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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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Tetrahedrons and spheres
ReticulatedPython   1
N 25 minutes ago by jb2015007
Let $OABC$ be a non-degenerate tetrahedron with $A=(a,0,0)$, $B=(0,b,0)$, $C=(0,0,c)$, and $O=(0,0,0).$ Let a sphere of radius $r$ be circumscribed about this tetrahedron. Prove that $$r^2+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 9\sqrt[3]{16}.$$
1 reply
+1 w
ReticulatedPython
44 minutes ago
jb2015007
25 minutes ago
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Geometry Transformation Problems
ReticulatedPython   6
N an hour ago by ReticulatedPython
Problem 1:
A regular hexagon of side length $1$ is rotated $360$ degrees about one side. The space through which the hexagon travels during the rotation forms a solid. Find the volume of this solid.

Problem 2:

A regular octagon of side length $1$ is rotated $360$ degrees about one side. The space through which the octagon travels through during the rotation forms a solid. Find the volume of this solid.

Source:Own

Hint

Useful Formulas
6 replies
ReticulatedPython
Apr 17, 2025
ReticulatedPython
an hour ago
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Area of Polygon
AIME15   46
N an hour ago by K1mchi_
The area of polygon $ ABCDEF$, in square units, is

IMAGE

\[ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 46 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 74 \]
46 replies
AIME15
Jan 12, 2009
K1mchi_
an hour ago
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Bogus Proof Marathon
pifinity   7605
N 3 hours ago by e_is_2.71828
Hi!
I'd like to introduce the Bogus Proof Marathon.

In this marathon, simply post a bogus proof that is middle-school level and the next person will find the error. You don't have to post the real solution :P

Use classic Marathon format: 
[hide=P#]a1b2c3[/hide]
[hide=S#]a1b2c3[/hide]


Example posts:

P(x)
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S(x)
P(x+1)
-----
Let's go!! Just don't make it too hard!
7605 replies
pifinity
Mar 12, 2018
e_is_2.71828
3 hours ago
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geometry problem
kjhgyuio   1
N 5 hours ago by vanstraelen
.........
1 reply
kjhgyuio
Today at 8:27 AM
vanstraelen
5 hours ago
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Inequalities
sqing   4
N Today at 1:16 PM by sqing
Let $x,y\ge 0$ such that $ 13(x^3+y^3) \leq 125(1+xy)$. Prove that
$$ k(x+y)-xy\leq 5(2k-5)$$Where $k\geq 5.6797. $
$$ 6(x+y)-xy\leq 35$$
4 replies
sqing
Yesterday at 1:04 PM
sqing
Today at 1:16 PM
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Inscribed Semi-Circle!!!
ehz2701   2
N Today at 10:53 AM by mathafou
A right triangle $ABC$ with legs $AB = a$ and $BC = b$ is drawn with a semicircle inscribed into the triangle. What is the smallest possible radius of the semi-circle and the largest possible radius?

2 replies
ehz2701
Sep 11, 2022
mathafou
Today at 10:53 AM
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geometry
carvaan   1
N Today at 10:52 AM by vanstraelen
OABC is a trapezium with OC // AB and ∠AOB = 37°. Furthermore, A, B, C all lie on the circumference of a circle centred at O. The perpendicular bisector of OC meets AC at D. If ∠ABD = x°, find last 2 digit of 100x.
1 reply
carvaan
Yesterday at 5:48 PM
vanstraelen
Today at 10:52 AM
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Inequalities
nhathhuyyp5c   1
N Today at 9:09 AM by Mathzeus1024
Let $a, b, c$ be non-negative real numbers such that $a^2 + b^2 + c^2 = 3$. Find the maximum and minimum values of the expression
\[ P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}. \]
1 reply
nhathhuyyp5c
Yesterday at 6:35 AM
Mathzeus1024
Today at 9:09 AM
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In a school of $800$ students, $224$ students play cricket, $240$ students play
Vulch   2
N Today at 8:12 AM by MathBot101101
Hello everyone,
In a school of $800$ students, $224$ students play cricket, $240$ students play hockey and $336$ students play basketball. $64$ students play both basketball and hockey, $80$ students play both cricket and basketball, $40$ students play both cricket and hockey, and $24$ students play all three: basketball, hockey, and cricket. Find the number of students who do not play any game.

