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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
official solution of IGO
ABCD1728   7
N 19 minutes ago by ABCD1728
Source: IGO official website
Where can I get the official solution of IGO for 2023 and 2024, there are some inhttps://imogeometry.blogspot.com/p/iranian-geometry-olympiad.html, but where can I find them on the official website, thanks :)
7 replies
ABCD1728
May 4, 2025
ABCD1728
19 minutes ago
Combo geo with circles
a_507_bc   10
N 22 minutes ago by EthanWYX2009
Source: 239 MO 2024 S8
There are $2n$ points on the plane. No three of them lie on the same straight line and no four lie on the same circle. Prove that it is possible to split these points into $n$ pairs and cover each pair of points with a circle containing no other points.
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a_507_bc
May 22, 2024
EthanWYX2009
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Vietnam TST #5
IMOStarter   2
N 28 minutes ago by cursed_tangent1434
Source: Vietnam TST 2022 P5
A fractional number $x$ is called pretty if it has finite expression in base$-b$ numeral system, $b$ is a positive integer in $[2;2022]$. Prove that there exists finite positive integers $n\geq 4$ that with every $m$ in $(\frac{2n}{3}; n)$ then there is at least one pretty number between $\frac{m}{n-m}$ and $\frac{n-m}{m}$
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Apr 27, 2022
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28 minutes ago
Squares consisting of digits 0, 4, 9
VicKmath7   4
N 43 minutes ago by NicoN9
Source: Bulgaria MO Regional round 2024, 9.3
A positive integer $n$ is called a $\textit{supersquare}$ if there exists a positive integer $m$, such that $10 \nmid m$ and the decimal representation of $n=m^2$ consists only of digits among $\{0, 4, 9\}$. Are there infinitely many $\textit{supersquares}$?
4 replies
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43 minutes ago
No more topics!
Indian Mathematical Olympiad 1986 - Problem 9
Leon   7
N Apr 14, 2021 by denery
Show that among all quadrilaterals of a given perimeter the square has the largest area.
7 replies
Leon
Aug 13, 2009
denery
Apr 14, 2021
Indian Mathematical Olympiad 1986 - Problem 9
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Leon
256 posts
#1 • 3 Y
Y by samrocksnature, Adventure10, Mango247
Show that among all quadrilaterals of a given perimeter the square has the largest area.
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Kunihiko_Chikaya
14514 posts
#2 • 3 Y
Y by vp2016, Adventure10, Mango247
Let the quadrilateral has side length $ a,\ b,\ c,\ d$ with $ p=\frac{a+b+c+d}{2}$,

we have $ [ABCD]=\sqrt{(p-a)(p-b)(p-c)(p-d)}=(\sqrt[4]{(p-a)(p-b)(p-c)(p-d)})^2$

$ \leq \left\{\frac{(p-a)+(p-b)+(p-c)+(p-d)}{4}\right\}^2=\frac{p^2}{4}$,

the equality holds when $ a=b=c=d$.
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Bugi
1857 posts
#3 • 4 Y
Y by Adventure10, Mango247, Mango247, Mango247
I think that formula holds only if the quadrilateral is cyclic.

But we can do like this:

Solution
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Kunihiko_Chikaya
14514 posts
#4 • 1 Y
Y by Adventure10
Bugi wrote:
I think that formula holds only if the quadrilateral is cyclic.

You are right.

Related problem

1969 Czechoslovakia Mathematical Olympiad

A convex quadrilateral $ ABCD$ with sides $ AB = a,\ BC = b,\ CD = c,\ DA = d$ and angles $ \alpha = \angle{DAB},\ \beta = \angle{ABC},\ \gamma = \angle{BCD},\ \delta = \angle{CDA}$
is given. Let $ \displaystyle s = \frac {a + b + c + d}{2}$ and $ P$ be the area of the quadrilateral. Prove that
\[ P^2 = (s - a)(s - b)(s - c)(s - d) - abcd\cos ^ 2 \displaystyle \frac {\alpha + \gamma}{2}.\]
Now the name of the nation changed,\ didn't that?

It occured to me that Ellipse has something to do with the solution.
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Mathias_DK
1312 posts
#5 • 3 Y
Y by Titttu, Adventure10, Mango247
Leon wrote:
Show that among all quadrilaterals of a given perimeter the square has the largest area.
Lemma 1:
Given a triangle $ \triangle ABC$, with $ BC$ and $ AB+AC$ fixed, then the area of $ \triangle ABC$ is largest when $ AB=AC$.
Proof:
The locus of all points $ P$ such that $ PB+PC = AB+AC$ is an ellipse. Since the area of $ \triangle PBC$ is the distance from the line $ BC$ to $ P$ times $ BC$, we see that $ \triangle PBC$ has the maximal area when $ P$ are furthest away from $ BC$. Because of symmetry this must be when $ P$ lies on the bisector of $ BC$, and hence $ AB=AC$.

Let $ \square ABCD$ be the quadrilateral with the largest area. Using lemma 1 on $ \triangle ABC$ we can wlog assume that $ AB=BC$. And using on $ \triangle ADC$: wlog $ AD=DC$. Hence $ \triangle BCD$ is concurrent to $ \triangle DAB$. So it is enough to find the maximal area of $ \triangle BCD$ when $ BC+CD$ is fixed. Using Lemma 1, we see that we can wlog assume that $ BC=CD$, so $ \square ABCD$ must be a rhombus. Let $ E$ be the projection of $ C$ on $ BD$. Then $ \triangle BEC$ is a rightangled triangle, with $ BC$ fixed. Now $ BE^2+EC^2 = BC^2$ is also fixed, and the area of $ \triangle BEC$ is $ \frac{1}{2}BE \times EC$. Since $ BE^2 + EC^2$ is fixed we see that $ BE \times EC$ is largest whenever $ BE=EC$.(Wellknown) Doing this for $ \triangle DEC$ also we see that $ \angle BCD = 90^\circ$, and since we could use this symmetric argument for all the angles in the quadrilateral, $ \square ABCD$ must be a square.
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Maharjun
109 posts
#6 • 3 Y
Y by lckihi, Adventure10, Mango247
I have made use of the result that if the sum of two, or more positive quantities is constant. then their product is maximum when they are all equal. For two quantities, it can be proven using A.M G.M as follows-

Click here if interested in proof of above inequality for two quantities

Step 1:(prove that the quadrilateral is a rhombus,using hero's formula for area of triangle)
solution

step 2 -(prove that the rhombus is a square, trivial)
solution
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guptaamitu1
656 posts
#7
Y by
Maharjun wrote:
I have made use of the result that if the sum of two, or more positive quantities is constant. then their product is maximum when they are all equal. For two quantities, it can be proven using A.M G.M as follows-

Click here if interested in proof of above inequality for two quantities

Step 1:(prove that the quadrilateral is a rhombus,using hero's formula for area of triangle)
solution

step 2 -(prove that the rhombus is a square, trivial)
solution

I think that this solution is wrong (because according to me, step 1 of your solution is not correct). Basically, you are there trying to show that if we replace $(a,b)$ by $({a + b\over 2},{a + b\over 2})$ then the area of the quadrilateral will increase. But that is not always true. Since if you do this replacement, then the value of $e$ also gets effected. So you cannot guarantee that the area of the quadrilateral will increase.
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denery
180 posts
#8
Y by
imagine a person using jensen" s inequality to prove it area of quadrilateral is a convex function :D
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