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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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0 replies
jwelsh
Jul 1, 2025
0 replies
Interesting Spiral
VitaPretor   0
2 minutes ago
a) We start at $(0,0)$ and walk $400$ feet north and turn $90$ degrees to the right.
We then walk 75% of $400$ or $300$ feet east and turn $90$ degrees to the right.
We next walk 75% of $300$ or $225$ feet south and turn $90$ degrees to the right.
We repeat the process indefinitely of walking 75% of the distance that we last walked and turning $90$ degrees to the right forming a spiral. What are the exact coordinates we approach after repeating the process indefinitely?

b) We start at $(0,0)$ and walk $x$ feet north and turn $90$ degrees to the right.
We then walk $x * y$ feet to the east and turn $90$ degrees to the right.
We next walk $x * y^2$ feet south and turn $90$ degrees to the right.
We repeat the process indefinitely of walking $y$ times the distance that we last walked and turning $90$ degrees to the right forming a spiral.
Assume $x$ > $0$ and $0$ < $y$ < $1$. If the point we eventually approach is $(50,60)$ find the ordered pair $(x,y)$.
0 replies
VitaPretor
2 minutes ago
0 replies
Problem3
samithayohan   119
N 36 minutes ago by ihategeo_1969
Source: IMO 2015 problem 3
Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma $ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90^{\circ}$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90^{\circ}$. Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different and lie on $\Gamma$ in this order.

Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.

Proposed by Ukraine
119 replies
samithayohan
Jul 10, 2015
ihategeo_1969
36 minutes ago
DIT(to) length chase :(
TestX01   4
N an hour ago by Entei
Source: 2024 UWUMO
Let $ABC$ be an acute triangle with circumcenter $O$. Let $D$ be a point on $(BOC)$, and let $OD$ intersect lines $AB,AC$ at $E,F$ respectively. Suppose the perpendicular line to $AB$ through $E$ intersects the perpendicular line to $AC$ through $F$ at $P$. Prove that $D,P$, and the reflection of $A$ over $O$ are collinear.
4 replies
TestX01
Jul 15, 2024
Entei
an hour ago
Problem 1
randomusername   79
N 2 hours ago by ihategeo_1969
Source: IMO 2015, Problem 1
We say that a finite set $\mathcal{S}$ of points in the plane is balanced if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is centre-free if for any three different points $A$, $B$ and $C$ in $\mathcal{S}$, there is no points $P$ in $\mathcal{S}$ such that $PA=PB=PC$.

(a) Show that for all integers $n\ge 3$, there exists a balanced set consisting of $n$ points.

(b) Determine all integers $n\ge 3$ for which there exists a balanced centre-free set consisting of $n$ points.

Proposed by Netherlands
79 replies
randomusername
Jul 10, 2015
ihategeo_1969
2 hours ago
Putnam 2013 A5
Kent Merryfield   11
N Jul 7, 2025 by quantam13
For $m\ge 3,$ a list of $\binom m3$ real numbers $a_{ijk}$ $(1\le i<j<k\le m)$ is said to be area definite for $\mathbb{R}^n$ if the inequality \[\sum_{1\le i<j<k\le m}a_{ijk}\cdot\text{Area}(\triangle A_iA_jA_k)\ge0\] holds for every choice of $m$ points $A_1,\dots,A_m$ in $\mathbb{R}^n.$ For example, the list of four numbers $a_{123}=a_{124}=a_{134}=1, a_{234}=-1$ is area definite for $\mathbb{R}^2.$ Prove that if a list of $\binom m3$ numbers is area definite for $\mathbb{R}^2,$ then it is area definite for $\mathbb{R}^3.$
11 replies
Kent Merryfield
Dec 9, 2013
quantam13
Jul 7, 2025
Putnam 2004 A5
Kent Merryfield   11
N Jun 30, 2025 by Ritwin
An $m\times n$ checkerboard is colored randomly: each square is independently assigned red or black with probability $\frac12.$ we say that two squares, $p$ and $q$, are in the same connected monochromatic region if there is a sequence of squares, all of the same color, starting at $p$ and ending at $q,$ in which successive squares in the sequence share a common side. Show that the expected number of connected monochromatic regions is greater than $\frac{mn}8.$
11 replies
Kent Merryfield
Dec 11, 2004
Ritwin
Jun 30, 2025
Interesting geometry walking problem
Hihihihihihihihihihihihi   3
N Jun 29, 2025 by greenturtle3141
Wondering if there is or is not an answer to a problem that I came up with a few months ago. Starting at the origin of the 2D plane is there a way walk to all integers coordinates moving only left up down and right (no diagonals) integer length such that all distances used are done so finitely many times.
To be clear if you go from (0,0) to (0,2) (0,1) is not covered nor are any other in-between points of any two coordinates otherwise you could just do a square spiral.
For the 1 dimensional case you can easily do it with the sequence 0,1,-1,2,-2... .
The 2 dimensional case can be seen as a sequence of gaussian integers to make it so we only have to deal with 1 number instead of 2 and since all gaussian integers need to be listed in the sequence the sequence can be seen as a bijection from the natural numbers to the gaussian integers. We know that a bijection must exist because of the square spiral but it must have the extra property that the difference in terms must =m or mi for some integer m. Please tell me where this isn't clear if it isn't.
3 replies
Hihihihihihihihihihihihi
Jun 29, 2025
greenturtle3141
Jun 29, 2025
Putnam 1972 B5
sqrtX   1
N Jun 25, 2025 by pikapika007
Source: Putnam 1972
Let $A,B,C$ and $D$ be non-coplanar points such that $\angle ABC=\angle ADC$ and $\angle BAD=\angle BCD$.
Show that $AB=CD$ and $AD=BC$.
1 reply
sqrtX
Feb 17, 2022
pikapika007
Jun 25, 2025
Putnam 2006 B1
Kent Merryfield   55
N Jun 17, 2025 by Jndd
Show that the curve $x^{3}+3xy+y^{3}=1$ contains only one set of three distinct points, $A,B,$ and $C,$ which are the vertices of an equilateral triangle.
55 replies
Kent Merryfield
Dec 4, 2006
Jndd
Jun 17, 2025
Two convex bodies with disjoint projections
Phorphyrion   1
N Jun 16, 2025 by GoldenBoy03
Source: 2023 IMC P9
We say that a real number $V$ is good if there exist two closed convex subsets $X$, $Y$ of the unit cube in $\mathbb{R}^3$, with volume $V$ each, such that for each of the three coordinate planes (that is, the planes spanned by any two of the three coordinate axes), the projections of $X$ and $Y$ onto that plane are disjoint.