Edit:
In the above problem,I just want to know that why the number of students who don't play any game shouldn't be 0, because,if we add 224,240 and 336 it comes out to be 800.I have solution,but I just want to know how to explain it without theoretically.Thank you!
2 replies
Vulch
Yesterday at 11:41 PM
MathBot101101
Today at 8:12 AM
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Inequalities
sqing   25
N Today at 3:58 AM by sqing
Let $ a,b $ be reals such that $ a^2+ab+b^2 =3$ . Prove that
$$ \frac{4}{ 3}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{11}{4 }$$$$ \frac{13}{ 4}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{2}{3 }$$$$ \frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq -\frac{17}{6 }$$$$ \frac{19}{ 6}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2}$$Let $ a,b $ be reals such that $ a^2-ab+b^2 =1 $ . Prove that
$$ \frac{3}{ 2}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }+ab \geq \frac{4}{15 }$$$$ \frac{14}{ 15}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }-ab \geq -\frac{1}{2 }$$$$ \frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq \frac{13}{42 }$$$$ \frac{41}{ 42}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2 }$$
25 replies
sqing
Apr 16, 2025
sqing
Today at 3:58 AM
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Three variables inequality
Headhunter   4
N Today at 3:18 AM by lbh_qys
$\forall a\in R$ ,$~\forall b\in R$ ,$~\forall c \in R$
Prove that at least one of $(a-b)^{2}$, $(b-c)^{2}$, $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$.

I assume that all are greater than it, but can't go more.
4 replies
Headhunter
Yesterday at 6:58 AM
lbh_qys
Today at 3:18 AM
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Indonesia Regional MO 2019 Part A
parmenides51   23
N Today at 2:08 AM by chinawgp
Indonesia Regional MO
Year 2019 Part A

Time: 90 minutes Rules


p1. In the bag there are $7$ red balls and $8$ white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ...


p2. Given a regular hexagon with a side length of $1$ unit. The area of the hexagon is ...


p3. It is known that $r, s$ and $1$ are the roots of the cubic equation $x^3 - 2x + c = 0$. The value of $(r-s)^2$ is ...


p4. The number of pairs of natural numbers $(m, n)$ so that $GCD(n,m) = 2$ and $LCM(m,n) = 1000$ is ...


p5. A data with four real numbers $2n-4$, $2n-6$, $n^2-8$, $3n^2-6$ has an average of $0$ and a median of $9/2$. The largest number of such data is ...


p6. Suppose $a, b, c, d$ are integers greater than $2019$ which are four consecutive quarters of an arithmetic row with $a <b <c <d$. If $a$ and $d$ are squares of two consecutive natural numbers, then the smallest value of $c-b$ is ...


p7. Given a triangle $ABC$, with $AB = 6$, $AC = 8$ and $BC = 10$. The points $D$ and $E$ lies on the line segment $BC$. with $BD = 2$ and $CE = 4$. The measure of the angle $\angle DAE$ is ...


p8. Sequqnce of real numbers $a_1,a_2,a_3,...$ meet $\frac{na_1+(n-1)a_2+...+2a_{n-1}+a_n}{n^2}=1$ for each natural number $n$. The value of $a_1a_2a_3...a_{2019}$ is ....


p9. The number of ways to select four numbers from $\{1,2,3, ..., 15\}$ provided that the difference of any two numbers at least $3$ is ...


p10. Pairs of natural numbers $(m , n)$ which satisfies $$m^2n+mn^2 +m^2+2mn = 2018m + 2019n + 2019$$are as many as ...


p11. Given a triangle $ABC$ with $\angle ABC =135^o$ and $BC> AB$. Point $D$ lies on the side $BC$ so that $AB=CD$. Suppose $F$ is a point on the side extension $AB$ so that $DF$ is perpendicular to $AB$. The point $E$ lies on the ray $DF$ such that $DE> DF$ and $\angle ACE = 45^o$. The large angle $\angle AEC$ is ...


p12. The set of $S$ consists of $n$ integers with the following properties: For every three different members of $S$ there are two of them whose sum is a member of $S$. The largest value of $n$ is ....


p13. The minimum value of $\frac{a^2+2b^2+\sqrt2}{\sqrt{ab}}$ with $a, b$ positive reals is ....


p14. The polynomial P satisfies the equation $P (x^2) = x^{2019} (x+ 1) P (x)$ with $P (1/2)= -1$ is ....


