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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next 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 an enriching 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][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
Prove the inequality
Butterfly   0
4 minutes ago

Let $a,b,c,d$ be positive real numbers. Prove $$a^2+b^2+c^2+d^2+abcd-3(a+b+c+d)+7\ge 0.$$
0 replies
Butterfly
4 minutes ago
0 replies
Inspired by SXJX (12)2022 Q1167
sqing   2
N 22 minutes ago by pooh123
Source: Own
Let $ a,b,c>0 $. Prove that$$\frac{kabc-1} {abc(a+b+c+8(2k-1))}\leq \frac{1}{16 }$$Where $ k>\frac{1}{2}.$
2 replies
+1 w
sqing
Yesterday at 4:01 AM
pooh123
22 minutes ago
Interesting inequalities
sqing   1
N 26 minutes ago by pooh123
Source: Own
Let $ a,b>0 $. Prove that
$$\frac{ab-1} {ab(a+b+2)} \leq \frac{1} {8}$$$$\frac{2ab-1} {ab(a+b+1)} \leq 6\sqrt 3-10$$
1 reply
sqing
Today at 4:24 AM
pooh123
26 minutes ago
Inequality with rational function
MathMystic33   4
N 27 minutes ago by ytChen
Source: Macedonian Mathematical Olympiad 2025 Problem 2
Let \( n > 2 \) be an integer, \( k > 1 \) a real number, and \( x_1, x_2, \ldots, x_n \) be positive real numbers such that \( x_1 \cdot x_2 \cdots x_n = 1 \). Prove that:

\[
\frac{1 + x_1^k}{1 + x_2} + \frac{1 + x_2^k}{1 + x_3} + \cdots + \frac{1 + x_n^k}{1 + x_1} \geq n.
\]
When does equality hold?
4 replies
MathMystic33
May 13, 2025
ytChen
27 minutes ago
Max min in geometry
son2007vn   0
Today at 5:33 AM
Given a triangle ABC and positive real numbers m, n, p, find the point M in the plane of the triangle such that m \cdot MA + n \cdot MB + p \cdot MC is minimized.
0 replies
son2007vn
Today at 5:33 AM
0 replies
EGMO help
mathprodigy2011   26
N Today at 5:24 AM by sunken rock
If we have a quadrilateral with 1 pair of parallel sides but the parallel sides are also equal, is that sufficient to stating the quadrilateral is a parallelogram. if it's not, please give a counter-example.
26 replies
mathprodigy2011
Sunday at 10:30 PM
sunken rock
Today at 5:24 AM
this problem pmo
derekli   2
N Today at 4:41 AM by Rabbit47
so I came across this aime #14 while practicing... i found the max value instead of min. ts pmo
2 replies
derekli
Today at 4:22 AM
Rabbit47
Today at 4:41 AM
Inequalities
sqing   9
N Today at 4:32 AM by sqing
Let $ a,b>0   $ . Prove that
$$ \frac{a}{a^2+a +2b+1}+ \frac{b}{b^2+2a +b+1}  \leq  \frac{2}{5} $$$$ \frac{a}{a^2+2a +b+1}+ \frac{b}{b^2+a +2b+1}  \leq  \frac{2}{5} $$
9 replies
sqing
May 13, 2025
sqing
Today at 4:32 AM
THREE People Meet at the SAME. TIME.
LilKirb   5
N Today at 3:13 AM by LilKirb
Three people arrive at the same place independently, at a random between $8:00$ and $9:00.$ If each person remains there for $20$ minutes, what's the probability that all three people meet each other?

I'm already familiar with the variant where there are only two people, where you Click to reveal hidden text It was an AIME problem from the 90s I believe. However, I don't know how one could visualize this in a Click to reveal hidden text Help on what to do?
5 replies
LilKirb
Yesterday at 1:06 PM
LilKirb
Today at 3:13 AM
collinear
spiralman   1
N Today at 2:32 AM by MathsII-enjoy
Given an acute triangle \( \triangle ABC \) with \( AB < AC \), inscribed in circle \( (O) \).
Let \( H \) be the orthocenter of triangle \( ABC \), and \( M \) be the midpoint of \( BC \).
A line passing through \( H \), parallel to \( AO \), intersects lines \( AB \) and \( AC \) at points \( D \) and \( E \), respectively.
Let \( K \) be the circumcenter of triangle \( ADE \). Prove that: Points \( H, K, M \) are collinear.

1 reply
spiralman
Yesterday at 5:50 PM
MathsII-enjoy
Today at 2:32 AM
shadow of a cylinder, shadow of a cone
vanstraelen   4
N Yesterday at 10:59 PM by mathafou

a) Given is a right cylinder of height $2R$ and radius $R$.
The sun shines on this solid at an angle of $45^{\circ}$.
What is the area of the shadow that this solid casts on the plane of the botom base?

b) Given is a right cone of height $2R$ and radius $R$.
The sun shines on this solid at an angle of $45^{\circ}$.
What is the area of the shadow that this solid casts on the plane of the base?
4 replies
vanstraelen
May 9, 2025
mathafou
Yesterday at 10:59 PM
Hard Problem help?
7168   5
N Yesterday at 10:22 PM by Shan3t
Suppose that a,b < or = 2010 are positive integers with b * Lcm(a,b) < or = a * gcd(a,b). Let N be the smallest positive integer value of n such that either nb/a and na/b are integers. What, over all a,b, is the greatest possible value of N?

THx!
5 replies
7168
Feb 25, 2011
Shan3t
Yesterday at 10:22 PM
Vieta's Relation
P162008   1
N Yesterday at 7:27 PM by vanstraelen
If $\alpha, \beta$ and $\gamma$ are the roots of the cubic equation $x^3 - x^2 - 2x + 1 = 0$ then compute $\sum_{cyc} (\alpha + \beta)^{1/3}.$
1 reply
P162008
Yesterday at 10:38 AM
vanstraelen
Yesterday at 7:27 PM
no of integer soultions of ||x| - 2020| < 5 - IOQM 2020-21 p5
parmenides51   12
N Yesterday at 7:19 PM by Yiyj
Find the number of integer solutions to $||x| - 2020| < 5$.
12 replies
parmenides51
Jan 18, 2021
Yiyj
Yesterday at 7:19 PM
Nice sequence problem.
mathlover1231   0
Apr 10, 2025
Source: Own
Scientists found a new species of bird called “N-coloured rainbow”. They also found out 3 interesting facts about the bird’s life: 1) every day, N-coloured rainbow is coloured in one of N colors.
2) every day, the color is different from yesterday (not every previous day, just yesterday).
3) there are no four days i, j, k, l in the bird’s life such that i<j<k<l with colours a, b, c, d respectively for which a=c ≠ b=d.
Find the greatest possible age (in days) of this bird as a function of N.
0 replies
mathlover1231
Apr 10, 2025
0 replies
Nice sequence problem.
G H J
G H BBookmark kLocked kLocked NReply
Source: Own
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mathlover1231
7 posts
#1
Y by
Scientists found a new species of bird called “N-coloured rainbow”. They also found out 3 interesting facts about the bird’s life: 1) every day, N-coloured rainbow is coloured in one of N colors.
2) every day, the color is different from yesterday (not every previous day, just yesterday).
3) there are no four days i, j, k, l in the bird’s life such that i<j<k<l with colours a, b, c, d respectively for which a=c ≠ b=d.
Find the greatest possible age (in days) of this bird as a function of N.
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