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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 2, 2025
0 replies
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For positive integers \( a, b, c \), find all possible positive integer values o
Jackson0423   2
N 10 minutes ago by ATM_
For positive integers \( a, b, c \), find all possible positive integer values of
\[ \frac{a}{b} + \frac{b}{c} + \frac{c}{a}. \]
2 replies
Jackson0423
an hour ago
ATM_
10 minutes ago
J
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Isosceles Triangle Geo
oVlad   2
N 11 minutes ago by SomeonesPenguin
Source: Romania Junior TST 2025 Day 1 P2
Consider the isosceles triangle $ABC$ with $\angle A>90^\circ$ and the circle $\omega$ of radius $AC$ centered at $A.$ Let $M$ be the midpoint of $AC.$ The line $BM$ intersects $\omega$ a second time at $D.$ Let $E$ be a point on $\omega$ such that $BE\perp AC.$ Let $N$ be the intersection of $DE$ and $AC.$ Prove that $AN=2\cdot AB.$
2 replies
1 viewing
oVlad
Yesterday at 9:38 AM
SomeonesPenguin
11 minutes ago
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IMO ShortList 1998, number theory problem 5
orl   63
N 15 minutes ago by ATM_
Source: IMO ShortList 1998, number theory problem 5
Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.
63 replies
orl
Oct 22, 2004
ATM_
15 minutes ago
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IMO Shortlist 2013, Number Theory #1
lyukhson   150
N 23 minutes ago by MuradSafarli
Source: IMO Shortlist 2013, Number Theory #1
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
\[ m^2 + f(n) \mid mf(m) +n \]
for all positive integers $m$ and $n$.
150 replies
lyukhson
Jul 10, 2014
MuradSafarli
23 minutes ago
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Might be the first equation marathon
steven_zhang123   34
N Apr 8, 2025 by rchokler
As far as I know, it seems that no one on HSM has organized an equation marathon before. Click to reveal hidden textSo why not give it a try? Click to reveal hidden text Let's start one!
Some basic rules need to be clarified:
$\cdot$ If a problem has not been solved within $5$ days, then others are eligible to post a new probkem.
$\cdot$ Not only simple one-variable equations, but also systems of equations are allowed.
$\cdot$ The difficulty of these equations should be no less than that of typical quadratic one-variable equations. If the problem involves higher degrees or more variables, please ensure that the problem is solvable (i.e., has a definite solution, rather than an approximate one).
$\cdot$ Please indicate the domain of the solution to the equation (e.g., solve in $\mathbb{R}$, solve in $\mathbb{C}$).
Here's an simple yet fun problem, hope you enjoy it :P :
P1
34 replies
steven_zhang123
Jan 20, 2025
rchokler
Apr 8, 2025
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2019 Back To School Mock AIME II #6 x - y = 3, x^5-y^5 = 408
parmenides51   2
N Mar 21, 2025 by CubeAlgo15
The value of $xy$ that satises $x - y = 3$ and $x^5-y^5 = 408$ for real $x$ and $y$ can be written as $\frac{-a + b \sqrt{c}}{d}$ where the greatest common divisor of positive integers $a$, $b$, and $d$ is $1$, and $c$ is not divisible by the square of any prime. Compute the value of $a + b + c + d$.
2 replies
parmenides51
Dec 16, 2023
CubeAlgo15
Mar 21, 2025
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solve the system of equations
Havu   4
N Mar 20, 2025 by LeoaB411
Solve the system of equations:
\[\begin{cases} 3x^2-2xy+3y^2+\dfrac{2}{x^2-2xy+y^2}=8\\ 2x+\dfrac{1}{x-y}=4 \end{cases}\]
4 replies
Havu
Mar 19, 2025
LeoaB411
Mar 20, 2025
J
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System of three equations - Iran First Round 2018, P14
Amir Hossein   2
N Mar 14, 2025 by ioannism45
For how many integers $k$ does the following system of equations has a solution other than $a=b=c=0$ in the set of real numbers? \begin{align*} \begin{cases} a^2+b^2=kc(a+b),\\ b^2+c^2 = ka(b+c),\\ c^2+a^2=kb(c+a).\end{cases}\end{align*}
2 replies
Amir Hossein
Mar 7, 2021
ioannism45
Mar 14, 2025
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System of Equations with GCD
MrHeccMcHecc   2
N Mar 11, 2025 by MrHeccMcHecc
Determine the sum of all possible values of $abc$ where $a,b,c$ are positive integers satisfying the equations $$\begin{cases} a= \gcd (b,c) + 3 \\ b= \gcd (c,a) + 3 \\ c= \gcd (a,b) + 3 \end{cases}$$
2 replies
MrHeccMcHecc
Mar 10, 2025
MrHeccMcHecc
Mar 11, 2025
J
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Finding Harmonic Mean of Roots
JasonMurong1   7
N Mar 3, 2025 by scrabbler94
Given polynomial x^3 − 18x^2 + 95x − 150, what is the harmonic mean of the roots?
A. 30/19
B. 290/95
C. 90/19
D. 133/95
E. NOTA


