[/quote]
,
and 
n
.
be a positive integer. A positive integer 
is called a 
digits in decimal representation (it cannot have leading zeros).
is a 
-![\[(20 + 25)^2 = 2025.\]](http://latex.artofproblemsolving.com/5/c/5/5c51bca485cea6c3750c48c09091eb5fd72b5192.png)
Find all 
-
be the set of all positive integers greater than or equal to 
and let 
be a function such that 
for all 
if 
,
players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least 
next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.
representation of 
?
be a fixed positive real number. Find the minimum possible value 
such that for any positive reals 
,
,
,
,
and 
divides 
.
and 
.
such that 
.
,
,
,
.
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 
.
![\[
\frac{5^{2000} - 1}{4}
\]](http://latex.artofproblemsolving.com/8/b/a/8ba5d1ed8bafd6653f9e6f351558a9976c43a57a.png)
is divided by 
.
has side length 
be 
.
and the circle centered at 
with radius 
occurs when 
for the function 
.
Configuration 
:
,
:
.
,
,
,
.
islands. One day, people decide to create island-states as follows. Each island randomly chooses one of the other five islands and builds a bridge between the two islands (it is possible for two bridges to be built between islands 
and 
.
.
,
for prime 
and 
for 
.
is a perfect power of 
be a triangle whose 
-
,
,
,
and the perimeter of 
,
of functions 
satisfying:
,
,
.
can be written as 
where 
are distinct primes, compute 
.
and that the first (leftmost) two digits of 
are 10. Compute the number of integers 
such that 
starts with either the digit 
or 
(in base 
)
be the circumcenter of 
be the feet of the altitudes from 
intersects 
.
(on the major arc 
,
lies on the line 
such that 
is perpendicular to 
.
and 
,
.
be the isosceles triangle with side lengths 
.
within this triangle, and compute 
,
denote the regular polygon with 
sides such that the vertices of 
denote the polygon that results after applying f a total of k times. The area of 
where 
for some positive integers 
where 
.
. This function can be expressed in the form 
for sequences of integers 
,
.
.
be the centroid and let the circumcenters of 
,
,
be 
,
and the midpoint of 
at 
.
,
is 
,
?
be a polynomial with nonnegative integer coefficients. If 
and 
,
is divided by 
Y by
Is there a way to prove
\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}=1-\frac{1}{{n+1)!}
without induction and using only combinatorial arguments?
Induction proof wasn't quite as pleasing for me.
\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}=1-\frac{1}{{n+1)!}
without induction and using only combinatorial arguments?
Induction proof wasn't quite as pleasing for me.
This post has been edited 1 time. Last edited by MathBot101101, Apr 21, 2025, 6:45 AM
Reason: im dum
Reason: im dum
![\[ \sum\limits_{i=1}^n \dfrac{i}{(i + 1)!} = 1 - \dfrac{1}{(n+1)!}. \]](http://latex.artofproblemsolving.com/2/8/8/288b06247fd042559aaa1e0c0c101b5b2eba7821.png)
This is indeed a simple induction exercise. As for combinatorial arguments, maybe we can rearrange this into ![\[ \frac{1}{(n+1)!} + \sum\limits_{i=1}^{n} \frac{i}{(i+1)!} = 1,\]](http://latex.artofproblemsolving.com/f/1/a/f1a775f27856ac90581ef10e3fb4f6faaee58a71.png)
but then I'm not sure how to get a probability that is clearly equal to the LHS and is also equal to 
gives that
(This is induction at heart.)
