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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
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Inequality => square
Rushil   12
N 7 minutes ago by ohiorizzler1434
Source: INMO 1998 Problem 4
Suppose $ABCD$ is a cyclic quadrilateral inscribed in a circle of radius one unit. If $AB \cdot BC \cdot CD \cdot DA \geq 4$, prove that $ABCD$ is a square.
12 replies
+1 w
Rushil
Oct 7, 2005
ohiorizzler1434
7 minutes ago
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p + q + r + s = 9 and p^2 + q^2 + r^2 + s^2 = 21
who   28
N 24 minutes ago by asdf334
Source: IMO Shortlist 2005 problem A3
Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.
28 replies
who
Jul 8, 2006
asdf334
24 minutes ago
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H not needed
dchenmathcounts   44
N an hour ago by Ilikeminecraft
Source: USEMO 2019/1
Let $ABCD$ be a cyclic quadrilateral. A circle centered at $O$ passes through $B$ and $D$ and meets lines $BA$ and $BC$ again at points $E$ and $F$ (distinct from $A,B,C$). Let $H$ denote the orthocenter of triangle $DEF.$ Prove that if lines $AC,$ $DO,$ $EF$ are concurrent, then triangle $ABC$ and $EHF$ are similar.

Robin Son
44 replies
dchenmathcounts
May 23, 2020
Ilikeminecraft
an hour ago
J
H
IZHO 2017 Functional equations
user01   51
N an hour ago by lksb
Source: IZHO 2017 Day 1 Problem 2
Find all functions $f:R \rightarrow R$ such that $$(x+y^2)f(yf(x))=xyf(y^2+f(x))$$, where $x,y \in \mathbb{R}$
51 replies
user01
Jan 14, 2017
lksb
an hour ago
J
No more topics!
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Counting Circles and Maximizing Edges
steven_zhang123   0
Today at 5:32 AM
Source: China TST 2001 Quiz 4 P2.1
Let the vertex set \( V \) of a graph be partitioned into \( h \) parts \( (V = V_1 \cup V_2 \cup \cdots \cup V_h) \), with \(|V_1| = n_1, |V_2| = n_2, \ldots, |V_h| = n_h \). If there is an edge between any two vertices only when they belong to different parts, the graph is called a complete \( h \)-partite graph, denoted as \( k(n_1, n_2, \ldots, n_h) \). Let \( n \) and \( r \) be positive integers, \( n \geq 6 \), \( r \leq \frac{2}{3}n \). Consider the complete \( r + 1 \)-partite graph \( k\left(\underbrace{1, 1, \ldots, 1}_{r}, n - r\right) \).

Answer the following questions:
1. Find the maximum number of disjoint circles (i.e., circles with no common vertices) in this complete \( r + 1 \)-partite graph.
2. Given \( n \), for all \( r \leq \frac{2}{3}n \), find the maximum number of edges in a complete \( r + 1 \)-partite graph \( k(1, 1, \ldots, 1, n - r) \) where no more than one circle is disjoint.
0 replies
steven_zhang123
Today at 5:32 AM
0 replies
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Counting Circles and Maximizing Edges
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Reveal topic tags
graph
graph theory
combinatorics
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Source: China TST 2001 Quiz 4 P2.1
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steven_zhang123
379 posts
steven_zhang123
#1 h  Today at 5:32 AM
Y by
Let the vertex set \( V \) of a graph be partitioned into \( h \) parts \( (V = V_1 \cup V_2 \cup \cdots \cup V_h) \), with \(|V_1| = n_1, |V_2| = n_2, \ldots, |V_h| = n_h \). If there is an edge between any two vertices only when they belong to different parts, the graph is called a complete \( h \)-partite graph, denoted as \( k(n_1, n_2, \ldots, n_h) \). Let \( n \) and \( r \) be positive integers, \( n \geq 6 \), \( r \leq \frac{2}{3}n \). Consider the complete \( r + 1 \)-partite graph \( k\left(\underbrace{1, 1, \ldots, 1}_{r}, n - r\right) \).

Answer the following questions:
1. Find the maximum number of disjoint circles (i.e., circles with no common vertices) in this complete \( r + 1 \)-partite graph.
2. Given \( n \), for all \( r \leq \frac{2}{3}n \), find the maximum number of edges in a complete \( r + 1 \)-partite graph \( k(1, 1, \ldots, 1, n - r) \) where no more than one circle is disjoint.
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