and 
,
in degrees as 
ranges over all real numbers.
and let 
(ie. 
with 
removed). Determine the number of injective* functions 
such that 
for all 
.
for all distinct 
and 
in the domain of 
possible triangles instead of 
)
be the centroid of 
.
as the centroid of 
and define 
and 
similarly. Points 
and 
are selected randomly from the sets 
,
,
,![$\frac{[G_AG_BG_c]}{[ABC]}.$](http://latex.artofproblemsolving.com/d/3/8/d38c1ec527c34f1e1519bba772a0765849e6fe7d.png)
chessboard, 
white knights and 
square vertically from the other, or one knight is 
Y by
Let a, b, c be positive real numbers. What is the minimum value of the following expression?
(18ca + 12bc + ab)² / (36abc² + 2ab²c + 3a²bc)
(18ca + 12bc + ab)² / (36abc² + 2ab²c + 3a²bc)
G
Topic
First Poster
Last Poster
k a June Highlights and 2025 AoPS Online Class Information
jlacosta 0
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!
Summer camps are starting this 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 a transformative 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)!
Be sure to mark your calendars for the following upcoming events:
[list][*]June 5th, Thursday, 7:30pm ET: Open Discussion with Ben Kornell and Andrew Sutherland, Art of Problem Solving's incoming CEO Ben Kornell and CPO Andrew Sutherland host an Ask Me Anything-style chat. Come ask your questions and get to know our incoming CEO & CPO!
[*]June 9th, Monday, 7:30pm ET, Game Jam: Operation Shuffle!, Come join us to play our second round of Operation Shuffle! If you enjoy number sense, logic, and a healthy dose of luck, this is the game for you. No specific math background is required; all are welcome.[/list]
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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[list][*]MATHCOUNTS/AMC 8 Basics
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[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!
Summer camps are starting this 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 a transformative 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)!
Be sure to mark your calendars for the following upcoming events:
[list][*]June 5th, Thursday, 7:30pm ET: Open Discussion with Ben Kornell and Andrew Sutherland, Art of Problem Solving's incoming CEO Ben Kornell and CPO Andrew Sutherland host an Ask Me Anything-style chat. Come ask your questions and get to know our incoming CEO & CPO!
[*]June 9th, Monday, 7:30pm ET, Game Jam: Operation Shuffle!, Come join us to play our second round of Operation Shuffle! If you enjoy number sense, logic, and a healthy dose of luck, this is the game for you. No specific math background is required; all are welcome.[/list]
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.
Introductory: Grades 5-10
Prealgebra 1 Self-Paced
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0 replies
Daily Problem Writing Practice
KSH31415 15
N
by KSH31415
I'm trying to get better at writing problems so I decided to challenge myself to write one problem for every day this month (June 2025). I will post them in this thread as well as edit this post with all of them in hide tags. If I can, I'll include a difficulty level in the form of AIME placement. If anybody wants to solve them and give feedback on the problem and/or my difficulty rating, please do!
June 1 (AIME P4)
A bag contains
red balls and blue balls. A draw consists of randomly selecting 
of the remaining balls from bag without replacement, and then setting them aside. A draw is called a match if the two balls have the same color. Compute the expected number of draws until either a match occurs or the bag is empty.
June 1 (AIME P4)
A bag contains
red balls and blue balls. A draw consists of randomly selecting 
of the remaining balls from bag without replacement, and then setting them aside. A draw is called a match if the two balls have the same color. Compute the expected number of draws until either a match occurs or the bag is empty.
15 replies
(own problem) integration+number theory
nithish_kumar138 0
Solve integration with limits 0 to π 1/(1+tanx)+1/(1+cotx)dx result is defined as k
N==K+i²+log10(k) +1/4k -sin(1.21×10^-5 k).
Then N yield a value close to following options
A.e(Eulers number or exponential)
B.π
C.c(speed of light)
..............Nithish kumar
N==K+i²+log10(k) +1/4k -sin(1.21×10^-5 k).
Then N yield a value close to following options
A.e(Eulers number or exponential)
B.π
C.c(speed of light)
..............Nithish kumar
0 replies
Geometry
Exoticbuttersowo 0
In an isosceles triangle ABC with base AC and interior cevian AM, such that MC = 2 MB, and a point L on AM, such that BLC = 90 degrees and MAC = 42 degrees. Determine LBC.
0 replies
(own problem) polynomials
nithish_kumar138 2
N
by Mathelets
When x^4-2x^2-3x^2+7x-2 is divided by x-2 the remainder is 'K'.
By using K find α,β,γ.
Let K= α²+β²+γ²
N=(α+β)²+(β+ γ)²+(γ+α)²
K'= α*β*γ/α+β.
Find K+N+K'....~Nithish
By using K find α,β,γ.
Let K= α²+β²+γ²
N=(α+β)²+(β+ γ)²+(γ+α)²
K'= α*β*γ/α+β.
Find K+N+K'....~Nithish
2 replies
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