Find $\sup \{V\mid V\ \text{is good}\}$.
1 reply
Phorphyrion
Aug 3, 2023
GoldenBoy03
Jun 16, 2025
What is a good book or website on Group Theory?
mathps7   1
N Jun 16, 2025 by Silver08
Hi everyone,

I am interested in learning Group Theory. I know a little bit about Group Theory... I know that it was developed by Evariste Galois to help him find the roots of polynomials. In particular, Galois proved that a polynomial of degree five (or higher) is not solvable by radicals in the general form. That is, P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f is not solvable by radicals. On the other hand, a polynomial of degree 2, 3, or 4 is solvable by radicals. This is a really interesting theorem in Group Theory. (It is often called the Abel-Ruffini theorem). One reason I want to learn Group Theory is so that I can prove this theorem and get a better understanding of polynomials.

I wanted to give you some background on why I want to learn Group Theory. (I would like to learn how to prove the Abel-Ruffini theorem, and I would also like to know how Group Theory can be used to solve Rubik's cubes.)

Having said all this, what books would you recommend for learning Group Theory? What are your favorite books on Group Theory? I notice that the AoPS Group Theory course uses a book called "Groups and Fields" by D. Jeremy Copeland. Do you know where I can find Jeremy Copeland's book? I searched it online, and also on Amazon, but I couldn't find it online, and I couldn't find it on Amazon.

If there are any good websites for learning Group Theory, I'd be interested to hear about these websites as well. I'm really looking for books and websites that teach Group Theory.

Thanks for reading my post. I appreciate any feedback.
1 reply
mathps7
Jun 16, 2025
Silver08
Jun 16, 2025
Geometry
krrispes   1
N Jun 4, 2025 by Mathzeus1024
Plz solve by using optimization method
1 reply
krrispes
Jan 19, 2025
Mathzeus1024
Jun 4, 2025
Classifying Math with Symbols Based on Behavior
Midevilgmer   2
N May 22, 2025 by Midevilgmer
I have been working on a new math idea that prioritizes the behavior of numbers, equations, and expressions rather than their exact values. Numbers are described using symbols like P (Positive), N (Negative), Z (Zero), I (Imaginary), and D (Decimal). The goal is to create a system that uses symbols that allows you to perform operations like P*D and P+N and determine the behavioral outcomes based on the properties involved. For example, instead of identifying a number as 3, you would describe it as a positive, odd, whole, prime number, allowing you work with those traits individually or together. I would like to mention that I already have created an Addition, Subtraction, Multiplication, Division, Square Root, Exponent, and Factorial table that shows how these different behaviors work in basic operations. Finally, I want to mention that my current background includes a knowledge of geometry, algebra, and a very little amount of calculus. Any thoughts or ideas would be appreciated.
2 replies
Midevilgmer
May 22, 2025
Midevilgmer
May 22, 2025
Collinearity in a Harmonic Configuration from a Cyclic Quadrilateral
kieusuong   0
May 15, 2025
Let \((O)\) be a fixed circle, and let \(P\) be a point outside \((O)\) such that \(PO > 2r\). A variable line through \(P\) intersects the circle \((O)\) at two points \(M\) and \(N\), such that the quadrilateral \(ANMB\) is cyclic, where \(A, B\) are fixed points on the circle.

