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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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function
BuiBaAnh   12
N 15 minutes ago by tom-nowy
Problem: Find all functions $f$: $Z->Z$ such that:
$f(xf(y)+f(x))=2f(x)+xy$ for all x,y E $Z$
12 replies
BuiBaAnh
Dec 26, 2014
tom-nowy
15 minutes ago
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IMO ShortList 2008, Number Theory problem 3
April   24
N 18 minutes ago by sansgankrsngupta
Source: IMO ShortList 2008, Number Theory problem 3
Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i + 1}) > a_{i - 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$.

Proposed by Morteza Saghafian, Iran
24 replies
April
Jul 9, 2009
sansgankrsngupta
18 minutes ago
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Find points with sames integer distances as given
nAalniaOMliO   1
N 25 minutes ago by Rohit-2006
Source: Belarus TST 2024
Points $A_1, \ldots A_n$ with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points $B_1, \ldots ,B_n$ with integer coordinates such that $A_iA_j=B_iB_j$ for every pair $1 \leq i \leq j \leq n$
N. Sheshko, D. Zmiaikou
1 reply
nAalniaOMliO
Jul 17, 2024
Rohit-2006
25 minutes ago
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Coincide
giangtruong13   2
N 28 minutes ago by giangtruong13
Source: Hanoi Specialized School's Math Test (Round 2 - Phase 1)
Let $ABCD$ be a trapezoid inscribed in circle $(O)$, $AD||BC, AD < BC$. Let $P$ is the symmetric point of $A$ across $BC$, $AP$ intersects $BC$ at $K$. Let $M$ is midpoint of $BC$ and $H$ is orthocenter of triangle $ABC$. On $BD$ take a point $F$ so that $AF||HM$. Prove that: $ FK,AC,PD$ coincide
2 replies
giangtruong13
Sunday at 4:05 PM
giangtruong13
28 minutes ago
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Product of all even divisors
girishpimoli   4
N 5 hours ago by williamxiao
$(1)$ Product of all even divisors of $9000$

$(2)$ If $4$ dice are rolled, Then number of ways of getting sum at least $13$ is
4 replies
girishpimoli
Yesterday at 2:13 PM
williamxiao
5 hours ago
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Basic geometry
AlexCenteno2007   3
N 5 hours ago by AlexCenteno2007
Given an isosceles triangle ABC with AB=BC, the inner bisector of Angle BAC And cut next to it BC in D. A point E is such that AE=DC. The inner bisector of the AED angle cuts to the AB side at the point F. Prove that the angle AFE= angle DFE
3 replies
AlexCenteno2007
Feb 9, 2025
AlexCenteno2007
5 hours ago
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BrUMO 2025 Team Round Problem 3
lpieleanu   1
N Today at 1:28 AM by franklin2013
Bruno and Brutus are running on a circular track with a $20$ foot radius. Bruno completes $5$ laps every hour, while Brutus completes $7$ laps every hour. If they start at the same point but run in opposite directions, how far along the track’s circumference (in feet) from the starting point are they when they meet for the sixth time? Note: Do not count the moment they start running as a meeting point.
1 reply
lpieleanu
Sunday at 11:05 PM
franklin2013
Today at 1:28 AM
J
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A problem with a rectangle
Raul_S_Baz   10
N Yesterday at 8:18 PM by undefined-NaN
On the sides AB and AD of the rectangle ABCD, points M and N are taken such that MB = ND. Let P be the intersection of BN and CD, and Q be the intersection of DM and CB. How can we prove that PQ || MN?
IMAGE
10 replies
Raul_S_Baz
Apr 26, 2025
undefined-NaN
Yesterday at 8:18 PM
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[CMC ARML 2020 I3] Unique Sequence
franchester   2
N Yesterday at 6:09 PM by CubeAlgo15
There is a unique nondecreasing sequence of positive integers $a_1$, $a_2$, $\ldots$, $a_n$ such that \[\left(a_1+\frac1{a_1}\right)\left(a_2+\frac1{a_2}\right)\cdots\left(a_n+\frac1{a_n}\right)=2020.\]Compute $a_1+a_2+\cdots+a_n$.

