satisfy that: 
satisfied 
Find the minimum of 
Y by
Source: Taichung P.S.1 math program tryouts
How many ordered pairs
are there such that 
and 
?
How many ordered pairs
are there such that 
and 
?G
Topic
First Poster
Last Poster
k a April Highlights and 2025 AoPS Online Class Information
jlacosta 0
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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Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. 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 events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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Prealgebra 1 Self-Paced
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0 replies
1 viewing
Problem nt
HopefullyMcNats2025 1
N
by HopefullyMcNats2025
How many positive integers from 1-100000 fit the conditions, they are not divisble by 3, 5, or 8 and that they are palindromes,
1 reply
Inequalities
nhathhuyyp5c 4
N
by maxamc
Prove that for all positive real numbers 
, the following inequality holds:
![\[
\sqrt{a + b} + \sqrt{b + c} + \sqrt{c + a} \geq \frac{4(ab + bc + ca)}{\sqrt{(a + b)(b + c)(c + a)}}
\]](//latex.artofproblemsolving.com/e/7/2/e72fd0f134289c359bc8766440babc52ae131961.png)
, the following inequality holds:![\[
\sqrt{a + b} + \sqrt{b + c} + \sqrt{c + a} \geq \frac{4(ab + bc + ca)}{\sqrt{(a + b)(b + c)(c + a)}}
\]](http://latex.artofproblemsolving.com/e/7/2/e72fd0f134289c359bc8766440babc52ae131961.png)
4 replies
Intermediate Counting
RenheMiResembleRice 4
N
by Apple_maths60
A coin is flipped, a 6-sided die numbered 1 through 6 is rolled, and a 10-sided die numbered 0
through 9 is rolled. What is the probability that the coin comes up heads and the sum of the
numbers that show on the dice is 8?
through 9 is rolled. What is the probability that the coin comes up heads and the sum of the
numbers that show on the dice is 8?
4 replies
Inequalities
sqing 2
N
by DAVROS
Let 
be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
2 replies
Might be the first equation marathon
steven_zhang123 33
N
by eric201291
As far as I know, it seems that no one on HSM has organized an equation marathon before. So why not give it a try? Let's start one!
Some basic rules need to be clarified:
If a problem has not been solved within 
days, then others are eligible to post a new probkem.
Not only simple one-variable equations, but also systems of equations are allowed.
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).
Please indicate the domain of the solution to the equation (e.g., solve in 
, solve in 
).
Here's an simple yet fun problem, hope you enjoy it :P :
Some basic rules need to be clarified:
If a problem has not been solved within 
days, then others are eligible to post a new probkem. Not only simple one-variable equations, but also systems of equations are allowed. 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). Please indicate the domain of the solution to the equation (e.g., solve in 
, solve in 
).Here's an simple yet fun problem, hope you enjoy it :P :
33 replies
Congruence
Ecrin_eren 2
N
by Ecrin_eren
Find the number of integer pairs (x, y) satisfying the congruence equation:
3y² + 3x²y + y³ ≡ 3x² (mod 41)
for 0 ≤ x, y < 41.
3y² + 3x²y + y³ ≡ 3x² (mod 41)
for 0 ≤ x, y < 41.
2 replies
How to judge a number is prime or not?
