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k a August Highlights and 2025 AoPS Online Class Information
jwelsh   0
Friday at 2:14 PM
CONGRATULATIONS to all the competitors at this year’s International Mathematical Olympiad (IMO)! The US Team took second place with 5 gold medals and 1 silver - we are proud to say that each member of the 2025 IMO team has participated in an AoPS WOOT (Worldwide Online Olympiad Training) class!

"As a parent, I'm deeply grateful to AoPS. Tiger has taken very few math courses outside of AoPS, except for a local Math Circle that doesn't focus on Olympiad math. AoPS has been one of the most important resources in his journey. Without AoPS, Tiger wouldn't be where he is today — especially considering he's grown up in a family with no STEM background at all."
— Doreen Dai, parent of IMO US Team Member Tiger Zhang

Interested to learn more about our WOOT programs? Check out the course page here or join a Free Scheduled Info Session. Early bird pricing ends August 19th!:
CodeWOOT Code Jam - Monday, August 11th
ChemWOOT Chemistry Jam - Wednesday, August 13th
PhysicsWOOT Physics Jam - Thursday, August 14th
MathWOOT Math Jam - Friday, August 15th

There is still time to enroll in our last wave of summer camps that start in August at the Virtual Campus, our video-based platform, for math and language arts! From Math Beasts Camp 6 (Prealgebra Prep) to AMC 10/12 Prep, you can find an informative 2-week camp before school starts. Plus, our math camps don’t have homework and cover cool enrichment topics like graph theory. Our language arts courses will build the foundation for next year’s challenges, such as Language Arts Triathlon for levels 5-6 and Academic Essay Writing for high school students.

Lastly, Fall is right around the corner! You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US. We’ve opened new Academy locations in San Mateo, CA, Pasadena, CA, Saratoga, CA, Johns Creek, GA, Northbrook, IL, and Upper West Side (NYC), New York.

Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.
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0 replies
jwelsh
Friday at 2:14 PM
0 replies
J
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Tricolor complete graphs
JustPostNorthKoreaTST   0
4 minutes ago
Source: 2015 North Korea Mathematical Olympiad P3
Consider a complete graph $K_n$ on $n$ vertices, where $n \ge 3$. Each edge is colored with one of three colors, and each color is used on at least one edge. Find the minimum positive integer $k$ such that for any such edge coloring and any color $C$ chosen from the three colors, it is possible to recolor at most $k$ edges to color $C$ so that the subgraph consisting of all edges of color $C$ is connected.
0 replies
JustPostNorthKoreaTST
4 minutes ago
0 replies
J
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How to solve this?
Mr._Calculator   0
17 minutes ago
In the diagram below, $ABC$ is a triangle, where the angle bisectors of $\angle ABC$ and $\angle ACB$ intersect at point $P$. Let points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, such that $DE$ passes through point $P$ and $\angle AED = \angle ABP$. If the areas of triangles $BDP$, $CEP$ and $BPC$ are equal to $18 \text{ cm}^2$, $36 \text{ cm}^2$ and $57 \text{ cm}^2$, respectively, then what is the area, in $\text{cm}^2$, of $ABC$?
0 replies
Mr._Calculator
17 minutes ago
0 replies
J
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Number Theory
JetFire008   1
N 26 minutes ago by blug
Source: Elementary Number Theory by David M. Burton
Modify Euclid's proof that there are infinitely many primes by assuming the existence of a largest prime $p$ and using the integer $N=p!+1$ to arrive at a contradiction.
1 reply
JetFire008
an hour ago
blug
26 minutes ago
J
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Partial products from p-2 numbers
JustPostNorthKoreaTST   2
N 27 minutes ago by navid
Source: 2015 North Korea Mathematical Olympiad P2
Let $p$ be an odd prime and $a_1,a_2,\ldots,a_{p-2}$ be positive integers (not necessarily different), such that for any $k \in \{1,2,\ldots,p-2\}$, we have $p \nmid a_k$ and $p \nmid (a_k^k-1)$. Show that one can pick some numbers from $a_1,a_2,\ldots,a_{p-2}$ such that their product is $\equiv 2 \pmod p$.
2 replies
JustPostNorthKoreaTST
3 hours ago
navid
27 minutes ago
J
No more topics!
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Parameter and 4 variables
mihaig   2
N Jun 3, 2025 by mihaig
Source: Own
Find the positive real constants $K$ such that
$$3\left(a^2+b^2+c^2+d^2\right)+4\left(abcd\right)^K\geq\left(a+b+c+d\right)^2$$for all $a,b,c,d\geq0$ satisfying $a+b+c+d\geq4.$
2 replies
mihaig
Jun 1, 2025
mihaig
Jun 3, 2025
J
Parameter and 4 variables
G H J
Reveal topic tags
inequalities
parameterization
L
G H BBookmark kLocked kLocked NReply
Source: Own
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mihaig
7555 posts
mihaig
#1 h  Jun 1, 2025, 9:33 AM
Y by
Find the positive real constants $K$ such that
$$3\left(a^2+b^2+c^2+d^2\right)+4\left(abcd\right)^K\geq\left(a+b+c+d\right)^2$$for all $a,b,c,d\geq0$ satisfying $a+b+c+d\geq4.$
Z K Y
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mihaig
7555 posts
mihaig
#2 h  Jun 1, 2025, 10:05 AM
Y by
Prove or disprove that
$$3\left(a^2+b^2+c^2+d^2\right)+4\left(abcd\right)^2\geq\left(a+b+c+d\right)^2$$for all $a,b,c,d\geq0$ satisfying $a+b+c+d\geq4.$
Z K Y
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mihaig
7555 posts
mihaig
#3 h  Jun 3, 2025, 9:26 AM
Y by
Prove or disprove that
$$3\left(a^2+b^2+c^2+d^2\right)+4\left(abcd\right)^3\geq\left(a+b+c+d\right)^2$$for all $a,b,c,d\geq0$ satisfying $a+b+c+d\geq4.$
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