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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.

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[*]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
J
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algebraic inequality
produit   0
3 minutes ago
Positive a, b, c satisfy a + b + c = ab + bc + ca. Prove that
a + b + c + 1 ⩾ 4abc.
0 replies
produit
3 minutes ago
0 replies
J
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pqr/uvw convert
Nguyenhuyen_AG   9
N 7 minutes ago by Rhapsodies_pro
Source: https://github.com/nguyenhuyenag/pqr_convert
Hi everyone,
As we know, the pqr/uvw method is a powerful and useful tool for proving inequalities. However, transforming an expression $f(a,b,c)$ into $f(p,q,r)$ or $f(u,v,w)$ can sometimes be quite complex. That's why I’ve written a program to assist with this process.
I hope you’ll find it helpful!

Download: pqr_convert

Screenshot:
IMAGE
IMAGE
9 replies
Nguyenhuyen_AG
Apr 19, 2025
Rhapsodies_pro
7 minutes ago
J
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interesting functional
Pomegranat   0
22 minutes ago
Source: I don't know sorry
Find all functions \( f : \mathbb{R}^+ \to \mathbb{R}^+ \) such that for all positive real numbers \( x \) and \( y \), the following equation holds:
\[ \frac{x + f(y)}{x f(y)} = f\left( \frac{1}{y} + f\left( \frac{1}{x} \right) \right) \]
0 replies
Pomegranat
22 minutes ago
0 replies
J
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Brilliant guessing game on triples
Assassino9931   0
an hour ago
Source: Al-Khwarizmi Junior International Olympiad 2025 P8
There are $100$ cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from $1$ to $100$. In a move, Madina is allowed to choose any $3$ cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?

Shubin Yakov, Russia
0 replies
Assassino9931
an hour ago
0 replies
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Number of real roots
girishpimoli   5
N 4 hours ago by mrgenius000
If $f(x)=x^2-2x$. Then number of real roots of $f(f(f(f(x))))=3$
5 replies
girishpimoli
Today at 3:44 AM
mrgenius000
4 hours ago
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Proving that the line passes through the midpoint.
MTA_2024   2
N 6 hours ago by Royal_mhyasd
Let $ABC$ be a triangle of orthocenter $H$. The circle of diameter $AC$ and the circumcircle of triangle $AHB$ intersect a second time in $K$.
Prove that the line $(CK)$ passes through the midpoint of segment $HB$.
2 replies
MTA_2024
May 7, 2025
Royal_mhyasd
6 hours ago
J
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Square number
linkxink0603   4
N Today at 3:29 AM by pooh123
Find m is positive interger such that m^4+3^m is square number
4 replies
linkxink0603
Yesterday at 11:20 AM
pooh123
Today at 3:29 AM
J
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Inequalities
sqing   7
N Today at 2:01 AM by sqing
Let $ a,b>0, a^2+ab+b^2 \geq 6 $. Prove that
$$a^4+ab+b^4\geq 10$$Let $ a,b>0, a^2+ab+b^2 \leq \sqrt{10} $. Prove that
$$a^4+ab+b^4 \leq 10$$Let $ a,b>0, a^2+ab+b^2 \geq \frac{15}{2} $. Prove that
$$ a^4-ab+b^4\geq 10$$Let $ a,b>0, a^2+ab+b^2 \leq \sqrt{10} $. Prove that
$$-\frac{1}{8}\leq a^4-ab+b^4\leq 10$$
7 replies
sqing
Thursday at 2:42 PM
sqing
Today at 2:01 AM
J
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Compilation of functions problems
Saucepan_man02   2
N Today at 12:45 AM by Saucepan_man02
Could anyone post some handout/compilation of problems related to functions (difficulty similar to AIME/ARML/HMMT etc)?

Thanks..
2 replies
Saucepan_man02
May 7, 2025
Saucepan_man02
Today at 12:45 AM
J
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How many triangles
Ecrin_eren   5
N Today at 12:10 AM by jasperE3


"Inside a triangle, 2025 points are placed, and each point is connected to the vertices of the smallest triangle that contains it. In the final state, how many small triangles are formed?"


5 replies
Ecrin_eren
May 2, 2025
jasperE3
Today at 12:10 AM
J
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Triangle on a tetrahedron
vanstraelen   2
N Yesterday at 7:51 PM by ReticulatedPython

Given a regular tetrahedron $(A,BCD)$ with edges $l$.
Construct at the apex $A$ three perpendiculars to the three lateral faces.
Take a point on each perpendicular at a distance $l$ from the apex such that these three points lie above the apex.
Calculate the lenghts of the sides of the triangle.
2 replies
vanstraelen
Yesterday at 2:43 PM
ReticulatedPython
Yesterday at 7:51 PM
J
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shadow of a cylinder, shadow of a cone
vanstraelen   2
N Yesterday at 6:33 PM by vanstraelen

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?
2 replies
vanstraelen
Yesterday at 3:08 PM
vanstraelen
Yesterday at 6:33 PM
J
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2023 Official Mock NAIME #15 f(f(f(x))) = f(f(x))
parmenides51   3
N Yesterday at 5:13 PM by jasperE3
How many non-bijective functions $f$ exist that satisfy $f(f(f(x))) = f(f(x))$ for all real $x$ and the domain of f is strictly within the set of $\{1,2,3,5,6,7,9\}$, the range being $\{1,2,4,6,7,8,9\}$?

Even though this is an AIME problem, a proof is mandatory for full credit. Constants must be ignored as we dont want an infinite number of solutions.
3 replies
parmenides51
Dec 4, 2023
jasperE3
Yesterday at 5:13 PM
J
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Geometry
AlexCenteno2007   3
N Yesterday at 4:18 PM by AlexCenteno2007
Let ABC be an acute triangle and let D, E and F be the feet of the altitudes from A, B and C respectively. The straight line EF and the circumcircle of ABC intersect at P such that F is between E and P, the straight lines BP and DF intersect at Q. Show that if ED = EP then CQ and DP are parallel.
3 replies
AlexCenteno2007
Apr 28, 2025
AlexCenteno2007
Yesterday at 4:18 PM
J
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J
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Math question involving chess
kjhgyuio   0
Apr 17, 2025
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0 replies
kjhgyuio
Apr 17, 2025
0 replies
J
Math question involving chess
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Reveal topic tags
Maths
chess
probability
wait probability wrong tag
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kjhgyuio
61 posts
kjhgyuio
#1 h  Apr 17, 2025, 8:40 AM
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