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Let 
and 
Prove that
and 
Prove that
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IMO Shortlist 2011, Algebra 7
orl 23
N
by bin_sherlo
Source: IMO Shortlist 2011, Algebra 7
Let 
and 
be positive real numbers satisfying 
and 
Prove that
![\[\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.\]](//latex.artofproblemsolving.com/0/a/b/0ab4b4e6c28a1ae67788525642661dbcd04d0bd8.png)
Proposed by Titu Andreescu, Saudi Arabia
and 
be positive real numbers satisfying 
and 
Prove that![\[\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.\]](http://latex.artofproblemsolving.com/0/a/b/0ab4b4e6c28a1ae67788525642661dbcd04d0bd8.png)
Proposed by Titu Andreescu, Saudi Arabia
23 replies
Find all real functions withf(x^2 + yf(z)) = xf(x) + zf(y)
Rushil 31
N
by Jakjjdm
Source: INMO 2005 Problem 6
Find all functions 
such that ![\[ f(x^2 + yf(z)) = xf(x) + zf(y) , \]](//latex.artofproblemsolving.com/2/b/a/2baee1d8464992fc38a16561b48b18f4b35807bd.png)
for all 
.
such that ![\[ f(x^2 + yf(z)) = xf(x) + zf(y) , \]](http://latex.artofproblemsolving.com/2/b/a/2baee1d8464992fc38a16561b48b18f4b35807bd.png)
for all 
.
31 replies
1000th Post!
PikaPika999 58
N
by K1mchi_
When I had less than 25 posts on AoPS, I saw many people create threads about them getting 1000th posts. I thought I would never hit 1000 posts, but here we are, this is my 1000th post.
As a lot of users like to do, I'll write my math story:
In conclusion, AoPS has helped me improve my math. I have also made many new friends on AoPS!
Finally, I would like to say thank you to all the new friends I made and all the instructors on AoPS that taught me!
As a lot of users like to do, I'll write my math story:
In conclusion, AoPS has helped me improve my math. I have also made many new friends on AoPS!
Finally, I would like to say thank you to all the new friends I made and all the instructors on AoPS that taught me!
58 replies
Math and AI 4 Girls
mkwhe 18
N
by K1mchi_
Hey everyone!
The 2025 MA4G competition is now open!
Apply Here: https://xmathandai4girls.submittable.com/submit
Visit https://www.mathandai4girls.org/ to get started!
Feel free to PM or email mathandai4girls@yahoo.com if you have any questions!
The 2025 MA4G competition is now open!
Apply Here: https://xmathandai4girls.submittable.com/submit
Visit https://www.mathandai4girls.org/ to get started!
Feel free to PM or email mathandai4girls@yahoo.com if you have any questions!
18 replies
real math problems
Soupboy0 54
N
by K1mchi_
Ill be posting questions once in a while. Here's the first question:
What fraction of numbers from
to 
have the digit 
and are divisible by 
?
What fraction of numbers from
to 
have the digit 
and are divisible by 
?
54 replies
Hello friends
bibidi_skibidi 6
N
by Thayaden
Now unfortunately I don't know the difficulty of the problems posted here but I'll try to replicate:
Bob has 20 apples and 19 oranges. How many ways can he split the fruits between 7 people if each person must have at least 1 apple and 2 oranges?
After looking at the other posters I realized just how bashy this is
Also I can only edit this message for now since new AoPS users can only send 6 messages every day
Bob has 20 apples and 19 oranges. How many ways can he split the fruits between 7 people if each person must have at least 1 apple and 2 oranges?
After looking at the other posters I realized just how bashy this is
Also I can only edit this message for now since new AoPS users can only send 6 messages every day
6 replies
Website to learn math
hawa 26
N
by K1mchi_
Hi, I'm kinda curious what website do yall use to learn math, like i dont find any website thats fun to learn math
26 replies
divisible by 111
aria123 5
N
by aria123
How many 6-digit natural numbers (with distinct digits) can be formed using the digits 2, 3, 4, 5, 6, and 7 that are divisible by 111?
5 replies
Marble Math
ilovebender 1
N
by EthanNg6
John has 10 marbles with the colors red, green, or blue, all either transparent or translucent. He arranged them in a circle with the following conditions:
3 marbles right beside each other cannot be the same color
The marble across from any marble (assuming John makes a perfect circle) cannot be the same color
Red cannot be in between 2 blues
Blue cannot be between 2 greens
Green cannot be between two Reds
How many different ways can John organize these marbles? State "Impossible" if you think there is no solution. State "Undefined" if one rule doesn't follow another
What if John arranged them into two rows with 5 marbles each all right beside each other. What about 5 rows with 2 marbles each?