mingzhehu 1
N
by scrabbler94
A=（10X1+1）(10X+1），X1，X∈N+
B=（10 X1+3）（10X+7），X∈N，X1∈N
C=(10 X1+9)(10X+9), X∈N，X1∈N
D=(10 X1+1)(10X+3), X1∈N+，X∈N
E=（10 X1+7）(10X+9），X∈N，X1∈N
F=（10 X1+1）（10X+7），X1∈N+，X∈N
G=（10 X1+3）（10X+9），X∈N，X1∈N
H=（10 X1+1）10X+9），X1∈N+，X∈N
I=（10 X1+3）（10X+3），X1∈N，X∈N
J=( 10X1+7)(10X+7),X∈N，X1∈N
For any natural number P∈{P=10N+1，n∈N}，make P=A or B or C
If P can make the roots of function group(ABC) without any root group completely made up of integer, P will be a prime
For any natural number P∈{P=10N+3，n∈N}，make P=D or E
If P can make the roots of function group(DE) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+7，n∈N}，make P=F or G
If P can make the roots of function group(FG) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+9，n∈N}，make P=H or I or J
If P can make the roots of function group(GIJ) without any root group completely made up
of integer, P will be a prime
B=（10 X1+3）（10X+7），X∈N，X1∈N
C=(10 X1+9)(10X+9), X∈N，X1∈N
D=(10 X1+1)(10X+3), X1∈N+，X∈N
E=（10 X1+7）(10X+9），X∈N，X1∈N
F=（10 X1+1）（10X+7），X1∈N+，X∈N
G=（10 X1+3）（10X+9），X∈N，X1∈N
H=（10 X1+1）10X+9），X1∈N+，X∈N
I=（10 X1+3）（10X+3），X1∈N，X∈N
J=( 10X1+7)(10X+7),X∈N，X1∈N
For any natural number P∈{P=10N+1，n∈N}，make P=A or B or C
If P can make the roots of function group(ABC) without any root group completely made up of integer, P will be a prime
For any natural number P∈{P=10N+3，n∈N}，make P=D or E
If P can make the roots of function group(DE) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+7，n∈N}，make P=F or G
If P can make the roots of function group(FG) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+9，n∈N}，make P=H or I or J
If P can make the roots of function group(GIJ) without any root group completely made up
of integer, P will be a prime
1 reply
inequality
revol_ufiaw 3
N
by MS_asdfgzxcvb
Prove that that for any real 
and natural number 
,
and natural number 
,
3 replies
What is an isogonal conjugate and why is it useful?
EaZ_Shadow 6
N
by maxamc
What is an isogonal conjugate and why is it useful? People use them in Olympiad geometry proofs but I don’t understand why and what is the purpose, as it complicates me because of me not understanding it.
6 replies
Any nice way to do this?
NamelyOrange 3
N
by pooh123
Source: Taichung P.S.1 math program tryouts
How many ordered pairs are there such that and ?
How many ordered pairs
are there such that and ?
3 replies
Any nice way to do this?
G
H
G
H
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#2
Apr 2, 2025, 1:33 PM
Y by
It’s the sum of floor(60/i) as i ranges from 1 to 60 (sorry for no latex)
This comes out to 261 I believe
This comes out to 261 I believe
This post has been edited 1 time. Last edited by HockeyMaster85, Apr 2, 2025, 2:14 PM
Reason: grammar
Reason: grammar
Z
K
Y
The post below has been deleted. Click to close.
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#3
Apr 2, 2025, 8:45 PM
Y by
HockeyMaster85 wrote:
It’s the sum of floor(60/i) as i ranges from 1 to 60 (sorry for no latex)
This comes out to 261 I believe
This comes out to 261 I believe
This is meant to be done by hand, so that's a bit painful...
Any other nice ways?
This post has been edited 1 time. Last edited by NamelyOrange, Apr 2, 2025, 8:45 PM
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Y by
NamelyOrange wrote:
HockeyMaster85 wrote:
It’s the sum of floor(60/i) as i ranges from 1 to 60 (sorry for no latex)
This comes out to 261 I believe
This comes out to 261 I believe
This is meant to be done by hand, so that's a bit painful...
Any other nice ways?
Consider the cases of
:If
then there are 
cases.If
then there are 
cases (must exclude 
).If
then there are 
cases (same reasoning as before).If
then there are 
cases.If
then there are 
cases.If
then there are 
cases.If
then there are 
cases.If
then no solution because 
.In total, there are
triples of 
.(the other guy forgot to exclude cases when
).
Z
K
Y
N Quick Reply
G
H
