[/quote]
,
be an integer and 
is a sequence of real numbers for which 
and:
Prove that all elements of the sequence are positive.
is made up of 64 small unit cubes. Exactly 16 of these small cubes must be colored red, subject to the following condition:
,
,
cubes, there must be exactly one red cube.
from the set of positive integers to the set of positive integers such that, for all positive integers 
and 
,![\[ a, f(b) \text{ and } f(b + f(a) - 1).\]](http://latex.artofproblemsolving.com/4/a/b/4ab97da27e05d0f9e7762f74d0d0bd2dc8b9842f.png)
,
holds for all positive reals 
and for every positive real 
,
,
,
be the real root of equation 
.
;
.
be an integer, 
a finite set of integers (not necessarily positive) and 
subsets of 
,
is 
.
elements.
Prove that
Where 
Prove that
Let 
Prove that
Let 
Prove that
be an acute triangle. Define 
as its incenter. Let 
and 
be the incircle's tangent points to 
and 
,
be the midpoint of 
.
be the intersection point of a perpendicular line passing through 
.
intersects the circumcircle of 
at 
.
intersects line 
at 
.
is cyclic.
for which there exists a sequence of distinct positive integers 
,
,
,
satisfying ![\[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\]](http://latex.artofproblemsolving.com/a/f/5/af594ff07aa2f89cb70ae669263558049556cf0e.png)
.
parts is a representation of n as a sum 
,
belong to 
.
parts with 
.![$\sqrt[3]{6n}$](http://latex.artofproblemsolving.com/2/0/1/201489b1f95ad200036698a606f712da7bb465b9.png)
different parts.
other knights where friendship is mutual. Prove that King Arthur can place some of his knights around a round table in such a way that every knight is friends with the 
people adjacent to him and between them there are at least 
friendships of knights that are not adjacent to each other.
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities 
we define:
A 
between 
,
,
and 
for every 
;
A 
between 
A 
between 
be the total number of pairs of cities in the country that are connected by direct flights. In terms of 
,
.
.
are fixed positive integers.
be the set of all functions 
with the following properties:
the following equality holds: 
such that
,
.
Y by
Ivan bought 
cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number 
. How many cats of each breed does he have?
cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number 
. How many cats of each breed does he have?This post has been edited 5 times. Last edited by jl_, Apr 23, 2025, 10:21 AM
Reason: Change source
Reason: Change source
each.
denote the number of cats of each breed. We know that 
and 
.![$$10=\frac{a+b+c+d+e}{5} \geq \sqrt[5]{abcde}=10.$$](http://latex.artofproblemsolving.com/5/3/c/53c2e29741af54049854cd86e84fc52424715f55.png)
This is precisely the equality case, which can only happen when 
.
