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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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Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

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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
Inequalities
sqing   9
N 2 minutes ago by sqing
Let $ a,b> 0 , \frac{a}{b^2}+\frac{b}{a^2}+\frac{56}{(a+b)^2} \leq 16.$ Prove that
$$ab(a+b) \geq 2$$Let $ a,b> 0 ,\frac{1}{a^2}+\frac{1}{b^2}+\frac{28}{(a+b)^2} \leq 9.$ Prove that
$$ab(a+b) \geq 2$$
9 replies
1 viewing
sqing
May 25, 2025
sqing
2 minutes ago
rare creative geo problem spotted in the wild
abbominable_sn0wman   3
N 2 hours ago by jasperE3
The following is the construction of the twindragon fractal.

Let $I_0$ be the solid square region with vertices at
\[
(0, 0), \left(\frac{1}{2}, \frac{1}{2}\right), (1, 0), \left(\frac{1}{2}, -\frac{1}{2}\right).
\]
Recursively, the region $I_{n+1}$ consists of two copies of $I_n$: one copy which is rotated $45^\circ$ counterclockwise around the origin and scaled by a factor of $\frac{1}{\sqrt{2}}$, and another copy which is also rotated $45^\circ$ counterclockwise around the origin and scaled by a factor of $\frac{1}{\sqrt{2}}$, and then translated by $\left(\frac{1}{2}, -\frac{1}{2}\right)$.

We have displayed $I_0$ and $I_1$ below.

Let $I_\infty$ be the limiting region of the sequence $I_0, I_1, \dots$.

The area of the smallest convex polygon which encloses $I_\infty$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Find $a + b$.
3 replies
abbominable_sn0wman
Yesterday at 6:04 PM
jasperE3
2 hours ago
Find the number of ordered triples (p,q,r)
Darealzolt   3
N 3 hours ago by ohiorizzler1434
Let \(p,q,r\) be prime numbers such that
\[
\frac{1}{pq}+\frac{1}{qr}+\frac{1}{pr}=\frac{1}{839}
\]Hence find the numbers of ordered triples \(\{p,q,r\}\)
3 replies
Darealzolt
Yesterday at 1:52 PM
ohiorizzler1434
3 hours ago
Great Geometry with Squares on sides of triangles
SomeonecoolLovesMaths   4
N 3 hours ago by ohiorizzler1434
Three squares are drawn on the sides of triangle \(ABC\) (i.e., the square on \(AB\) has \(AB\) as one of its sides and lies outside \(ABC\)). Show that the lines drawn from the vertices \(A\), \(B\), and \(C\) to the centers of the opposite squares are concurrent.

IMAGE
4 replies
SomeonecoolLovesMaths
May 22, 2025
ohiorizzler1434
3 hours ago
Algebraic Manipulation
Darealzolt   1
N 3 hours ago by ohiorizzler1434
It is known that \(a,b \in \mathbb{R}\) that satisfies
\[
a^3+b^3=1957
\]\[
(a+b)(a+1)(b+1)=2014
\]Hence, find the value of \(a+b\)
1 reply
Darealzolt
4 hours ago
ohiorizzler1434
3 hours ago
Vieta Substitution
Darealzolt   0
4 hours ago
Let \(\alpha,\beta,\gamma,\delta\) be the roots of the equation \(x^4-3x^3+6x^2+5x-25\). Hence, find the value of \(Z\) if
\[
Z=\frac{\alpha+\beta+\gamma+\delta}{\alpha(\beta+\gamma+\delta)+\beta(\gamma+\delta)}+\alpha\beta\gamma\delta[\alpha\beta(\gamma+\delta)+\gamma\delta(\alpha+\beta)]
\]
0 replies
Darealzolt
4 hours ago
0 replies
Floor Function Series
Darealzolt   2
N 4 hours ago by pieMax2713
Let \( \lfloor x \rfloor \) denote the greatest integer less than or equal to \(x\). Hence find the value of \(M\), if
\[
M = \left\lfloor \frac{1^2}{3} \right\rfloor + \left\lfloor \frac{2^2}{3}  \right\rfloor + \left\lfloor \frac{3^2}{3}  \right\rfloor + \dots + \left\lfloor \frac{39^2}{3}  \right\rfloor + \left\lfloor \frac{40^2}{3}  \right\rfloor
\]
2 replies
Darealzolt
5 hours ago
pieMax2713
4 hours ago
Vincentian Numbers
Darealzolt   0
5 hours ago
A number is called \(Vincentian\) if within that number exists a digit \(k \in \{1,2,3,4,5,6,7\}\) that appears exactly \(k^2\) times in that number, hence find the number of \(Vincentian\) that consist of 4 digits (Numbers may contain a 0)
0 replies
Darealzolt
5 hours ago
0 replies
9 Isogonal and isotomic conjugates
V0305   13
N 5 hours ago by ohiorizzler1434
1. Do you think isogonal conjugates should be renamed to angular conjugates?
2. Do you think isotomic conjugates should be renamed to cevian conjugates?