Am I supposed to used the sum of the roots, the sum of the products of the roots twice at a time, and the product of the roots to form a system of equations to find the roots? I tried that and I'm stuck. Is there a trick to this problem? Or should I just find the roots via rational root theorem? Could someone please help me? Thanks.
7 replies
JasonMurong1
Mar 2, 2025
scrabbler94
Mar 3, 2025
J
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Very Nice equations
steven_zhang123   6
N Feb 27, 2025 by eric201291
Solve this:$\left\{\begin{matrix} x^{2023}+y^{2023}+z^{2023}=2023 \\ x^{2024}+y^{2024}+z^{2024}=2024 \\ x^{2025}+y^{2025}+z^{2025}=2025 \end{matrix}\right.$
6 replies
steven_zhang123
Oct 13, 2024
eric201291
Feb 27, 2025
J
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Hard system of Equations
William_Mai   10
N Feb 18, 2025 by William_Mai
Find the real solutions of the following system of equations:

$\begin{cases} x^4 = 3y + 2 \\ y^4 = 3z + 2 \\ z^4 = 3x + 2 \end{cases}$

Source: https://www.facebook.com/share/p/1MrR3u8VTU/
10 replies
William_Mai
Feb 17, 2025
William_Mai
Feb 18, 2025
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2023 Christmas Mock AIME #8 3x3 complex non linear system
parmenides51   2
N Feb 5, 2025 by Vivaandax
Let $a$, $b$, and $c$ be complex numbers such that they satisfy these equations:
$$abc + 4a^4 = 3$$$$abc + 2d^4 = 2$$$$abc +\frac{3c^4}{4}=-1$$If the maximum of $\left|\frac{a^2b^2c^2+1}{abc}\right|$ can be expressed as $\frac{p+\sqrt{q}}{r}$ for positive integers $p$, $q$, and $r$, find the minimum possible value of $p + q + r$.
2 replies
parmenides51
Jan 16, 2024
Vivaandax
Feb 5, 2025
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System of Equation: The Blue Waterfall
hypertension   4
N Jan 27, 2025 by MihaiT
Can u solve "The Blue Waterfall" System of equations?
x+y+z+w=7
y*z+w=7
x*y+z=5
z*w-x=2
For real and complex!
The ones who will make it will win my love!
Thanks a million,
George
4 replies
hypertension
Jan 26, 2025
MihaiT
Jan 27, 2025
J
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J
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Conditional upper bound for binomial coeff
Miquel-point   0
Apr 6, 2025
Source: Romanian IMO TST 1981, Day 3 P3
Let $n>r\geqslant 3$ be two integers and $d$ be a positive integer such that $nd\geqslant \dbinom{n+r}{r+1}$. Show that \[(n-t)(d-t)>\dbinom{n-t+r}{r+1},\]for $t=1,2,\ldots,n-1$

Vasile Brânzănescu
0 replies
Miquel-point
Apr 6, 2025
0 replies
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Conditional upper bound for binomial coeff
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Reveal topic tags
algebra
binomial coefficients
inequalities
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Source: Romanian IMO TST 1981, Day 3 P3
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Miquel-point
465 posts
Miquel-point
#1 h  Apr 6, 2025, 6:39 PM • 1 Y
Y by PikaPika999
Let $n>r\geqslant 3$ be two integers and $d$ be a positive integer such that $nd\geqslant \dbinom{n+r}{r+1}$. Show that \[(n-t)(d-t)>\dbinom{n-t+r}{r+1},\]for $t=1,2,\ldots,n-1$

Vasile Brânzănescu
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