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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
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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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

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[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
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[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
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0 replies
jlacosta
Mar 2, 2025
0 replies
Perfect Squares, Infinite Integers and Integers
steven_zhang123   0
4 minutes ago
Source: China TST 2001 Quiz 5 P1
For which integer \( h \), are there infinitely many positive integers \( n \) such that \( \lfloor \sqrt{h^2 + 1} \cdot n \rfloor \) is a perfect square? (Here \( \lfloor x \rfloor \) denotes the integer part of the real number \( x \)?
0 replies
steven_zhang123
4 minutes ago
0 replies
f(f(x)+y)+f(x+y)=2x+2f(y)
parmenides51   3
N 6 minutes ago by Burmf
Source: 2015 AGCN Competition p1 by bobthesmartypants https://artofproblemsolving.com/community/c5h1128876p5232794
Find all functions $f:\mathbb{R}_{\ge 0}\to \mathbb{R}_{\ge 0}$ satisfying$$f(f(x)+y)+f(x+y)=2x+2f(y)$$
3 replies
1 viewing
parmenides51
Dec 5, 2023
Burmf
6 minutes ago
Help to prove an inequality
JK1603JK   1
N 8 minutes ago by jawadkaleem
Source: unknown
If a,b,c\ge 0: ab+bc+ca=1 then prove \frac{a\left(b+c+2\right)}{bc+2a}+\frac{b\left(c+a+2\right)}{ca+2b}+\frac{c\left(a+b+2\right)}{ab+2c}\ge 3
* Please help me convert it to latex form. Thank you.
1 reply
1 viewing
JK1603JK
24 minutes ago
jawadkaleem
8 minutes ago
2^a + 3^b + 5^c = n!
togrulhamidli2011   2
N 12 minutes ago by togrulhamidli2011
\[
\text{Find all non-negative integers } (a, b, c, n) \text{ such that}
\]\[
2^a + 3^b + 5^c = n!
\]
2 replies
togrulhamidli2011
21 minutes ago
togrulhamidli2011
12 minutes ago
No more topics!
Easy [contraction on the euclidean plane]
Omid Hatami   9
N Jun 5, 2016 by P-H-David-Clarence
Source: Iran 2004
$f:\mathbb{R}^2 \to \mathbb{R}^2$ is injective and surjective. Distance of $X$ and $Y$ is not less than distance of $f(X)$ and $f(Y)$. Prove for $A$ in plane:
\[ S(A) \geq S(f(A))\]
where $S(A)$ is area of $A$
9 replies
Omid Hatami
Sep 14, 2004
P-H-David-Clarence
Jun 5, 2016
Easy [contraction on the euclidean plane]
G H J
G H BBookmark kLocked kLocked NReply
Source: Iran 2004
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Omid Hatami
1275 posts
#1 • 3 Y
Y by Adventure10, Mango247, and 1 other user
$f:\mathbb{R}^2 \to \mathbb{R}^2$ is injective and surjective. Distance of $X$ and $Y$ is not less than distance of $f(X)$ and $f(Y)$. Prove for $A$ in plane:
\[ S(A) \geq S(f(A))\]
where $S(A)$ is area of $A$
Z K Y
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Sailor
256 posts
#2 • 2 Y
Y by Adventure10, Mango247
Maybe it's time to post your solution, since you say it's easy...
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sam-n
793 posts
#3 • 2 Y
Y by Adventure10, Mango247
Hint : u should solve it with circle packing :D
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DusT
297 posts
#4 • 2 Y
Y by Adventure10, Mango247
Maybe the best idea is to break it in small squares, or any regular figure..., but I think the problem is the function is not continuous.......
Also, here arises a question, is A a contour of the figure, or is A the entire figure, because some continuity problems appear....
What do you think, sam-n ????
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Simo_the_Wolf
106 posts
#5 • 2 Y
Y by Adventure10, Mango247
Take a point P inside A.
Take all the distances from P to all points of the perimeter of A.

we take $k=max[d(P,f(B))/d(P,B)]$ where B is a point of the perimeter.

We know that k<1 and all points of f(A) are in A' where A' is A dilatate about P of the coefficient k. So S[f(A)]<=S(A') but S(A')=kS(A)<S(A) and so
S[f(A)]<S(A)

c.v.d.
Z K Y
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Pascual2005
1160 posts
#6 • 3 Y
Y by Adventure10, Adventure10, Mango247
what is circle packing?
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Omid Hatami
1275 posts
#7 • 2 Y
Y by Adventure10, Mango247
Well for any figure $B$ you must first put some cirlces in $B$ that they are distinct and sum of their areas is $S(B)- \epsilon$.
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Omid Hatami
1275 posts
#8 • 2 Y
Y by Adventure10, Mango247
Well consider circles with centers $f(A_1),f(A_2),...,f(A_n)(A_i \in A)$ with radius $r_1,...,r_n$ that are distinct and sum of ther areas is $S(f(A))- \epsilon$
Now draw circles with centers $A_1,...,A_n$ and radius $r_1,...,r_n$ .
Prove they are distinct and are in $A$ so for every $\epsilon$:
\[ S(A) \geq S(f(A))-\epsilon\]
And this proves the problem
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Omid Hatami
1275 posts
#9 • 2 Y
Y by Adventure10, Mango247
For circle packing you must use this that for every $\epsilon$ there are finite squares in $A$ sum of their area is at least $S(A) - \epsilon$
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P-H-David-Clarence
92 posts
#10 • 2 Y
Y by Adventure10, Mango247
Simo_the_Wolf wrote:
Take a point P inside A.
Take all the distances from P to all points of the perimeter of A.

we take $k=max[d(P,f(B))/d(P,B)]$ where B is a point of the perimeter.

We know that k<1 and all points of f(A) are in A' where A' is A dilatate about P of the coefficient k. So S[f(A)]<=S(A') but S(A')=kS(A)<S(A) and so
S[f(A)]<S(A)

c.v.d.

But the figure may not be convex so why f(A) is in A'?
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