[/quote]
,
board, such that Batman and Robin are on the same diagonal (picture). In each round, first The Joker moves to an adjacent cell (having a common side), without exiting the board. Then in the same round Batman and Robin move to an adjacent cell. The Joker wins if he reaches the fourth "target" vertex cell (marked T). Batman and Robin win if they catch The Joker i.e. at least one of them is on the same cell as The Joker.
Prove that
Where 
Prove that
Let 
Prove that
Let 
Prove that
be the foot of perpendicular from 
to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle 
.
with centre 
passes through 
and 
at 
and 
respectively. Let 
be the foot of altitude from 
,
be the midpoint of 
is equidistant from 
,
.
with integer coefficients such that for any positive integer 
,
is positive and 
such that
-
,
be a subset of 
vertices of the polygon. Show that if 
,
consisting of ![\[ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.\]](http://latex.artofproblemsolving.com/d/7/c/d7c9b2e32b9ed6d0c16be523df5acd83a7b782b2.png)
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
denote the set of positive real numbers. Find all functions 
such that for each 
,
satisfying 
is called tasty if for every ordered pair of positive integers 
with 
at least one of 
and 
is a terminating decimal. Do there exist infinitely many tasty integers?
![\[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]](http://latex.artofproblemsolving.com/e/7/f/e7fb8c20a43535b5aaed5ab254f1ef043263c62b.png)
true?
denote the set of all polynomials in three variables 
with integer coefficients. Let 
denote the subset of 
with 
.
for all non-negative integers 
satisfying 
.
of integer numbers, we define 
as the sum of the elements of 
whose intersection is empty, whose union is the set 
and such that the product 
is a perfect square.
be the set of real numbers. Let 
be a function such that ![\[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\]](http://latex.artofproblemsolving.com/5/d/3/5d3476fd9195d085d0030d6036a9fde1cf6756a7.png)
for every 
.
.
for every 
or 
for every 
Y by
Define act <Castle> as following:
For rectangle with dimensions i * j, doing <Castle> means to change its dimensions to (i+p) * (j+q) where p,q is a natural number smaller than 3.
Define 1*1 rectangle as "C0" rectangle, and define "Cn" ("n" is a natural number) as a rectangle that can be created with "n" <Castle>s.
Plus, there is a constraint for "Cn" rectangle. The constraint is that "Cn" rectangle's area must be bigger than n^2 and be same or smaller than (n+1)^2. (n^2 < Area =< (n+1)^2)
Let all "C20" rectangle's area's sum be A, and let all "C20" rectangles perimeter's sum be B.
What is A-B?
For rectangle with dimensions i * j, doing <Castle> means to change its dimensions to (i+p) * (j+q) where p,q is a natural number smaller than 3.
Define 1*1 rectangle as "C0" rectangle, and define "Cn" ("n" is a natural number) as a rectangle that can be created with "n" <Castle>s.
Plus, there is a constraint for "Cn" rectangle. The constraint is that "Cn" rectangle's area must be bigger than n^2 and be same or smaller than (n+1)^2. (n^2 < Area =< (n+1)^2)
Let all "C20" rectangle's area's sum be A, and let all "C20" rectangles perimeter's sum be B.
What is A-B?
This post has been edited 1 time. Last edited by jaydenkaka, Apr 23, 2025, 2:23 PM
