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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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[*]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 in triangle
Nguyenhuyen_AG   0
3 minutes ago
Let $a,b,c$ be the lengths of the sides of a triangle. Prove that
\[\frac{1}{(a-4b)^2}+\frac{1}{(b-4c)^2}+\frac{1}{(c-4a)^2} \geqslant \frac{1}{ab+bc+ca}.\]
0 replies
Nguyenhuyen_AG
3 minutes ago
0 replies
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D,E,F are collinear.
TUAN2k8   2
N 7 minutes ago by TUAN2k8
Source: Own
Help me with this:
2 replies
TUAN2k8
May 28, 2025
TUAN2k8
7 minutes ago
J
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Combinatorial identity
MehdiGolafshan   4
N 14 minutes ago by watery
Let $n$ is a positive integer. Prove that
$$\sum_{k=0}^{n-1}\frac{1}{k+1}\binom{n-1}{k} = \frac{2^n-1}{n}.$$
4 replies
MehdiGolafshan
Jan 16, 2023
watery
14 minutes ago
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JBMO Shortlist 2023 G7
Orestis_Lignos   7
N an hour ago by tilya_TASh
Source: JBMO Shortlist 2023, G7
Let $D$ and $E$ be arbitrary points on the sides $BC$ and $AC$ of triangle $ABC$, respectively. The circumcircle of $\triangle ADC$ meets for the second time the circumcircle of $\triangle BCE$ at point $F$. Line $FE$ meets line $AD$ at point $G$, while line $FD$ meets line $BE$ at point $H$. Prove that lines $CF, AH$ and $BG$ pass through the same point.
7 replies
Orestis_Lignos
Jun 28, 2024
tilya_TASh
an hour ago
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24th PMO, Qualifying Stage #7
elpianista227   1
N an hour ago by elpianista227
Suppose $a, b, c$ are the roots of the polynomial $x^3 + 2x^2 + 2$. Let $f$ be the unique monic polynomial whose roots are $a^2, b^2, c^2$. Find $f(1)$.
1 reply
elpianista227
an hour ago
elpianista227
an hour ago
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27th Philippine Mathematical Olympiad Area Stage #5
Siopao_Enjoyer   1
N an hour ago by Siopao_Enjoyer
Find the sum of the cubes of the roots of the polynomial p(x)=x^3-x^2+2x-3.
1 reply
Siopao_Enjoyer
an hour ago
Siopao_Enjoyer
an hour ago
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Help me please
dssdgeww   1
N an hour ago by whwlqkd
Prove that there exists a positive integer n with 2024 prime divisors such that n| 2^n + 1
1 reply
dssdgeww
3 hours ago
whwlqkd
an hour ago
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[PMO20 Qualifying I.13] Log raised to Log
LilKirb   1
N 3 hours ago by LilKirb
Find the sum of the solutions to the logarithmic equation:
\[ x^{\log{x}} = 10^{2 - 3\log{x} + 2(\log{x})^2}\]where $\log{x}$ is the logarithm of $x$ to the base $10$
1 reply
LilKirb
3 hours ago
LilKirb
3 hours ago
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my brain isn't working :(
missmaialee   41
N 3 hours ago by sl1345961
Compute $(-1)^{11}-1^{10}+2^9+(-2)^8$.
41 replies
missmaialee
Yesterday at 9:45 PM
sl1345961
3 hours ago
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Inequalitis
sqing   0
4 hours ago
Let $ a,b,c\geq 0 , a^2+b^2+c^2 =3.$ Prove that
$$a^3 +b^3 +c^3 +\frac{11}{5}abc \leq \frac{26}{5}$$
0 replies
sqing
4 hours ago
0 replies
J
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Algebraic Manipulation
Darealzolt   4
N 4 hours ago by jasperE3
It is known that \(a,b \in \mathbb{R}\) that satisfies
\[ a^3+b^3=1957 \]\[ (a+b)(a+1)(b+1)=2014 \]Hence, find the value of \(a+b\)
4 replies
Darealzolt
Yesterday at 4:01 AM
jasperE3
4 hours ago
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not obvious trig identity!
mathmax001   1
N 4 hours ago by joeym2011
Problem ( trigonometry )
Let $ x \in \mathbb{R} $ and n a positive integer $ n >=1 $, Show that : $$ \tan\left({\frac{(n+1)x}{2}}\right)= \frac{\sum_{k=1}^n{\sin kx}}{\sum_{k=1}^n{\cos kx}} $$
Here is my take in this video: https://youtu.be/DBPyHNqk0GI?si=9r-YDuwv794AGe1p
1 reply
mathmax001
5 hours ago
joeym2011
4 hours ago
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confused
greenplanet2050   3
N Today at 12:19 AM by mathprodigy2011
um something weird happened today

