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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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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Find the area enclosed by the curve |z|^2 + |z^2 - 2i| = 16
mqoi_KOLA   1
N a minute ago by EeEApO
Find the area of the Argand plane enclosed by the curve $$ |z|^2 + |z^2 - 2i| = 16.$$
1 reply
+1 w
mqoi_KOLA
4 hours ago
EeEApO
a minute ago
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TST Junior Romania 2025
ant_   1
N 16 minutes ago by wassupevery1
Source: ssmr
Consider the isosceles triangle $ABC$, with $\angle BAC > 90^\circ$, and the circle $\omega$ with center $A$ and radius $AC$. Denote by $M$ the midpoint of side $AC$. The line $BM$ intersects the circle $\omega$ for the second time in $D$. Let $E$ be a point on the circle $\omega$ such that $BE \perp AC$ and $DE \cap AC = {N}$. Show that $AN = 2AB$.
1 reply
ant_
Yesterday at 5:01 PM
wassupevery1
16 minutes ago
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Very tight inequalities
KhuongTrang   2
N 44 minutes ago by SunnyEvan
Source: own
Problem. Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=1.$ Prove that $$\color{black}{\frac{1}{35a+12b+2}+\frac{1}{35b+12c+2}+\frac{1}{35c+12a+2}\ge \frac{4}{39}.}$$$$\color{black}{\frac{1}{4a+9b+6}+\frac{1}{4b+9c+6}+\frac{1}{4c+9a+6}\le \frac{2}{9}.}$$When does equality hold?
2 replies
KhuongTrang
May 17, 2024
SunnyEvan
44 minutes ago
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Sum of First, Second, and Third Powers
Brut3Forc3   47
N an hour ago by cubres
Source: 1973 USAMO Problem 4
Determine all roots, real or complex, of the system of simultaneous equations
\begin{align*} x+y+z &= 3, \\ x^2+y^2+z^2 &= 3, \\ x^3+y^3+z^3 &= 3.\end{align*}
47 replies
Brut3Forc3
Mar 7, 2010
cubres
an hour ago
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Ihave a minor issue.
CovertQED   0
2 hours ago
The area of triangle ABC is 18，sin2A +sin2B =4sinAsinB.Find the minimum perimeter of triangle ABC.
0 replies
CovertQED
2 hours ago
0 replies
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one very nice!
MihaiT   1
N 3 hours ago by MihaiT
Given $m_1$ weights, each weighing $k_1$ and another $m_2$ weights with $k_2$ each. Write a algorithm that determines the ways in which a scale can be balanced with a weight $X$ on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers $m_1,k_1,m_2,k_2,X$. What will be displayed after three successive runs: 5,2,5,1,4 | 5,2,5,1,11 | 5,2,5,1,20?

One answer is possible:
a)10;5;0;
b)20;7;0;
c)20;7;1;
d)10;10;0;
e)10;7;0;
f)20;5;0,
1 reply
MihaiT
Mar 31, 2025
MihaiT
3 hours ago
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Geometry problem about Euler line
lgx57   2
N 4 hours ago by pooh123
If the Euler line of a triangle is parallel to one side of the triangle, what is the relationship between the sides of this triangle?

The relationship between the angles of this triangle
2 replies
1 viewing
lgx57
Apr 9, 2025
pooh123
4 hours ago
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Inequalities
sqing   5
N Today at 9:16 AM by sqing
Let $ a,b,c,d\geq 0 ,a-b+d=21 $ and $ a+3b+4c=101 $. Prove that
$$ - \frac{1681}{3}\leq ab - cd \leq 820$$$$ - \frac{16564}{9}\leq ac -bd \leq 420$$$$ - \frac{10201}{48}\leq ad- bc \leq\frac{1681}{3}$$
5 replies
sqing
Yesterday at 3:53 AM
sqing
Today at 9:16 AM
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JEE Related ig?
mikkymini2   10
N Today at 4:08 AM by Idiot_of_the64squares
Hey everyone,

Just wanted to see if there are any other JEE aspirants on this forum currently prepping for it[mention year if you can]

I am actually entering 10th this year and have decided to try for it...So this year is just going to go in me strengthening my math (IOQM level (heard its enough till Mains part, so will start from there) for the problem solving part, and learn some topics from 11th and 12th as well)