Please answer truthfully :)

Credit to Stead for this renaming idea
13 replies
V0305
May 26, 2025
ohiorizzler1434
5 hours ago
Prove atleast one from a,b,c is 2
Darealzolt   2
N Today at 1:25 AM by sqing
Let \(a,b,c\) be real numbers, such that
\[
a^2+b^2+c^2+abc=5
\]\[
a+b+c=3
\]Prove that atleast one of the numbers \(a,b,c\) is equal to \( 2\).
2 replies
Darealzolt
Yesterday at 11:31 AM
sqing
Today at 1:25 AM
Interesting Geometry
captainmath99   4
N Yesterday at 8:01 PM by captainmath99
Let ABC be a right triangle such that $\angle{C}=90^\circ, CA=6, CB=4$. A circle O with center C has a radius of 2. Let P be a point on the circle O.

a)What is the minimum value of $(AP+\dfrac{1}{2}BP)$?
Answer Check

b) What is the minimum value of $(\dfrac{1}{3}AP+BP)$?
Answer Check
4 replies
captainmath99
May 25, 2025
captainmath99
Yesterday at 8:01 PM
Looking for even one person to study math.
abduqahhor_math   2
N Yesterday at 6:25 PM by EaZ_Shadow
Hi guys,I am looking for a person to study math topics related to olympiad.I have just finished 10th grade
2 replies
abduqahhor_math
Yesterday at 5:22 PM
EaZ_Shadow
Yesterday at 6:25 PM
Inequalities
lgx57   0
Yesterday at 3:55 PM
Let $a,b,c,d,e \ge 0$,$\sum \dfrac{1}{a+4}=1$.Prove that:
$$\sum \dfrac{a}{a^2+4} \le 1$$
Let $x,y,z>0$.Prove that:
$$\sum (y+z)\sqrt{\dfrac{yz}{(z+x)(y+x)}} \ge x+y+z$$
0 replies
lgx57
Yesterday at 3:55 PM
0 replies
Find the sum of all the products a_i a_j
Darealzolt   1
N Yesterday at 2:13 PM by alexheinis
Among the 100 constants \( \{ a_1,a_2,a_3,\dots,a_{100} \} \),there are \(39\) equal to \( -1\), and \(61\) equal to \(1\). Find the sum of all the products \(a_i a_j\) , where \(a \leq i < j \leq 100\).
1 reply
Darealzolt
Yesterday at 11:24 AM
alexheinis
Yesterday at 2:13 PM
ez problem....
Cobedangiu   4
N Apr 19, 2025 by iniffur
Let $x,y \in Z$ and $xy \cancel \vdots7$
Find $n \in Z^+$.
$x^2+y^2+xy=7^n$
4 replies
Cobedangiu
Apr 18, 2025
iniffur
Apr 19, 2025
ez problem....
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Cobedangiu
70 posts
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Let $x,y \in Z$ and $xy \cancel \vdots7$
Find $n \in Z^+$.
$x^2+y^2+xy=7^n$
This post has been edited 1 time. Last edited by Cobedangiu, Apr 18, 2025, 11:08 AM
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Cobedangiu
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Cobedangiu wrote:
Let $x,y \in Z$ and $xy \cancel \vdots7$
Find $n \in Z^+$.
$x^2+y^2+xy=7^n$

no :<?
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Lankou
1406 posts
#3
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What does the symbol $\cancel\vdots$ mean?
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giangtruong13
150 posts
#4 • 1 Y
Y by Lankou
Lankou wrote:
What does the symbol $\cancel\vdots$ mean?
It means $ 7 \cancel | xy $
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iniffur
538 posts
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$$x^2+xy+y^2=7^n~~~~~~~~~~~~~~~~~~(1)$$
In its present wording the problem does not require to find out the triplets $~(x,~y,n)~$,

but rather the powers of seven which can be represented as $~~x^2+xy+y^2,~~$ with the proviso that

$~7\nmid xy~~$

It can be derived from the literature regarding $~~x^2+xy+y^2=N~~$ that if $~(x,y)=(a,b),~(c,d)~~$ are

two solutions to $~x^2+xy+y^2=7^n,~~~(x,y)=(ac+bc+bd,~ad-bc)~~$ is a further solution by virtue

of the multiplicative property of numbers under the form $~~x^2+xy+y^2=N~~$ (see (*) below)

Therefore, it is sufficient to depart from two solutions, one selected from the group consisting of:

$x^2+xy+y^2=7~~\Longrightarrow (a,b)=(1,2), (2,1), (-1,3), (-2,3), (-3,2), (-3,1), (-2, -1), (-1,-2), (1,-3), (2, -3), (3,-2),$

$(3, -1)$

and the second one selected from the group consisting of:

$ x^2+xy+y^2=49~~\Longrightarrow (c,d)= (3,5), (5,3), (-3,8), (-5,8), (-8,5), (-8,3), (-5,-3),(-3,-5), (-8,3), (8,-5)$

to derive further solutions corresponding to any power of seven by applying the formula:

$(x,y)=(ac+bc+bd,~ad-bc)$

Of course, care should be taken that $~~7\nmid ac+bc+bd,~ad-bc~~$

Example 1

$a=1, b=2, c=3, d=5\Longrightarrow (1*3+2*3+2*5=19,~~1*5-2*3=-1)\Longrightarrow 19^2-19+1=343=7^3$

Example 2

$a=-3, b=1, c=19, d=-1\Longrightarrow (-39,-16)\Longrightarrow 39^2+39*16+16^2=2401=7^4$

Example 3

$a=8, b=-5, c=19, d=-1\Longrightarrow (62,~87)\Longrightarrow 62^2+62*87+87^2=16807=7^5$

And so on.

It might be possible to generalize this approach to the power of other primes (at first sight 3 and 13 could

work).



(*)Click to reveal hidden text
This post has been edited 1 time. Last edited by iniffur, Apr 20, 2025, 1:12 PM
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