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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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Gergonne point Harmonic quadrilateral
niwobin   4
N 15 minutes ago by on_gale
Triangle ABC has incircle touching the sides at D, E, F as shown.
AD, BE, CF concurrent at Gergonne point G.
BG and CG cuts the incircle at X and Y, respectively.
AG cuts the incircle at K.
Prove: K, X, D, Y form a harmonic quadrilateral. (KX/KY = DX/DY)
4 replies
niwobin
May 17, 2025
on_gale
15 minutes ago
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NCG Returns!
blacksheep2003   64
N 19 minutes ago by SomeonecoolLovesMaths
Source: USEMO 2020 Problem 1
Which positive integers can be written in the form \[\frac{\operatorname{lcm}(x, y) + \operatorname{lcm}(y, z)}{\operatorname{lcm}(x, z)}\]for positive integers $x$, $y$, $z$?
64 replies
blacksheep2003
Oct 24, 2020
SomeonecoolLovesMaths
19 minutes ago
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Binomial stuff
Arne   2
N 33 minutes ago by Speedysolver1
Source: Belgian IMO preparation
Let $p$ be prime, let $n$ be a positive integer, show that \[ \gcd\left({p - 1 \choose n - 1}, {p + 1 \choose n}, {p \choose n + 1}\right) = \gcd\left({p \choose n - 1}, {p - 1 \choose n}, {p + 1 \choose n + 1}\right). \]
2 replies
Arne
Apr 4, 2006
Speedysolver1
33 minutes ago
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Geometry hard problem
noneofyou34   1
N 38 minutes ago by Lil_flip38
Let ABC be a triangle with incircle Γ. The tangency points of Γ with sides BC, CA, AB are A1, B1, C1 respectively. Line B1C1 intersects line BC at point A2. Similarly, points B2 and C2 are constructed. Prove that the perpendicular lines from A2, B2, C2 to lines AA1, BB1, CC1 respectively are concurret.
1 reply
noneofyou34
Today at 3:13 PM
Lil_flip38
38 minutes ago
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Docked 4 points Help
sadas123   9
N 3 hours ago by ethan2011
In school we had this beginners like middle school contest, but we had to right down our solution kind of like usajmo except no proofs. It was also graded out of 7 but I got 4 Points docked for this question. what was my problem??? But I kind of had to rush the solution on this question because there was another problem before this that was like 1000x times harder.

Question:The solutions to the equation x^3-13x^2+ax−48=0 are all positive whole numbers. What is $a$?


Solution: We can see that we can use Vieta's formulas to find that the product of the roots is $48$, and the sum of the roots is $13$. So we need to find a combination of integers that multiply to $48$ and add up to $13$. Let's call the roots of the equation p, q, and r. From Vieta's, we get that $p+q+r=-13$ and $pqr = -48$. Looking at the factors of $48$, which is $2^4*3$, we try to split the numbers in a way that gives us the correct sum and product. Trying 3, -2, and -8, we see that they add up to $-13$ and multiply to $-48$, so they work. That means the roots of the polynomial are -3, -2, and -8, and the factorization is $(x-3)(x-2)(x-8)$. Multiplying it out, we get $x^3-13x^2+46x-48$, so we find that a = 46.
9 replies
sadas123
Yesterday at 4:06 PM
ethan2011
3 hours ago
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System of Equations
P162008   1
N 4 hours ago by alexheinis
If $a,b$ and $c$ are complex numbers such that

$\sum_{cyc} ab = 23$

$\frac{a}{c + a} + \frac{b}{a + b} + \frac{c}{b + c} = -1$

$\frac{a^2b}{b + c} + \frac{b^2c}{c + a} + \frac{c^2a}{a + b} = 202$

Compute $\sum_{cyc} a^2.$
1 reply
P162008
Today at 10:25 AM
alexheinis
4 hours ago
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collinear
spiralman   0
4 hours ago
Given an acute triangle \( \triangle ABC \) with \( AB < AC \), inscribed in circle \( (O) \).
Let \( H \) be the orthocenter of triangle \( ABC \), and \( M \) be the midpoint of \( BC \).
A line passing through \( H \), parallel to \( AO \), intersects lines \( AB \) and \( AC \) at points \( D \) and \( E \), respectively.
Let \( K \) be the circumcenter of triangle \( ADE \). Prove that: Points \( H, K, M \) are collinear.

0 replies
spiralman
4 hours ago
0 replies
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THREE People Meet at the SAME. TIME.
LilKirb   1
N 5 hours ago by hellohi321
Three people arrive at the same place independently, at a random between $8:00$ and $9:00.$ If each person remains there for $20$ minutes, what's the probability that all three people meet each other?