Proposed by lminsl
2 replies
franchester
May 29, 2020
CubeAlgo15
Yesterday at 6:09 PM
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Geometry Basic
AlexCenteno2007   2
N Yesterday at 5:58 PM by mathafou
Let $ABC$ be an isosceles triangle such that $AC=BC$. Let $P$ be a dot on the $AC$ side.
The tangent to the circumcircle of $ABP$ at point $P$ intersects the circumcircle of $BCP$ at $D$. Prove that CD$ \parallel$AB
2 replies
AlexCenteno2007
Yesterday at 12:11 AM
mathafou
Yesterday at 5:58 PM
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trigonogeometry 2024 TMC AIME Mock #15
parmenides51   6
N Yesterday at 5:28 PM by NamelyOrange
Let $\vartriangle ABC$ have angles $ \alpha, \beta$ and $\gamma$ such that $\cos (\alpha) = \frac1 3$ and $\cos (\beta) = \frac{1}{17}$ . Moreover, suppose that the product of the side lengths of the triangle is equal to its area. Let $(ABC)$ denote the circumcircle of $ABC$. Let $AO$ intersect $(BOC)$ at $D$, $BO$ intersect $(COA)$ at $ E$, and $CO$ intersect $(AOB)$ at $F$. If the area of $DEF$ can be written as $\frac{p\sqrt{r}}{q}$ for relatively prime integers $p$ and $q$ and squarefree $r$, find the sum of all prime factors of $q$, counting multiplicities (so the sum of prime factors of $48$ is $2 + 2 + 2 + 2 + 3 = 11$), given that $q$ has $30$ divisors.
6 replies
parmenides51
Apr 26, 2025
NamelyOrange
Yesterday at 5:28 PM
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Range of a trigonometric function
Saucepan_man02   3
N Yesterday at 5:26 PM by rchokler
Find the range of the function: $f(x)=\frac{\sin^2 x+\sin x-1}{\sin^2 x-\sin x+2}$.
3 replies
Saucepan_man02
Yesterday at 4:44 PM
rchokler
Yesterday at 5:26 PM
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hmmt quadratic power of a prime
martianrunner   3
N Yesterday at 4:54 PM by martianrunner
I was practicing problems and came across one as such:

"Find all integers $x$ such that $2x^2 + x-6$ is a positive integral power of a prime positive integer."

I mean after factoring I don't really know where to go...

A hint would be appreciated, and if you want to solve it, please hide your solutions!

Thanks :)
3 replies
martianrunner
Yesterday at 5:11 AM
martianrunner
Yesterday at 4:54 PM
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Geometry Angle Chasing
Sid-darth-vater   4
N Yesterday at 4:05 PM by mathafou
Is there a way to do this without drawing obscure auxiliary lines? (the auxiliary lines might not be obscure I might just be calling them obscure)

For example I tried rotating triangle MBC 80 degrees around point C (so the BC line segment would now lie on segment AC) but I couldn't get any results. Any help would be appreciated!
4 replies
Sid-darth-vater
Apr 21, 2025
mathafou
Yesterday at 4:05 PM
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Problem with perimeters
mateivld   6
N Nov 30, 2020 by parmenides51
Source: belarus national olympiad
Suppose that there is a point X in the plane of a given quadrilateral ABCD such that the perimeters of ABX, BCX, CDX, ADX are equal. Prove that the quadrilateral is cyclic.
6 replies
mateivld
Jan 10, 2006
parmenides51
Nov 30, 2020
J
Problem with perimeters
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Reveal topic tags
geometry
perimeter
geometry unsolved
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Source: belarus national olympiad
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mateivld
208 posts
mateivld
#1 h  Jan 10, 2006, 10:54 AM • 2 Y
Y by Adventure10, Mango247
Suppose that there is a point X in the plane of a given quadrilateral ABCD such that the perimeters of ABX, BCX, CDX, ADX are equal. Prove that the quadrilateral is cyclic.
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darij grinberg
6555 posts
darij grinberg
#2 h  Jan 10, 2006, 12:13 PM • 2 Y
Y by Adventure10, Mango247
It is not cyclic, but circumscribed. The proof is trivial: Since the perimeters of the triangles ABX, BCX, CDX, DAX are equal, we have AB + AX + BX = BC + BX + CX = CD + CX + DX = DA + DX + AX, so that

(AB + CD) + (AX + BX + CX + DX) = (AB + AX + BX) + (CD + CX + DX) = (BC + BX + CX) + (DA + DX + AX)
= (BC + DA) + (AX + BX + CX + DX),

so that AB + CD = BC + DA, and thus the quadrilateral ABCD is circumscribed.

Darij
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mateivld
208 posts
mateivld
#3 h  Jan 11, 2006, 10:34 AM • 2 Y
Y by Adventure10, Mango247
Well I obtained that it is circumscribed,but that is all you can get out of it?The statement is that it is also cyclic.
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darij grinberg
6555 posts
darij grinberg
#4 h  Jan 11, 2006, 10:39 AM • 2 Y
Y by Adventure10, Mango247
mateivld wrote:
The statement is that it is also cyclic.

Then it is wrong...

Darij
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mateivld
208 posts
mateivld
#5 h  Jan 11, 2006, 10:54 AM • 2 Y
Y by Adventure10, Mango247
Can tou tell me why?I don't think that it is wrong.
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shobber
3498 posts
shobber
#6 h  Jan 11, 2006, 11:09 AM • 2 Y
Y by Adventure10, Mango247
AB=BC=CD=DA. Then ABCD is not always cyclic.
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parmenides51
30650 posts
parmenides51
#7 h  Nov 30, 2020, 5:32 PM
Y by
(a) Suppose that there is a point $X$ in the plane of a given convex quadrilateral $ABCD$ such that the perimeters of triangles $ABX,BCX,CDX, DAX$ are equal. Show that $ABCD$ is a tangential quadrilateral.

(b) If a convex $ABCD$ is tangential, does there necessarily exist a point $X$ such that the perimeters of triangles $ABX,BCX,CDX,DAX$ are equal?

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This post has been edited 5 times. Last edited by parmenides51, Dec 16, 2022, 12:03 PM
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