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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.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

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.

Be sure to mark your calendars for the following events:
[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.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]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!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
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0 replies
jlacosta
Mar 2, 2025
0 replies
IMO 2009, Problem 5
orl   86
N 22 minutes ago by Ilikeminecraft
Source: IMO 2009, Problem 5
Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths
\[ a, f(b) \text{ and } f(b + f(a) - 1).\]
(A triangle is non-degenerate if its vertices are not collinear.)

Proposed by Bruno Le Floch, France
86 replies
orl
Jul 16, 2009
Ilikeminecraft
22 minutes ago
IMO 2023 P2
799786   88
N 24 minutes ago by Frd_19_Hsnzde
Source: IMO 2023 P2
Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E \neq A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$. Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$.
88 replies
799786
Jul 8, 2023
Frd_19_Hsnzde
24 minutes ago
Diagonals BD,CE concurrent with diameter AO in cyclic ABCDE
WakeUp   10
N 29 minutes ago by zhenghua
Source: Romanian TST 2002
Let $ABCDE$ be a cyclic pentagon inscribed in a circle of centre $O$ which has angles $\angle B=120^{\circ},\angle C=120^{\circ},$ $\angle D=130^{\circ},\angle E=100^{\circ}$. Show that the diagonals $BD$ and $CE$ meet at a point belonging to the diameter $AO$.

Dinu Șerbănescu
10 replies
WakeUp
Feb 5, 2011
zhenghua
29 minutes ago
Parallel lines in two-circle configuration
Tintarn   3
N 39 minutes ago by zhenghua
Source: Francophone 2024, Senior P3
Let $ABC$ be an acute triangle, $\omega$ its circumcircle and $O$ its circumcenter. The altitude from $A$ intersects $\omega$ in a point $D \ne A$ and the segment $AC$ intersects the circumcircle of $OCD$ in a point $E \ne C$. Finally, let $M$ be the midpoint of $BE$. Show that $DE$ is parallel to $OM$.
3 replies
Tintarn
Apr 4, 2024
zhenghua
39 minutes ago
No more topics!
Minimum number of values in the union of sets
bnumbertheory   4
N Oct 16, 2023 by math90
Source: Simon Marais Mathematics Competition 2023 Paper A Problem 3
For each positive integer $n$, let $f(n)$ denote the smallest possible value of $$|A_1 \cup A_2 \cup \dots \cup A_n|$$where $A_1, A_2, A_3 \dots A_n$ are sets such that $A_i \not\subseteq A_j$ and $|A_i| \neq |A_j|$ whenever $i \neq j$. Determine $f(n)$ for each positive integer $n$.
4 replies
bnumbertheory
Oct 14, 2023
math90
Oct 16, 2023
Minimum number of values in the union of sets
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G H BBookmark kLocked kLocked NReply
Source: Simon Marais Mathematics Competition 2023 Paper A Problem 3
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bnumbertheory
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#1
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For each positive integer $n$, let $f(n)$ denote the smallest possible value of $$|A_1 \cup A_2 \cup \dots \cup A_n|$$where $A_1, A_2, A_3 \dots A_n$ are sets such that $A_i \not\subseteq A_j$ and $|A_i| \neq |A_j|$ whenever $i \neq j$. Determine $f(n)$ for each positive integer $n$.
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alexheinis
10471 posts
#2
Y by
First we derive some small values and then we use two lemmata to show that $f(n)=n+2$ for $n\ge 3$.
We call a collection of sets good if the condition of the problem is satisfied, hence no inclusions and distinct cardinalities.
- we have $f(1)=0$ if you consider this case.
- suppose $A_1,A_2\subset [1,2]$ is good . Then the $|A_k|\in \{1,2\}$ are distinct, hence one of them equals 2. Contradiction. With $\{1\},\{2,3\}$ we have $f(2)=3$.
- suppose $A_1,A_2,A_3\subset [1,4]$ is good. Then the $|A_k|\in \{1,2,3\}$ are distinct, hence wlog $A_3=\{4\}$ and $A_1,A_2\subset \{1,2,3\}$ with cardinalities 2,3. Hence one of them is $\{1,2,3\}$, contradiction. With $\{1,4\},\{2,3,4\},\{5\}$ we see that $f(3)=5$.

Lemma 1. Suppose $A_1,\cdots, A_n\subset [1,n+2]$ are good with cardinalities $1,\cdots,n$. Then we can construct the same for $n+2$. Proof: let $B_k:=A_k\cup \{n+3\}, B_{n+1}:=[1,n+2],B_{n+2}=\{n+4\}$.

With induction base $n=2,3$ we find that $f(n)\le n+2$ for all $n\ge 2$.

Lemma 2. Suppose that $n\ge 3$ and $f(n)\le n+1$. Then also $f(n-1)\le n$.
Proof: suppose that $A_1,\cdots,A_n\subset [1,n+1]$ is good. The $|A_k|\in [1,n]$ are distinct hence wlog $A_n=\{n+1\}$. Then $A_1,\cdots, A_{n-1}\subset [1,n]$ are good.

Now suppose that $f(n)\le n+1$ for some $n\ge 3$. Then $n>3$ and we can use Lemma 2 to obtain $f(n-1)\le n$ etc until $f(3)\le 4$. Contradiction, hence $f(n)=n+2$ for $n\ge 3$.
This post has been edited 2 times. Last edited by alexheinis, Oct 15, 2023, 8:37 PM
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math90
1474 posts
#3
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Above: nice solution. BTW, you can prove that $f(n)\ge n+2$ for $n\ge 3$ directly: suppose that $A_1,\cdots,A_n\subset [1,n+1]$ is good. Then since $A_i\not\subset A_j$ and $|A_i|\ne |A_j|$ for $i\ne j$ and $n\ge 3$, the sizes of $A_i$ are $1,\ldots,n$. Hence we can assume WLOG $A_n=\{n+1\}$. Then $A_1,\cdots, A_{n-1}\subset [1,n]$ are good of sizes $2,\ldots,n$. WLOG $A_{n-1}=\{1,\ldots,n\}$. But then $A_{n-2}\subset A_{n-1}$ (here we use the assumption $n\ge 3$), contradiction.
This post has been edited 4 times. Last edited by math90, Oct 16, 2023, 10:17 PM
Reason: Thanks @below
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alexheinis
10471 posts
#5 • 1 Y
Y by math90
@math90: Thank you. What you write is nearly correct, we don't have inclusions in a good collection hence $A_i\subset A_j$ is not true when $i<j$. The cardinalities are distinct, however, and that's why we have $1,\cdots,n$ for the cardinalities. The rest of your argument is fine.
This post has been edited 1 time. Last edited by alexheinis, Oct 16, 2023, 9:33 PM
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math90
1474 posts
#6
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Thanks, corrected now.
This post has been edited 1 time. Last edited by math90, Oct 16, 2023, 10:16 PM
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