p15. Look at a chessboard measuring $19 \times 19$ square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of $k$ coins on the chessboard where each coin is only loaded exactly on one square and each square can contain coins or blanks. At each turn. You must select exactly one plot that holds the minimum number of coins in the number of neighbors of the plot and then you must give exactly one coin to each neighbor of the selected plot. The game ends if you are no longer able to select squares with the intended conditions. The smallest number of $k$ so that the game never ends for any initial square selection is ....
23 replies
parmenides51
Nov 11, 2021
chinawgp
Today at 2:08 AM
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k VOLUNTEERING OPPORTUNITIES OPEN TO HIGH/MIDDLE SCHOOLERS
im_space_cadet   13
N Today at 12:30 AM by im_space_cadet
Hi everyone!
Do you specialize in contest math? Do you have a passion for teaching? Do you want to help leverage those college apps? Well, I have something for all of you.

I am im_space_cadet, and during the fall of last year, I opened my non-profit DeltaMathPrep which teaches students preparing for contest math the problem-solving skills they need in order to succeed at these competitions. Currently, we are very much understaffed and would greatly appreciate the help of more tutors on our platform.

Each week on Saturday and Wednesday, we meet once for each competition: Wednesday for AMC 8 and Saturday for AMC 10 and we go over a past year paper for the entire class. On both of these days, we meet at 9PM EST in the night.

This is a great opportunity for anyone who is looking to have a solid activity to add to their college resumes that requires low effort from tutors and is very flexible with regards to time.

This is the link to our non-profit for anyone who would like to view our initiative:
https://www.deltamathprep.org/

If you are interested in this opportunity, please send me a DM on AoPS or respond to this post expressing your interest. I look forward to having you all on the team!

Thanks,
im_space_cadet
13 replies
im_space_cadet
Yesterday at 2:27 PM
im_space_cadet
Today at 12:30 AM
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STATE SOLUTIONS AND STUFF DROPPED!!!
Soupboy0   53
N Apr 7, 2025 by fruitmonster97
https://www.mathcounts.org/resources/past-competitions
53 replies
Soupboy0
Apr 4, 2025
fruitmonster97
Apr 7, 2025
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STATE SOLUTIONS AND STUFF DROPPED!!!
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Reveal topic tags
MATHCOUNTS
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Soupboy0
337 posts
Soupboy0
#40 h  Apr 5, 2025, 1:08 PM
Y by
i memorized sprint 20 before the competition:skull:
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MathPerson12321
3704 posts
MathPerson12321
#41 h  Apr 5, 2025, 3:12 PM
Y by
@3above correct, this works for one time things and mostly is an intro to expected value.

You have to consider the other factors as well.
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superhuman233
473 posts
superhuman233
#42 h  Apr 5, 2025, 5:59 PM
Y by
i was so lucky that i guessed sprint 20 right otherwise i wouldn't have made nats

Click to reveal hidden text
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ilikemath247365
243 posts
ilikemath247365
#43 h  Apr 5, 2025, 6:00 PM
Y by
Bro, sprint 20 took the most time for me in the first 20. I can't believe that I actually did coordinate geometry on the real test on this problem!

I also almost trolled this problem. I miscalculated by somehow getting the square root of 49/25 but I clutched and caught my mistake to change it to the square root of 81/25 which is 9/5.
This post has been edited 1 time. Last edited by ilikemath247365, Apr 5, 2025, 6:01 PM
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Andyluo
929 posts
Andyluo
#44 h  Apr 5, 2025, 6:02 PM
Y by
superhuman233 wrote:
i was so lucky that i guessed sprint 20 right otherwise i wouldn't have made nats

Click to reveal hidden text

how is that guessing lmao, it's how you're supposed to solve it. honestly belongs in first 10 questions
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jb2015007
1915 posts
jb2015007
#45 h  Apr 5, 2025, 6:16 PM • 1 Y
Y by PikaPika999
20 was didnt take too long for me
js had to draw the altitude and do some coord bashing then pythag then ur done
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JamesYMath
12 posts
JamesYMath
#46 h  Apr 6, 2025, 12:34 AM • 1 Y
Y by PikaPika999
You could've also used the altitude theorem for #20 but I guess that's just similar triangles
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PikaPika999
1245 posts
PikaPika999
#47 h  Apr 6, 2025, 12:44 AM
Y by
MathPerson12321 wrote:
c_double_sharp wrote:
s11 was dumb i added wrong
s18 was dumb how did i not see that
s23 was dumb how did i not see that
s24 was dumb how am i not able to read properly
s27 was dumb like the official solution was assume rectangle
t6 was dumb like i did not calcbash enough terms
t8 was dumb how did i not see that