Define the following:
- \(G = AM \cap BN\),
- \(T = AN \cap BM\),
- \(PJ\) is the tangent from \(P\) to the circle \((O)\), and \(J\) is the point of tangency.

**Problem:**
Prove that for all such configurations:
1. The points \(T\), \(G\), and \(J\) are collinear.
2. The line \(TG\) is perpendicular to chord \(AB\).
3. As the line through \(P\) varies, the point \(G\) traces a fixed straight line, which is parallel to the isogonal conjugate axis (the so-called *isotropic line*) of the centers \(O\) and \(P\).

---

### Outline of a Synthetic Proof:

**1. Harmonic Configuration:**
- Since \(A, N, M, B\) lie on a circle, their cross-ratio is harmonic:
\[
  (ANMB) = -1.
  \]- The intersection points \(G = AM \cap BN\), and \(T = AN \cap BM\) form a well-known harmonic setup along the diagonals of the quadrilateral.

**2. Collinearity of \(T\), \(G\), \(J\):**
- The line \(PJ\) is tangent to \((O)\), and due to harmonicity and projective duality, the polar of \(G\) passes through \(J\).
- Thus, \(T\), \(G\), and \(J\) must lie on a common line.

**3. Perpendicularity:**
- Since \(PJ\) is tangent at \(J\) and \(AB\) is a chord, the angle between \(PJ\) and chord \(AB\) is right.
- Therefore, line \(TG\) is perpendicular to \(AB\).

**4. Quasi-directrix of \(G\):**
- As the line through \(P\) varies, the point \(G = AM \cap BN\) moves.
- However, all such points \(G\) lie on a fixed line, which is perpendicular to \(PO\), and is parallel to the isogonal (or isotropic) line determined by the centers \(O\) and \(P\).

---

**Further Questions for Discussion:**
- Can this configuration be extended to other conics, such as ellipses?
- Is there a pure projective geometry interpretation (perhaps using polar reciprocity)?
- What is the locus of point \(T\), or of line \(TG\), as \(P\) varies?

*This configuration arose from a geometric investigation involving cyclic quadrilaterals and harmonic bundles. Any insights, counterexamples, or improvements are warmly welcomed.*
0 replies
kieusuong
May 15, 2025
0 replies
Geometry
Arytva   2
N Jun 6, 2025 by MathsII-enjoy
Source: Source?
Let two circles \(\omega_1\) and \(\omega_2\) meet at two distinct points \(X\) and \(Y\). Choose any line \(\ell\) through \(X\), and let \(\ell\) meet \(\omega_1\) again at \(A\) (other than \(X\)) and meet \(\omega_2\) again at \(B\). On \(\omega_1\), let \(M\) be the midpoint of the minor arc \(AY\) (i.e., the point on \(\omega_1\) such that \(\angle AMY\) subtends the arc \(AY\)), and on \(\omega_2\) let \(N\) be the midpoint of the minor arc \(BY\). Prove that
\[
MN \parallel \text{(radical axis of } \omega_1, \omega_2).
\]
2 replies
Arytva
Jun 6, 2025
MathsII-enjoy
Jun 6, 2025
Geometry
G H J
G H BBookmark kLocked kLocked NReply
Source: Source?
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Arytva
122 posts
#1
Y by
Let two circles \(\omega_1\) and \(\omega_2\) meet at two distinct points \(X\) and \(Y\). Choose any line \(\ell\) through \(X\), and let \(\ell\) meet \(\omega_1\) again at \(A\) (other than \(X\)) and meet \(\omega_2\) again at \(B\). On \(\omega_1\), let \(M\) be the midpoint of the minor arc \(AY\) (i.e., the point on \(\omega_1\) such that \(\angle AMY\) subtends the arc \(AY\)), and on \(\omega_2\) let \(N\) be the midpoint of the minor arc \(BY\). Prove that
\[
MN \parallel \text{(radical axis of } \omega_1, \omega_2).
\]
This post has been edited 1 time. Last edited by Arytva, Jun 6, 2025, 9:30 AM
Z K Y
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aaravdodhia
2665 posts
#2
Y by
Not true, unless I'm missing something?
Attachments:
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MathsII-enjoy
97 posts
#3
Y by
aaravdodhia wrote:
Not true, unless I'm missing something?

I see $X,M,N$ collinear and its lemma can be proved easily.
Z K Y
N Quick Reply
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