, find the length of . be the sum of the smallest positive perfect squares minus the sum of the smallest positive odd numbers. What is the largest prime factor of ?
and two congruent equilateral triangles placed inside the square, one in each corner and with one overlapping side. What is the distance between the marked vertices? in each unit square of a two-by-two grid, such that no two isosceles triangles share an edge? One valid construction is shown on the left, followed by an invalid construction on the right., followed by Dr. Drescher writing another on the second turn. Each turn afterwards, they take the two newest numbers on the board and concatenate them, writing the resulting number of the board. For instance, the first few numbers on the board are , , 
, 
, 
, 
How many turns does it take for them to write a number which is divisible by 
?
). We have 
distribute the 
factors of 
among the values. Continuing for all primes gives the product 
. For the example, the answer is 
. The number of unordered pairs would be messy because you have to do casework with pairs containing repeated values.
, how many pairs of 
numbers multiply to 
,
.
,
numbers that multiply to 
,
,
then their product far exceed 
)
I think?