totally agree

fr tho

:rotfl:
This post has been edited 1 time. Last edited by PikaPika999, Apr 6, 2025, 12:44 AM
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giratina3
494 posts
giratina3
#48 h  Apr 6, 2025, 1:44 AM • 1 Y
Y by PikaPika999
I choked #7 on sprint because I thought for some stupid reason that the maximum area of a triangle with perimeter 24 was a right triangle :wallbash:

Probably could have gone to nationals with that question. Just saying...

Sigh... guess I just wasn't able to make nationals because I'm too nervous during the test... which is exactly what happened last year too.

So close to nationals... yet so far...
This post has been edited 1 time. Last edited by giratina3, Apr 6, 2025, 1:45 AM
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martianrunner
169 posts
martianrunner
#49 h  Apr 7, 2025, 4:17 AM
Y by
i had seen some extremely trivial solution for p4 target that didn't even require a calculator but i forgot it now

does anyone have a non-bash solution for that one
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derekwang2048
1225 posts
derekwang2048
#50 h  Apr 7, 2025, 4:23 AM
Y by
i just solved the system of equations and it was pretty straightforward, i realized how to do it with like 2 minutes left
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Bummer12345
135 posts
Bummer12345
#51 h  Apr 7, 2025, 1:01 PM
Y by
martianrunner wrote:
i had seen some extremely trivial solution for p4 target that didn't even require a calculator but i forgot it now

does anyone have a non-bash solution for that one

here are 5 stupid solutions to target #4

sol 1 (sort of bash, most logical)

sol 2 (cheese, no bash)

sol 3 (graphing calc)

sol 4 (even more cheese)

sol 5 (sol 2+3 equals 5)
This post has been edited 3 times. Last edited by Bummer12345, Apr 7, 2025, 1:09 PM
Reason: latex
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c_double_sharp
307 posts
c_double_sharp
#52 h  Apr 7, 2025, 1:31 PM • 1 Y
Y by MathPerson12321
p8 is 27/2 confirmed but why does the official solution work
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happypi31415
742 posts
happypi31415
#53 h  Apr 7, 2025, 4:48 PM • 2 Y
Y by Aaronjudgeisgoat, MathPerson12321
The official solution works because the coins follow a Binomial Distribution, which means that we have that $E[x]=np$ where $E[x]$ is the expected value, $p$ is the probability, and $n$ is the number of trials. This also follows from linearity of expectation on each of the groups of $3$. Then, setting $E[x]=1$ and $p=\frac{2}{27}$ gives the desired answer. :)

Alternatively, another easy solution is to use recursion: suppose that we want the expected number of rolls, $x$, until an event happens with probability $p$. Then, we have that $x=p+(x+1)(1-p)$ which rearranges to the desired $x=\frac{1}{p}$.

edit: thinking about it now im not completely sure this is valid because expected amount of rolls between HTH and TTT is different but whatever
This post has been edited 3 times. Last edited by happypi31415, Apr 7, 2025, 8:32 PM
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fruitmonster97
2478 posts
fruitmonster97
#54 h  Apr 7, 2025, 5:29 PM
Y by
target seems interesting
personal difficulty ratings: 1 is trivial, 10 is nats t8(from a year not named 2024 or 2023)
t1: 1, trivial, just average 9876 and 5432.
t2: 1.5 to 2, trivial but annoying.
First two seem hard to silly, esp. with 1 being so easy.
t3: 1 to 1.5, also trivial.
t4: 2 to 2.5, too easy for the placement.
Another very sweepable round, as intended. Nowhere near the difficulty of last year's t4(which was the only one i got wrong last year).
t5: 1 to 1.5, just floor(202/24)
t6: 6 to 7, imo perfect placement. nontrivial. Albeit you don't even have to find the intended solution, simply approximate 2^2025-1 to 2^2025 and plugging the first ~13 terms into a calc should give a good approximation.
t7: 3 to 3.5, fair placement. simple crt.
t8: 4 to 9, really just depends on your perception. there are definitely people who just assumed you can always reciprocate, but there are definitely people who tried much, much harder methods.
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