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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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Problem 5
blug   1
N 9 minutes ago by atdaotlohbh
Source: Polish Math Olympiad 2025 Finals P5
Convex quadrilateral $ABCD$ is described on a circle $\omega$, and is not a trapezius inscribed in a circle. Let the tangency points of $\omega$ and $AB, BC, CD, DA$ be $K, L, M, N$ respectively. A circle with a center $I_K$, different from $\omega$ is tangent to the segement $AB$ and lines $AD, BC$. A circle with center $I_L$, different from $\omega$ is tangent to segment $BC$ and lines $AB, CD$. A circle with center $I_M$, different from $\omega$ is tangent to segment $CD$ and lines $AD, BC$. A circle with center $I_N$, different from $\omega$ is tangent to segment $AD$ and lines $AB, CD$. Prove that the lines $I_KK, I_LL, I_MM, I_NN$ are concurrent.
1 reply
blug
Yesterday at 12:08 PM
atdaotlohbh
9 minutes ago
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¿10^n-1 is a divisor of 11^n-1?
EmersonSoriano   1
N 15 minutes ago by Filipjack
Source: 2017 Peru Southern Cone TST P2
Determine if there exists a positive integer $n$ such that $10^n - 1$ is a divisor of $11^n - 1$.
1 reply
EmersonSoriano
2 hours ago
Filipjack
15 minutes ago
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D1018 : Can you do that ?
Dattier   1
N 17 minutes ago by Dattier
Source: les dattes à Dattier
We can find $A,B,C$, such that $\gcd(A,B)=\gcd(C,A)=\gcd(A,2)=1$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$.

For example :

$C=20$
$A=47650065401584409637777147310342834508082136874940478469495402430677786194142956609253842997905945723173497630499054266092849839$

$B=238877301561986449355077953728734922992395532218802882582141073061059783672634737309722816649187007910722185635031285098751698$

Can you find $A,B,C$ such that $A>3$ is prime, $C,B \in (\mathbb Z/A\mathbb Z)^*$ with $o(C)=(A-1)/2$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$?
1 reply
Dattier
Mar 24, 2025
Dattier
17 minutes ago
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A special family of subsets of {1, 2, ..., 100}.
EmersonSoriano   0
an hour ago
Source: 2017 Peru Southern Cone TST P10
Miguel has a list consisting of several subsets of 10 elements from the set ${1,2,\dots,100}$. He says to Cecilia: "If you pick any subset of 10 elements from ${1,2,\dots,100}$, it will be disjoint with at least one subset from my list." What is the smallest possible number of subsets Miguel's list can have if what he tells Cecilia is true?
0 replies
EmersonSoriano
an hour ago
0 replies
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No more topics!
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ABC and A_{2021}C_{2021}C are similar , sequence of angle bisectors
parmenides51   1
N May 27, 2022 by RagvaloD
Source: 2021 SPbU Qualifying round, grades 10-11 p6 - Saint Petersburg State University School Olympiad
In triangle $ABC$ with sides $AB = 5$, $BC = 6$ and $AC = 4$, $AA_1$ is bisector of angle $\angle BAC$. Further, in triangle $AA_1C$, the bisector $A_1C_1$ of angle $\angle AA_1C$ is drawn, in triangle $C_1A_1C$ the bisector $C_1A_2$ of angle $\angle A_1C_1C$ is drawn, in triangle $C_1A_2C$, the bisector $A_2C_2$ of angle $\angle C_1A_2C$, $...$, in triangle $C_{2020}A_{2020}C$ the bisector $C_{2020}A_{2021}$ of angle $\angle A_{2020}C_{2020}C$ is drawn, in triangle $C_{2020}A_{2021}C$ the bisector $A_{2021}C_{2021}$ of angle $\angle C_{2020}A_{2021}C$ is drawn. Prove that triangles $ABC$ and $A_{2021}C_{2021}C$ are similar and find the ratio of similarity of these triangles.
1 reply
parmenides51
May 27, 2022
RagvaloD
May 27, 2022
J
ABC and A_{2021}C_{2021}C are similar , sequence of angle bisectors
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Reveal topic tags
ratio
similar triangles
similarity
angle bisector
geometry
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Source: 2021 SPbU Qualifying round, grades 10-11 p6 - Saint Petersburg State University School Olympiad
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parmenides51
30629 posts
parmenides51
#1 h  May 27, 2022, 9:22 PM
Y by
In triangle $ABC$ with sides $AB = 5$, $BC = 6$ and $AC = 4$, $AA_1$ is bisector of angle $\angle BAC$. Further, in triangle $AA_1C$, the bisector $A_1C_1$ of angle $\angle AA_1C$ is drawn, in triangle $C_1A_1C$ the bisector $C_1A_2$ of angle $\angle A_1C_1C$ is drawn, in triangle $C_1A_2C$, the bisector $A_2C_2$ of angle $\angle C_1A_2C$, $...$, in triangle $C_{2020}A_{2020}C$ the bisector $C_{2020}A_{2021}$ of angle $\angle A_{2020}C_{2020}C$ is drawn, in triangle $C_{2020}A_{2021}C$ the bisector $A_{2021}C_{2021}$ of angle $\angle C_{2020}A_{2021}C$ is drawn. Prove that triangles $ABC$ and $A_{2021}C_{2021}C$ are similar and find the ratio of similarity of these triangles.
This post has been edited 2 times. Last edited by parmenides51, May 28, 2022, 5:34 AM
Reason: title
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RagvaloD
4894 posts
RagvaloD
#2 h  May 27, 2022, 10:33 PM
Y by
$A_1C=\frac{BC*AC}{AB+AC}=\frac{8}{3}$
$\cos ACB=\frac{AC^2+BC^2-AB^2}{2AC*BC}=\frac{9}{16}$
$AA_1^2=AC^2+A_1C^2-2AC*A_1C*\cos ACA_1=\frac{100}{9} \to AA_1=\frac{10}{3}$
So $AB:BC:CA=5:6:4=AA_1:AC:CA_1 \to \triangle ABC \sim \triangle A_1AC$ with $k=\frac{3}{2}$
$\triangle A_1AC \sim \triangle C_1A_1C$
Continue that way we will have that $triangle ABC \sim A_{2021}C_{2021}C$ with $k= \left(\frac{3}{2}\right)^{4042}$
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