I'm already familiar with the variant where there are only two people, where you Click to reveal hidden text It was an AIME problem from the 90s I believe. However, I don't know how one could visualize this in a Click to reveal hidden text Help on what to do?
1 reply
LilKirb
Today at 1:06 PM
hellohi321
5 hours ago
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Vieta's Formula.
BlackOctopus23   6
N Today at 2:00 PM by Shan3t
Can someone help me understand Vieta's Formula? I am currently learning it for my class. I learned that for a polynomial of degree $n$, all the roots added will give $-\frac{a_{n-1}}{a_n}$. I also learned that if every single root, multiplies every single root, it will give $\frac{a_{n-2}}{a_n}$. I also learned that if all the roots are multiplied, it will give $-\frac{a_0}{a_n}$. Is this right? And is there any purpose for these equations?
6 replies
BlackOctopus23
Yesterday at 11:10 PM
Shan3t
Today at 2:00 PM
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Number of elements in Set
girishpimoli   1
N Today at 1:34 PM by alexheinis
Let $A=\left\{1,2,3,4,5,6,7\right\}$ and $B=\left\{3,6,7,9\right\}.$ Then the number of elements in the set ${C⊆A:C∩B=ϕ}$ is
1 reply
girishpimoli
Today at 1:23 PM
alexheinis
Today at 1:34 PM
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geometry
luckvoltia.112   1
N Today at 11:37 AM by MathsII-enjoy
ChGiven an acute triangle ABC inscribed in circle $(O)$ The altitudes $BE, CF$ , intersect
each other at $H$. The tangents at $B$ and $C $of $(O)$ intersect at $S$. Let $M $be the midpoint of $BC$. $EM$ intersects $SC$
at $I$, $FM$ intersects $SB$ at $J.$
a) Prove that the points $I, S, M, J$ lie on the same circle.
b) The circle with diameter $AH$ intersects the circle $(O)$ at the second point $T.$ The line $AH$ intersects
$(O)$ at the second point $K$. Prove that $S,K,T$ are collinear.
1 reply
luckvoltia.112
Yesterday at 3:04 PM
MathsII-enjoy
Today at 11:37 AM
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System of Equations
P162008   0
Today at 10:48 AM
If $a,b$ and $c$ are complex numbers such that

$(a + b)(b + c) = 1$

$(a - b)^2 + (a^2 - b^2)^2 = 85$

$(b - c)^2 + (b^2 - c^2)^2 = 75$

Compute $(a - c)^2 + (a^2 - c^2)^2.$
0 replies
P162008
Today at 10:48 AM
0 replies
J
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System of Equations
P162008   0
Today at 10:43 AM
If $a,b$ and $c$ are real numbers such that

$\prod_{cyc} (a + b) = abc$

$\prod_{cyc} (a^3 + b^3) = (abc)^3$

Compute the value of $abc.$
0 replies
P162008
Today at 10:43 AM
0 replies
J
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System of Equations
P162008   0
Today at 10:34 AM
If $a,b$ and $c$ are complex numbers such that

$\frac{ab}{b + c} + \frac{bc}{c + a} + \frac{ca}{a + b} = -9$

$\frac{ab}{c + a} + \frac{bc}{a + b} + \frac{ca}{b + c} = 10$

Compute $\frac{a}{c + a} + \frac{b}{a + b} + \frac{c}{b + c}.$
0 replies
P162008
Today at 10:34 AM
0 replies
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J
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Find set of all points for a regular triangular pyramid
orl   4
N Sep 24, 2017 by parmenides51
Source: First Zhautykov Olympiad 2005, Problem 3
Let SABC be a regular triangular pyramid. Find the set of all points $ D (D! = S)$ in the space satisfing the equation $ |cos ASD - 2cosBSD - 2 cos CSD| = 3$.
4 replies
orl
Dec 22, 2008
parmenides51
Sep 24, 2017
J
Find set of all points for a regular triangular pyramid
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Reveal topic tags
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3D geometry
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Source: First Zhautykov Olympiad 2005, Problem 3
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orl
3647 posts
orl
#1 h  Dec 22, 2008, 12:46 AM • 2 Y
Y by Adventure10, Mango247
Let SABC be a regular triangular pyramid. Find the set of all points $ D (D! = S)$ in the space satisfing the equation $ |cos ASD - 2cosBSD - 2 cos CSD| = 3$.
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Vo Duc Dien
341 posts
Vo Duc Dien
#2 h  Mar 14, 2011, 11:49 PM • 2 Y
Y by Adventure10, Mango247
Did you mean (D!= S) for D ≠ S?

Please clarify!
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gold46
595 posts
gold46
#3 h  Jan 14, 2012, 6:36 PM • 2 Y
Y by Adventure10, Mango247
some one could give me solution ?
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ank 1729
272 posts
ank 1729
#4 h  Jul 29, 2017, 3:47 PM • 2 Y
Y by Adventure10, Mango247
Anyone? Very interesting problem in my opinion.
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parmenides51
30653 posts
parmenides51
#5 h  Sep 24, 2017, 9:30 PM • 1 Y
Y by Adventure10
a better restatement of the problem

Let ${SABC}$. be a regular triangular pyramid (${SA=SB=SC}$. and ${AB=BC=CA)}$ ). Find the locus of all points ${D \, (D\ne S)}$. in the space that satisfy the equation ${ |cos \delta_A -2cos \delta_B - 2cos \delta_C | = 3 }$. where the angle ${\delta_X=\angle XSD}$ for each ${X \in \{ A,B,C\} }$.
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