[/quote]
,
is defined by the formula![\[ a_{i + 1} = \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]](http://latex.artofproblemsolving.com/5/c/2/5c2ac5789507cdeaabc16284447f28294dd4fbdf.png)
here 
is an arbitrary real number, 
denotes the greatest integer not exceeding 
,
.
for 
sufficiently large.
be a function defined on 
satisfying
for all 
for all 
such that 
.
.
,
sums 
with 
,
for which there exist 
and 
,
is divisible by 
.
,
is divisible by 
.
be an integer. Ivan writes the numbers 
each on different cards. He then shuffles these 
cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.
of integers having the following property:
such that 
is divisible by 
for all positive integers 
and 
Prove 
such that there exists a pair 
of positive integers, such that 
is not divisible by the cube of any prime, and 
and have answers
is an isosceles trapezoid with 
.
passing through 
and 
intersects the side 
and the diagonal 
at points 
and 
respectively. Tangent to 
at 
.
be the number of five-element subsets that can be chosen from the set of the first 
natural numbers so that at least two of the five numbers are consecutive. Find the remainder when 
.
be the number of ways in which Ed can arrange the 
marbles to satisfy the requirement. Find the remainder when 
Y by
Charlyn walks completely around the boundary of a square whose sides are each 
km long. From any point on her path she can see exactly 
km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?
Most people jsut subtract the inner square 3*3 but why are there no semicircle areas in the empty spot in the middle?
km long. From any point on her path she can see exactly 
km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?
Most people jsut subtract the inner square 3*3 but why are there no semicircle areas in the empty spot in the middle?
This post has been edited 1 time. Last edited by Spacepandamath13, Jun 6, 2025, 2:08 AM
