and 
Prove that
Let 
Prove that
year today and has 
posts as of now (excluding this) and has received 
upvotes.
months and it is probably the biggest exam I have ever given. So yeah I need to prepare for that and I think I should not spend much time on AoPS anymore as of now. I will probably join AoPS after 
months. I know no one asked but just in case you guys may be surprised as to why are there no more 
,
and the median 
divide 
into three equal parts. If 
,
is?
.
as 
.
,
.
,
.
,
.
.
.
be the circumcenter, then 
,
,
are colinear. Therefore, 
or 
.
and 
coincide, and the angle is not trisected. Thus, 
is a right-angled triangle. Consequently, the altitude divides the 
angle into 
and 
,
,
.
and 
.
,
be a triangle in which 
.
,
to meet 
at 
.
and 
.
equals?
.
(Pythagorean Theorem). Thus 
and 
(Pythagorean Theorem).
be a square of side length 
be points in the interiors of the sides 
,
then find the length of 
and 
.
as 
,
and 
.
.
be located in the 
p
,
(i).
,
(ii).
.
.
and 
be the images of 
and 
respectively. The distance between the circumcenters of triangles 
is?
and 
and 
.
then 
and 
.
.
is just 
.
.
and 
right angled triangle thus the circumradius is 
.
.
places to choose form and 
.
people. Let 
be the number of cars that will be used and 
be the number of people per car. What is the smallest value of 
,
.
.
is 
.
or 
.
and 
thus resulting the final 
.
was quite natural. Now the main thing is observing that 
satisfy the equation 
.
is bijective.
such that 
.
But a function can't give two distinct values, unless 
this is not true so 
,
To obtain any 
just let 
.
Thus 
which satisfy for all 
for 
![\[f(x)=x^2\cdot f\left(\frac{1}{x}\right)=x^2\cdot f\left(1+\frac{1-x}{x}\right)=x^2+x^2\cdot f\left(\frac{1-x}{x}\right)=x^2+x^2\cdot \frac{(1-x)^2}{x^2}\cdot f\left(\frac{x}{1-x}\right)=\]](http://latex.artofproblemsolving.com/4/d/a/4da809b44b08f413bde69c4df27874c14bc7e4b8.png)
![\[x^2+(1-x)^2\cdot f\left(\frac{1}{1-x}-1\right)=x^2+(1-x)^2\cdot f\left(\frac{1}{1-x}\right)-(1-x)^2=x^2-(1-x)^2+f(1-x)=2x+f(-x)=2x-f(x)\]](http://latex.artofproblemsolving.com/3/d/d/3ddf8024f0a68b7279bee29f51e58e913fd1d3aa.png)
and hence 
holds for all 
which easily extends to 
such that
and 
be the assertion 
and so 
and 
since 
is non constant;
and so :
and so 
implies 
which gives immediately 
,
and 
which indeed is a solution.
implies 
which gives immediately 
and 
and 
which indeed is a solution.
implies 
which gives immediately 
which indeed is a solution.
and 
Prove that
Where 
Y by Foxellar
Let 
then prove 
then prove 
G
Topic
First Poster
Last Poster
k a March Highlights and 2025 AoPS Online Class Information
jlacosta 0
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.
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0 replies
Inequalities
sqing 13
N
by ReticulatedPython
Let 
and 
. Prove that
Equality holds when 
or 
or 
or
and 
. Prove that
Equality holds when 
or 
or 
or
13 replies
no of triangles in a 3x3 grid (by 9 points) (HOMC 2016 J Q7)
parmenides51 8
N
by rchokler
Nine points form a grid of size 
. How many triangles are there with 
vertices at these points?
. How many triangles are there with 
vertices at these points?
8 replies
combinations, probability
Chanome 2
N
by musclebaby
Given a fair 
-sided die with sides 
, consider the following game:
1. Roll the die. If the roll results in, you win immediately.
2. Otherwise, roll again. However, if the second roll is not greater than the previous roll, you lose.
3. Continue rolling until either:
- You roll, in which case you win.
- Or, your current roll is not greater than your previous roll, in which case you lose.
For example, when
:
- Rolls
: Win
- Rolls
: Lose
- Rolls
: Lose
- Rolls
: Win
Find a formula to find the probability of winning for any given.
