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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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[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.
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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
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Inequality
MathsII-enjoy   7
N 4 minutes ago by sqing
A interesting problem generalized :-D
7 replies
+1 w
MathsII-enjoy
Saturday at 1:59 PM
sqing
4 minutes ago
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F.E....can you solve it?
Jackson0423   0
25 minutes ago
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\left(\frac{x^2 - f(x)}{f(x) - 1}\right) = x \]for all real numbers \( x \) satisfying \( f(x) \neq 1 \).
0 replies
Jackson0423
25 minutes ago
0 replies
J
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Functional equation
Math-wiz   25
N an hour ago by Adywastaken
Source: IMOC SL A1
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$,
$$f(xy+f(x))=f(xf(y))+x$$
25 replies
Math-wiz
Dec 15, 2019
Adywastaken
an hour ago
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Nice numer theory
GeoArt   5
N an hour ago by Primeniyazidayi
$p$ is a prime number, $m, x, y$ are natural numbers ($m, x, y > 1$). It is known that $\frac{x^p + y^p}{2}$ $=$ $(\frac{x+y}{2} )^m$. Prove that $p = m$.
5 replies
GeoArt
Jan 7, 2021
Primeniyazidayi
an hour ago
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IOQM 2022-23 P-7
lifeismathematics   2
N 2 hours ago by Adywastaken
Find the number of ordered pairs $(a,b)$ such that $a,b \in \{10,11,\cdots,29,30\}$ and
$\hspace{1cm}$ $GCD(a,b)+LCM(a,b)=a+b$.
2 replies
lifeismathematics
Oct 30, 2022
Adywastaken
2 hours ago
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Inequalities
sqing   7
N 3 hours ago by sqing
Let $ a,b,c>0 $ and $ a+b\leq 16abc. $ Prove that
$$ a+b+kc^3\geq\sqrt[4]{\frac{4k} {27}}$$$$ a+b+kc^4\geq\frac{5} {8}\sqrt[5]{\frac{k} {2}}$$Where $ k>0. $
$$ a+b+3c^3\geq\sqrt{\frac{2} {3}}$$$$ a+b+2c^4\geq \frac{5} {8}$$
7 replies
sqing
Yesterday at 12:46 PM
sqing
3 hours ago
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Inequalities
sqing   10
N 4 hours ago by sqing
Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$$ (1-abc) (1-a)(1-b)(1-c) \ge 208 $$$$ (1+abc) (1-a)(1-b)(1-c) \le -224 $$$$(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840 $$
10 replies
sqing
Jul 12, 2024
sqing
4 hours ago
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China MO 1996 p1
math_gold_medalist28   1
N 4 hours ago by MathsII-enjoy
Let ABC be a triangle with orthocentre H. The tangent lines from A to the circle with diameter BC touch this circle at P and Q. Prove that H, P and Q are collinear.
1 reply
math_gold_medalist28
May 2, 2025
MathsII-enjoy
4 hours ago
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If it is an integer then perfect square
Ecrin_eren   1
N 4 hours ago by Pal702004


"Let a, b, c, d be non-zero digits, and let abcd and dcba represent four-digit numbers.

Show that if the number abcd / dcba is an integer, then that integer is a perfect square."



1 reply
Ecrin_eren
May 1, 2025
Pal702004
4 hours ago
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A Collection of Good Problems from my end
SomeonecoolLovesMaths   6
N 5 hours ago by SomeonecoolLovesMaths
This is a collection of good problems and my respective attempts to solve them. I would like to encourage everyone to post their solutions to these problems, if any. This will not only help others verify theirs but also perhaps bring forward a different approach to the problem. I will constantly try to update the pool of questions.

The difficulty level of these questions vary from AMC 10 to AIME. (Although the main pool of questions were prepared as a mock test for IOQM over the years)

Problem 1

Problem 2

Problem 3

Problem 4
6 replies
SomeonecoolLovesMaths
Yesterday at 8:16 AM
SomeonecoolLovesMaths
5 hours ago
J
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parallelogram in a tetrahedron
vanstraelen   0
Today at 6:43 AM
Given a tetrahedron $ABCD$ and a plane $\mu$, parallel with the edges $AC$ and $BD$.
$AB \cap \mu=P$.
a) Prove: the intersection of the tetrahedron with the plane is a parallelogram.
b) If $\left|AC\right|=14,\left|BD\right|=7$ and $\frac{\left|PA\right|}{\left|PB\right|}=\frac{3}{4}$,
calculates the lenghts of the sides of this parallelogram.
0 replies
vanstraelen
Today at 6:43 AM
0 replies
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Arithmetic Series and Common Differences
4everwise   6
N Today at 2:12 AM by epl1
For each positive integer $k$, let $S_k$ denote the increasing arithmetic sequence of integers whose first term is $1$ and whose common difference is $k$. For example, $S_3$ is the sequence $1,4,7,10,...$. For how many values of $k$ does $S_k$ contain the term $2005$?
6 replies
4everwise
Nov 10, 2005
epl1
Today at 2:12 AM
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find number of elements in H
Darealzolt   0
Today at 1:50 AM
If \( H \) is the set of positive real solutions to the system
\[ x^3 + y^3 + z^3 = x + y + z \]\[ x^2 + y^2 + z^2 = xyz \]then find the number of elements in \( H \).
0 replies
Darealzolt
Today at 1:50 AM
0 replies
J
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old problem from an open contest
Darealzolt   0
Today at 1:41 AM
Given that $a, b \in \mathbb{R}$ satisfy
\[ a + \frac{1}{a + 2015} = b - 4030 + \frac{1}{b - 2015} \]and $|a - b| > 5000$. Determine the value of
\[ \frac{ab}{2015} - a + b. \]
0 replies
Darealzolt
Today at 1:41 AM
0 replies
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Number theory
MuradSafarli   1
N Apr 29, 2025 by Sadigly
Prove that for any natural number \( n \) :

\[ 1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3). \]
1 reply
MuradSafarli
Apr 29, 2025
Sadigly
Apr 29, 2025
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Number theory
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Reveal topic tags
number theory
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MuradSafarli
109 posts
MuradSafarli
#1 h  Apr 29, 2025, 7:39 PM
Y by
Prove that for any natural number \( n \) :

\[ 1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3). \]
This post has been edited 1 time. Last edited by MuradSafarli, Apr 29, 2025, 7:41 PM
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Sadigly
159 posts
Sadigly
#3 h  Apr 29, 2025, 8:28 PM
Y by
Just try to write it like $\text{something something}\binom{8n+3}{4n+1}$
This post has been edited 1 time. Last edited by Sadigly, Apr 29, 2025, 8:32 PM
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