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, Mar 21, 2025, 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 
.Distributing cupcakes
by KevinYang2.71, Mar 21, 2025, 12:00 PM
Let 
and 
be positive integers with 
. There are cupcakes of different flavors arranged around a circle and people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person , it is possible to partition the circle of cupcakes into groups of consecutive cupcakes so that the sum of 's scores of the cupcakes in each group is at least 
. Prove that it is possible to distribute the cupcakes to the people so that each person receives cupcakes of total score at least with respect to .
and 
be positive integers with 
. There are cupcakes of different flavors arranged around a circle and people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person , it is possible to partition the circle of cupcakes into groups of consecutive cupcakes so that the sum of 's scores of the cupcakes in each group is at least 
. Prove that it is possible to distribute the cupcakes to the people so that each person receives cupcakes of total score at least with respect to .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, Mar 20, 2025, 12:09 PM
Base 2n of n^k
by KevinYang2.71, Mar 20, 2025, 12:01 PM
Let 
and 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
, the digits in the base-
representation of 
are all greater than .
and 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
, the digits in the base-
representation of 
are all greater than .Gunn Math Competition
by the_math_prodigy, Mar 8, 2025, 8:13 PM
Attachments:
This post has been edited 7 times. Last edited by AoPSSheriff, Mar 18, 2025, 4:24 PM
Reason: Removed discord link.
Reason: Removed discord link.
Portia's vs. Lara's school
by MathArt4, Feb 5, 2021, 5:47 PM
Portia’s high school has 
times as many students as Lara’s high school. The two high schools have a total of
students. How many students does Portia’s high school have?
times as many students as Lara’s high school. The two high schools have a total of
students. How many students does Portia’s high school have?
This post has been edited 4 times. Last edited by nsato, Dec 30, 2021, 8:19 PM
Triangle in a square
by fortenforge, Feb 6, 2013, 5:41 PM
Square 
has side length 
. Point 
is on 
, and the area of 
is 
. What is 
?
![[asy]
pair A,B,C,D,E;
A=(0,0);
B=(0,50);
C=(50,50);
D=(50,0);
E = (30,50);
draw(A--B);
draw(B--E);
draw(E--C);
draw(C--D);
draw(D--A);
draw(A--E);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
label("A",A,SW);
label("B",B,NW);
label("C",C,NE);
label("D",D,SE);
label("E",E,N);
[/asy]](//latex.artofproblemsolving.com/d/9/a/d9ad7d16a15397560a5daa045621239464589d6e.png)
has side length 
. Point 
is on 
, and the area of 
is 
. What is 
?
![[asy]
pair A,B,C,D,E;
A=(0,0);
B=(0,50);
C=(50,50);
D=(50,0);
E = (30,50);
draw(A--B);
draw(B--E);
draw(E--C);
draw(C--D);
draw(D--A);
draw(A--E);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
label("A",A,SW);
label("B",B,NW);
label("C",C,NE);
label("D",D,SE);
label("E",E,N);
[/asy]](http://latex.artofproblemsolving.com/d/9/a/d9ad7d16a15397560a5daa045621239464589d6e.png)
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