BOMBARDIRO CROCODILO VS TRALALERO TRALALA
by LostDreams, Mar 21, 2025, 12:11 PM
Let 
be a positive integer, and let 
be nonnegative integers such that 
Prove that
![\[
\sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}.
\]](//latex.artofproblemsolving.com/c/4/9/c49d0ccfef69402d05dc44d721172498c605cf7e.png)
Note: 
for all nonnegative integers 
.
be a positive integer, and let 
be nonnegative integers such that 
Prove that![\[
\sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}.
\]](http://latex.artofproblemsolving.com/c/4/9/c49d0ccfef69402d05dc44d721172498c605cf7e.png)
Note: 
for all nonnegative integers 
.This post has been edited 6 times. Last edited by LostDreams, 2 hours ago
funny title placeholder
by pikapika007, Mar 21, 2025, 12:10 PM
Let 
be a set of integers with the following properties:
contains all positive integers.
be a set of integers with the following properties:
.- If
and 
, then 
. - If for some
, 
is composite, then all positive divisors of are in .
.
and 
,
.
,
is composite, then all positive divisors of contains all positive integers.This post has been edited 1 time. Last edited by pikapika007, Today at 12:12 PM
Reason: wrong year
Reason: wrong year
usamOOK geometry
by KevinYang2.71, Mar 21, 2025, 12:00 PM
Let 
be the orthocenter of acute triangle 
, let 
be the foot of the altitude from 
to 
, and let 
be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.
be the orthocenter of acute triangle 
, let 
be the foot of the altitude from 
to 
, and let 
be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.combo j3 :blobheart:
by rhydon516, Mar 20, 2025, 12:08 PM
Let 
and be positive integers, and let 
be a 
grid of unit squares.
A domino is a
or 
rectangle. A subset of grid squares in is domino-tileable if dominoes can be placed to cover every square of exactly once with no domino extending outside of . Note: The empty set is domino tileable.
An up-right path is a path from the lower-left corner of to the upper-right corner of formed by exactly 
edges of the grid squares.
Determine, with proof, in terms of and , the number of up-right paths that divide into two domino-tileable subsets.
and be positive integers, and let 
be a 
grid of unit squares.A domino is a
or 
rectangle. A subset of grid squares in is domino-tileable if dominoes can be placed to cover every square of exactly once with no domino extending outside of . Note: The empty set is domino tileable.An up-right path is a path from the lower-left corner of
to the upper-right corner of formed by exactly 
edges of the grid squares.Determine, with proof, in terms of
and , the number of up-right paths that divide into two domino-tileable subsets.This post has been edited 1 time. Last edited by rhydon516, Yesterday at 12:09 PM
Day Before Tips
by elasticwealth, Mar 19, 2025, 12:09 AM
This post has been edited 2 times. Last edited by elasticwealth, Mar 19, 2025, 4:15 AM
USAJMO problem 3: Inequality
by BOGTRO, Apr 24, 2012, 9:57 PM
Let 
be positive real numbers. Prove that 
.
be positive real numbers. Prove that 
.The ones who are crazy enough to think they can change the world are the ones who do.
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