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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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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Problem 5
blug   1
N 38 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
38 minutes ago
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¿10^n-1 is a divisor of 11^n-1?
EmersonSoriano   1
N 44 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
3 hours ago
Filipjack
44 minutes ago
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D1018 : Can you do that ?
Dattier   1
N an hour 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
an hour ago
J
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A special family of subsets of {1, 2, ..., 100}.
EmersonSoriano   0
2 hours 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
2 hours ago
0 replies
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6 lines concurrent, angle bisectors, tangents, circumcircle, incircle related
parmenides51   1
N Jul 1, 2021 by cadaeibf
Source: 2011 Kyiv TST2 10.2 11.2 for Ukraine MO
The bisectors of the angles $A, B$ and $C$ of the triangle $ABC$ intersect the circle circumscribed around this triangle S_1 (O, R) at points $A_2, B_2, C_2$, respectively. Tangents to the circle $S_1$ at points $A_2, B_2, C_2$ intersect between at points $A_3, B_3, C_3$ (points $A$ and $A_3$ lie on one side of the line $BC$, similarly for others points). Let the circle $S_2 (I, r)$ be inscribed in the triangle $ABC$ touches its sides at points $A_1, B_1, C_1$, respectively (point $A_1 \in BC$, similarly for other points). Prove that lines $A_1A_2$, $B_1B_2$, $C_1C_2$, $AA_3$, $BB_3$, $CC_3$ intersect at one point.
1 reply
parmenides51
Jun 30, 2021
cadaeibf
Jul 1, 2021
J
6 lines concurrent, angle bisectors, tangents, circumcircle, incircle related
G H J
Reveal topic tags
geometry
concurrency
concurrent
L
G H BBookmark kLocked kLocked NReply
Source: 2011 Kyiv TST2 10.2 11.2 for Ukraine MO
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parmenides51
30629 posts
parmenides51
#1 h  Jun 30, 2021, 10:41 PM • 1 Y
Y by HWenslawski
The bisectors of the angles $A, B$ and $C$ of the triangle $ABC$ intersect the circle circumscribed around this triangle S_1 (O, R) at points $A_2, B_2, C_2$, respectively. Tangents to the circle $S_1$ at points $A_2, B_2, C_2$ intersect between at points $A_3, B_3, C_3$ (points $A$ and $A_3$ lie on one side of the line $BC$, similarly for others points). Let the circle $S_2 (I, r)$ be inscribed in the triangle $ABC$ touches its sides at points $A_1, B_1, C_1$, respectively (point $A_1 \in BC$, similarly for other points). Prove that lines $A_1A_2$, $B_1B_2$, $C_1C_2$, $AA_3$, $BB_3$, $CC_3$ intersect at one point.
This post has been edited 1 time. Last edited by parmenides51, Jun 30, 2021, 10:41 PM
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cadaeibf
700 posts
cadaeibf
#2 h  Jul 1, 2021, 2:54 PM
Y by
Since $A_2$ is the midpoint of arc $BA_2C$, the tangent $B_3A_2C_3$ is parallel to line $BA_1C$ and cyclically. Therefore $\Delta ABC$ is directly similar to $A_3B_3C_3$. In particular their contact triangles $\Delta A_1B_1C_1$ and $\Delta A_2B_2C_2$ are also similar. Therefore, there exist a direct homothety between the pairs of triangles, and this implies that there exist a point $X$, such that $X\in A_1A_2,B_1B_2,C_1C_2,AA_3,BB_3,CC_3$ (and also $X\in OI$), and so we are done.
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