I was doing the 2002 aime ii and i tried #9

I used PIE with $(2^{10}-1)-(\text{Number of times there are n same elements})$

so for like 1 same element i did $2^9 \cdot \dbinom{10}{1}$ cause there are 10 ways to choose 1 element that will be repeated. Similarly for 2 same elements it would be $2^8 \cdot \dbinom{10}{2}$

So if $A_n=2^{10-n} \cdot \dbinom{10}{n},$ the answer would be $(2^{10}-1)-([A_1+A_3+A_5+A_7+A_9]-[A_2+A_4+A_6+A_8+A_{10}].$ But this number turned out to be $0.$

Later when looking at the solution, i found out that the correct number was $28501.$ But I realized that $A_2+A_4+A_6+A_8+A_{10}=28501.$ So I was really confused of why i got the right answer somehow in my calculations.

Can someone explain why this happened? Thanks! :)
3 replies
greenplanet2050
Yesterday at 6:29 PM
mathprodigy2011
Today at 12:19 AM
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Easy one
irregular22104   2
N Yesterday at 10:27 PM by trangbui
Given two positive integers $a,b$ written on the board. We apply the following rule: At each step, we will add all the numbers that are the sum of the two numbers on the board so that the sum does not appear on the board. For example, if the two initial numbers are $2,5$; then the numbers on the board after step 1 are $2,5,7$; after step 2 are $2,5,7,9,12;...$
1) With $a = 3$; $b = 12$, prove that the number 2024 cannot appear on the board.
2) With $a = 2$; $b = 34$, prove that the number 2024 can appear on the board.
2 replies
irregular22104
May 6, 2025
trangbui
Yesterday at 10:27 PM
J
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Compute the least value of f(p) - ISL 1986
Amir Hossein   2
N Aug 24, 2012 by mihai miculita
A tetrahedron $ABCD$ is given such that $AD = BC = a; AC = BD = b; AB\cdot CD = c^2$. Let $f(P) = AP + BP + CP + DP$, where $P$ is an arbitrary point in space. Compute the least value of $f(P).$
2 replies
Amir Hossein
Aug 31, 2010
mihai miculita
Aug 24, 2012
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Compute the least value of f(p) - ISL 1986
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Reveal topic tags
geometry
3D geometry
tetrahedron
circumcircle
IMO Shortlist
minimization
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Amir Hossein
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#1 h  Aug 31, 2010, 10:30 AM • 1 Y
Y by Adventure10
A tetrahedron $ABCD$ is given such that $AD = BC = a; AC = BD = b; AB\cdot CD = c^2$. Let $f(P) = AP + BP + CP + DP$, where $P$ is an arbitrary point in space. Compute the least value of $f(P).$
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mihai miculita
666 posts
mihai miculita
#2 h  Aug 23, 2012, 9:14 AM • 3 Y
Y by Amir Hossein, Adventure10, Mango247
1). If $|AB|=|CD|=c;\, |AC|=|BD|=b,\,|AD|=|BC|=a$ then, $ABCD$ isoscele tetrahedron.

2).
$f(P)=|PA|+|PB|+|PC|+|PD|=min\Leftrightarrow \{\begin{array}{c} \angle{APB}\equiv\angle{CPD}\\ \angle{APC}\equiv\angle{BPD}\\ \angle{APD}\equiv\angle{BPC}.\end{array}$
$1) \mbox{ and } 2)\Rightarrow f(P)=min\Leftrightarrow P=G(barycenter)=O(circumcenter)\Leftrightarrow \\ \Leftrightarrow |PA|=|PB|=|PC|=|PD|=R (circumradius)\Rightarrow \\ \Rightarrow f(P)=4R=\sqrt{2(a^2+b^2+c^2)}.$
This post has been edited 1 time. Last edited by mihai miculita, Aug 24, 2012, 5:08 AM
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mihai miculita
666 posts
mihai miculita
#3 h  Aug 24, 2012, 5:02 AM • 3 Y
Y by Amir Hossein, Adventure10, Mango247
$\mbox{Let }M\mbox{-the midpoint of }[AB] \mbox{ and } N\mbox{-midpoint of }[CD]; \mbox{ then: }MN\perp AB;CD.$
$\mbox{In general if }|AB|\neq|CD|,\mbox{ then: } \boxed{f(P)=\mbox{minim, for the point }P_0\in[MN]; \dfrac{|P_0M|}{|P_0N|}=\dfrac{|AB|}{|CD|}\neq 1}.$
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