It would be great to connect with others who are going through the same thing - share study strategies, tips, resources, discuss, and maybe even form study groups(not sure how to tho :maybe: ) and motivate each other ig?. :D
So yea, cya later
10 replies
mikkymini2
Apr 10, 2025
Idiot_of_the64squares
Today at 4:08 AM
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Inequalities
sqing   0
Today at 3:33 AM
Let $ a,b,c\in [0,1] $ . Prove that
$$(a+b+c)\left(\frac{1}{a^2+3}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{19}{4}$$$$(a+b+c)\left(\frac{1}{a^2+ 4}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{23}{5}$$$$(a+b+c)\left(\frac{1}{a^2+ \frac{5}{2}}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{34}{7}$$$$(a+b+c)\left(\frac{1}{a^2+ \frac{7}{2}}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{14}{3}$$
0 replies
sqing
Today at 3:33 AM
0 replies
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lcm(1,2,3,...,n)
lgx57   5
N Today at 3:09 AM by Kempu33334
Let $M=\operatorname{lcm}(1,2,3,\cdots,n)$.Estimate the range of $M$.
5 replies
lgx57
Apr 9, 2025
Kempu33334
Today at 3:09 AM
J
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Inequality
math2000   7
N Today at 2:59 AM by imnotgoodatmathsorry
Let $a,b,c>0$.Prove that $\dfrac{1}{(a+b)\sqrt{(a+2c)(b+2c)}}>\dfrac{3}{2(a+b+c)^2}$
7 replies
math2000
Jan 22, 2021
imnotgoodatmathsorry
Today at 2:59 AM
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How to prove one-one function
Vulch   5
N Today at 2:56 AM by jasperE3
Hello everyone,
I am learning functional equations.
To prove the below problem one -one function,I have taken two non-negative real numbers $ (1,2)$ from the domain $\Bbb R_{*},$ and put those numbers into the given function f(x)=1/x.It gives us 1=1/2.But it's not true.So ,it can't be one-one function.But in the answer,it is one-one function.Would anyone enlighten me where is my fault? Thank you!
5 replies
Vulch
Yesterday at 8:03 PM
jasperE3
Today at 2:56 AM
J
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Let a,b,c > 0 such that a+b+c=3. Prove that $ \frac{a^2}{a^2-2a+4} + \frac{b^2
bo_ngu_toan   3
N Today at 2:13 AM by imnotgoodatmathsorry
Let a,b,c > 0 such that a+b+c=3. Prove that $ \frac{a^2}{a^2-2a+4} + \frac{b^2}{b^2-2b+4} + \frac{c^2}{c^2-2c+4} \leq 1$
3 replies
bo_ngu_toan
Jun 4, 2023
imnotgoodatmathsorry
Today at 2:13 AM
J
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J
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Minimum area and perimeter
cjquines0   0
Jun 19, 2016
Source: Canada Repêchage 2016/2
Let $P = (7, 1)$ and let $O = (0, 0)$.

(a) If $S$ is a point on the line $y = x$ and $T$ is a point on the horizontal $x$-axis so that $P$ is on the line segment $ST$, determine the minimum possible area of triangle $OST$.

(b) If $U$ is a point on the line $y = x$ and $V$ is a point on the horizontal $x$-axis so that $P$ is on the line segment $UV$, determine the minimum possible perimeter of triangle $OUV$.
0 replies
cjquines0
Jun 19, 2016
0 replies
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Minimum area and perimeter
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Reveal topic tags
geometry
algebra
perimeter
area
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Source: Canada Repêchage 2016/2
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cjquines0
510 posts
cjquines0
#1 h  Jun 19, 2016, 1:30 AM • 2 Y
Y by Adventure10, Mango247
Let $P = (7, 1)$ and let $O = (0, 0)$.

(a) If $S$ is a point on the line $y = x$ and $T$ is a point on the horizontal $x$-axis so that $P$ is on the line segment $ST$, determine the minimum possible area of triangle $OST$.

(b) If $U$ is a point on the line $y = x$ and $V$ is a point on the horizontal $x$-axis so that $P$ is on the line segment $UV$, determine the minimum possible perimeter of triangle $OUV$.
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