Post your answer down below
I just thought of this question right off the top of my head, and I didn't have time to do it, but I'd love to see your answers!
Edit: I just realized "What if John arranged them into two rows with 5 marbles each all right beside each other. What about 5 rows with 2 marbles each?" Is a bit confusing, what I meant was that if you arrange the marbles 2 by 5 (one on the top), the arrow points to what is across from that marble. Same with the 5 by 2, the arrows point that is across.
3 marbles right beside each other cannot be the same color
The marble across from any marble (assuming John makes a perfect circle) cannot be the same color
Red cannot be in between 2 blues
Blue cannot be between 2 greens
Green cannot be between two Reds
How many different ways can John organize these marbles? State "Impossible" if you think there is no solution. State "Undefined" if one rule doesn't follow another
What if John arranged them into two rows with 5 marbles each all right beside each other. What about 5 rows with 2 marbles each?
Post your answer down below
I just thought of this question right off the top of my head, and I didn't have time to do it, but I'd love to see your answers!
Edit: I just realized "What if John arranged them into two rows with 5 marbles each all right beside each other. What about 5 rows with 2 marbles each?" Is a bit confusing, what I meant was that if you arrange the marbles 2 by 5 (one on the top), the arrow points to what is across from that marble. Same with the 5 by 2, the arrows point that is across.
1 reply
9 Was the 2025 AMC 8 harder or easier than last year?
Sunshine_Paradise 185
N
by valisaxieamc
Also what will be the DHR?
185 replies
Math Problem I cant figure out how to do without bashing
equalsmc2 2
N
by EthanNg6
Hi,
I cant figure out how to do these 2 problems without bashing. Do you guys have any ideas for an elegant solution? Thank you!
Prob 1.
An RSM sports field has a square shape. Poles with letters M, A, T, H are located at the corners of the square (see the diagram). During warm up, a student starts at any pole, runs to another pole along a side of the square or across the field along diagonal MT (only in the direction from M to T), then runs to another pole along a side of the square or along diagonal MT, and so on. The student cannot repeat a run along the same side/diagonal of the square in the same direction. For instance, she cannot run from M to A twice, but she can run from M to A and at some point from A to M. How many different ways are there to complete the warm up that includes all nine possible runs (see the diagram)? One possible way is M-A-T-H-M-H-T-A-M-T (picture attached)
Prob 2.
In the expression 5@5@5@5@5 you replace each of the four @ symbols with either +, or, or x, or . You can insert one or more pairs of parentheses to control the order of operations. Find the second least whole number that CANNOT be the value of the resulting expression. For example, each of the numbers 25=5+5+5+5+5 and 605+(5+5)×5+5 can be the value of the resulting expression.
Prob 3. (This isnt bashing I don't understand how to do it though)
Suppose BC = 3AB in rectangle ABCD. Points E and F are on side BC such that BE = EF = FC. Compute the sum of the degree measures of the four angles EAB, EAF, EAC, EAD.
P.S. These are from an RSM olympiad. The answers
I cant figure out how to do these 2 problems without bashing. Do you guys have any ideas for an elegant solution? Thank you!
Prob 1.
An RSM sports field has a square shape. Poles with letters M, A, T, H are located at the corners of the square (see the diagram). During warm up, a student starts at any pole, runs to another pole along a side of the square or across the field along diagonal MT (only in the direction from M to T), then runs to another pole along a side of the square or along diagonal MT, and so on. The student cannot repeat a run along the same side/diagonal of the square in the same direction. For instance, she cannot run from M to A twice, but she can run from M to A and at some point from A to M. How many different ways are there to complete the warm up that includes all nine possible runs (see the diagram)? One possible way is M-A-T-H-M-H-T-A-M-T (picture attached)
Prob 2.
In the expression 5@5@5@5@5 you replace each of the four @ symbols with either +, or, or x, or . You can insert one or more pairs of parentheses to control the order of operations. Find the second least whole number that CANNOT be the value of the resulting expression. For example, each of the numbers 25=5+5+5+5+5 and 605+(5+5)×5+5 can be the value of the resulting expression.
Prob 3. (This isnt bashing I don't understand how to do it though)
Suppose BC = 3AB in rectangle ABCD. Points E and F are on side BC such that BE = EF = FC. Compute the sum of the degree measures of the four angles EAB, EAF, EAC, EAD.
P.S. These are from an RSM olympiad. The answers
2 replies
3 var inquality
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#2
Apr 8, 2025, 7:41 PM
Y by
Dear sqing,
I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics?
Thank you
I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics?
Thank you
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