-sided die with sides 
, consider the following game:1. Roll the die. If the roll results in
, you win immediately.2. Otherwise, roll again. However, if the second roll is not greater than the previous roll, you lose.
3. Continue rolling until either:
- You roll
, in which case you win.- Or, your current roll is not greater than your previous roll, in which case you lose.
For example, when
:- Rolls
: Win- Rolls
: Lose- Rolls
: Lose- Rolls
: WinFind a formula to find the probability of winning for any given
.
2 replies
4-digit evens with 0,1,2,3,4,5 (Puerto Rico TST 2024.6)
Equinox8 0
Find the sum of all 
-digit even numbers that can be written using the digits 
and 
. Digits can be repeated in a number.
-digit even numbers that can be written using the digits 
and 
. Digits can be repeated in a number.
0 replies
Geometry Problem
JetFire008 3
N
by cj13609517288
Equilateral 
is drawn externally on side 
of 
. Point 
is taken on 
. Find 
if 
.
is drawn externally on side 
of 
. Point 
is taken on 
. Find 
if 
.
3 replies
how to name the numbers on unit circle
Miranda2829 1
N
by Lankou
I am bit confused on how do you call the numbers on unit circle
sin of 30 is 1/2 cosine of 30 is square root 3/2, etc...
is this call trig ratio?, trig functions? or coordinate?
thanks
sin of 30 is 1/2 cosine of 30 is square root 3/2, etc...
is this call trig ratio?, trig functions? or coordinate?
thanks
1 reply
Goodbye AoPS
SomeonecoolLovesMaths 17
N
by SomeonecoolLovesMaths
There are math questions at the end of the post.
And as a result I am quitting AoPS (atleast as of now)
I will probably update my blog(?).
So yeah this concludes this post(?). I probably have not written whatever I wanted to but AoPS has been basically the website I have been the most active on for the past year so leaving this hurts. (Not like I am going for forever lol)
Anyways as promised here are a "few" (not so original) questions.
----------------------------
And as a result I am quitting AoPS (atleast as of now)
I will probably update my blog(?).
So yeah this concludes this post(?). I probably have not written whatever I wanted to but AoPS has been basically the website I have been the most active on for the past year so leaving this hurts. (Not like I am going for forever lol)
Anyways as promised here are a "few" (not so original) questions.
----------------------------
17 replies
Help me please
ntu0301 1
N
by alexheinis
Determine all integers 
that satisfy the following condition: For every integer k such that 
there always exists a positive integer 
that is divisible by n and 
. 
: sum of elements of
that satisfy the following condition: For every integer k such that 
there always exists a positive integer 
that is divisible by n and 
. 
: sum of elements of
1 reply
a+b+c=3 ine
jokehim 5
N
by LearnMath_105
Problem. Given non-negative real numbers 
satisfying 
Prove that 
Proposed by Phan Ngoc Chau
satisfying 
Prove that 
Proposed by Phan Ngoc Chau
5 replies
Number Theory
Foxellar 3
N
by Pal702004
**Problem:**
Define
as the sum of the digits of a natural number . Also, define
![\[
\{x\} \text{ as the fractional part of } x.
\]](//latex.artofproblemsolving.com/b/f/7/bf7b4cc20cbdcee755c73f967c6d231a0896b4b4.png)
The smallest natural number such that
![\[
\left\{\frac{n}{s(n)}\right\} = \frac{1}{6}
\]](//latex.artofproblemsolving.com/e/9/3/e9324df5f1fe12512254c2a8e273629b6b824126.png)
is 
.
Define
as the sum of the digits of a natural number . Also, define![\[
\{x\} \text{ as the fractional part of } x.
\]](http://latex.artofproblemsolving.com/b/f/7/bf7b4cc20cbdcee755c73f967c6d231a0896b4b4.png)
The smallest natural number such that![\[
\left\{\frac{n}{s(n)}\right\} = \frac{1}{6}
\]](http://latex.artofproblemsolving.com/e/9/3/e9324df5f1fe12512254c2a8e273629b6b824126.png)
is 
.
3 replies
inequalities
LuvThoConBoiRoi 9
N
by sqing
With are all real positive number that: 
. Prove that:
are all real positive number that: 
. Prove that:
9 replies
An inequality
G
H
G
H
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by the QM-AM inequality, so
,
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N Quick Reply
G
H
