Summer is a great time to explore cool problems to keep your skills sharp!  Schedule a class today!

G
Topic
First Poster
Last Poster
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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]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]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

Prealgebra 2
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Tuesday, Jun 10 - Aug 26

Calculus
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!


MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT

Programming

Introduction to Programming with Python
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
May 1, 2025
0 replies
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
3^n + 61 is a square
VideoCake   28
N 33 minutes ago by Jupiterballs
Source: 2025 German MO, Round 4, Grade 11/12, P6
Determine all positive integers \(n\) such that \(3^n + 61\) is the square of an integer.
28 replies
VideoCake
May 26, 2025
Jupiterballs
33 minutes ago
Centroid, altitudes and medians, and concyclic points
BR1F1SZ   5
N 36 minutes ago by AshAuktober
Source: Austria National MO Part 1 Problem 2
Let $\triangle{ABC}$ be an acute triangle with $BC > AC$. Let $S$ be the centroid of triangle $ABC$ and let $F$ be the foot of the perpendicular from $C$ to side $AB$. The median $CS$ intersects the circumcircle $\gamma$ of triangle $\triangle{ABC}$ at a second point $P$. Let $M$ be the point where $CS$ intersects $AB$. The line $SF$ intersects the circle $\gamma$ at a point $Q$, such that $F$ lies between $S$ and $Q$. Prove that the points $M,P,Q$ and $F$ lie on a circle.

(Karl Czakler)
5 replies
BR1F1SZ
May 5, 2025
AshAuktober
36 minutes ago
An easy number theory problem
TUAN2k8   0
2 hours ago
Source: Own
Find all positive integers $n$ such that there exist positive integers $a$ and $b$ with $a \neq b$ satifying the condition that,
$1) \frac{a^n}{b} + \frac{b^n}{a}$ is an integer.
$2) \frac{a^n}{b} + \frac{b^n}{a} | a^{10}+b^{10}$.
0 replies
TUAN2k8
2 hours ago
0 replies
Polynomial having infinitely many prime divisors
goodar2006   12
N 2 hours ago by quantam13
Source: Iran 3rd round 2011-Number Theory exam-P1
$P(x)$ is a nonzero polynomial with integer coefficients. Prove that there exists infinitely many prime numbers $q$ such that for some natural number $n$, $q|2^n+P(n)$.

Proposed by Mohammad Gharakhani
12 replies
goodar2006
Sep 19, 2012
quantam13
2 hours ago
Reflected point lies on radical axis
Mahdi_Mashayekhi   4
N 2 hours ago by SimplisticFormulas
Source: Iran 2025 second round P4
Given is an acute and scalene triangle $ABC$ with circumcenter $O$. $BO$ and $CO$ intersect the altitude from $A$ to $BC$ at points $P$ and $Q$ respectively. $X$ is the circumcenter of triangle $OPQ$ and $O'$ is the reflection of $O$ over $BC$. $Y$ is the second intersection of circumcircles of triangles $BXP$ and $CXQ$. Show that $X,Y,O'$ are collinear.
4 replies
Mahdi_Mashayekhi
Apr 19, 2025
SimplisticFormulas
2 hours ago
IMO Shortlist 2013, Number Theory #3
lyukhson   49
N 2 hours ago by lakshya2009
Source: IMO Shortlist 2013, Number Theory #3
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
49 replies
lyukhson
Jul 10, 2014
lakshya2009
2 hours ago
classical triangle geo - points on circle
Valentin Vornicu   63
N 2 hours ago by endless_abyss
Source: USAMO 2005, problem 3, Zuming Feng
Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on its side $BC$. Construct a point $C_{1}$ in such a way that the convex quadrilateral $APBC_{1}$ is cyclic, $QC_{1}\parallel CA$, and $C_{1}$ and $Q$ lie on opposite sides of line $AB$. Construct a point $B_{1}$ in such a way that the convex quadrilateral $APCB_{1}$ is cyclic, $QB_{1}\parallel BA$, and $B_{1}$ and $Q$ lie on opposite sides of line $AC$. Prove that the points $B_{1}$, $C_{1}$, $P$, and $Q$ lie on a circle.
63 replies
Valentin Vornicu
Apr 21, 2005
endless_abyss
2 hours ago
easy number theory sequnce problem
skellyrah   3
N 2 hours ago by grupyorum
Source: simillar to 2016 Greece,Team Selection Test,Problem
Define the sequnce ${(a_n)}_{n\ge0}$ by $a_0=3$ and $a_n=2a_{n-1}+1$
Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.
3 replies
skellyrah
2 hours ago
grupyorum
2 hours ago
Finding a subsquare from the main square
goodar2006   2
N 2 hours ago by quantam13
Source: Iran 3rd round 2012-Special Lesson exam-Part 2-P4
Prove that if $n$ is large enough, in every $n\times n$ square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by $1391$.
2 replies
goodar2006
Sep 15, 2012
quantam13
2 hours ago
Three sets having the same color
goodar2006   2
N 2 hours ago by quantam13
Source: Iran 3rd round 2012-Special Lesson exam-Part 2-P3
Prove that if $n$ is large enough, then for each coloring of the subsets of the set $\{1,2,...,n\}$ with $1391$ colors, two non-empty disjoint subsets $A$ and $B$ exist such that $A$, $B$ and $A\cup B$ are of the same color.
2 replies
goodar2006
Sep 15, 2012
quantam13
2 hours ago
1000 points with distinct pairwise distances
goodar2006   2
N 2 hours ago by quantam13
Source: Iran 3rd round 2012-Special Lesson exam-Part1-P3
Prove that if $n$ is large enough, among any $n$ points of plane we can find $1000$ points such that these $1000$ points have pairwise distinct distances. Can you prove the assertion for $n^{\alpha}$ where $\alpha$ is a positive real number instead of $1000$?
2 replies
goodar2006
Jul 27, 2012
quantam13
2 hours ago
JBMO Shortlist 2023 N3
Orestis_Lignos   10
N 2 hours ago by AylyGayypow009
Source: JBMO Shortlist 2023, N3
Let $A$ be a subset of $\{2,3, \ldots, 28 \}$ such that if $a \in A$, then the residue obtained when we divide $a^2$ by $29$ also belongs to $A$.

Find the minimum possible value of $|A|$.
10 replies
Orestis_Lignos
Jun 28, 2024
AylyGayypow009
2 hours ago
Random concyclicity in a square config
Maths_VC   6
N 3 hours ago by Assassino9931
Source: Serbia JBMO TST 2025, Problem 1
Let $M$ be a random point on the smaller arc $AB$ of the circumcircle of square $ABCD$, and let $N$ be the intersection point of segments $AC$ and $DM$. The feet of the tangents from point $D$ to the circumcircle of the triangle $OMN$ are $P$ and $Q$ , where $O$ is the center of the square. Prove that points $A$, $C$, $P$ and $Q$ lie on a single circle.
6 replies
Maths_VC
May 27, 2025
Assassino9931
3 hours ago
Maxi-inequality
giangtruong13   1
N 3 hours ago by giangtruong13
Let $a,b,c >0$ and $a+b+c=2abc$. Find max: $$P= \sum_{cyc} \frac{a+2}{\sqrt{6(a^2+2)}}$$
1 reply
giangtruong13
Yesterday at 3:42 PM
giangtruong13
3 hours ago
Strange angle condition and concyclic points
lminsl   129
N May 27, 2025 by Aiden-1089
Source: IMO 2019 Problem 2
In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$.

Prove that points $P,Q,P_1$, and $Q_1$ are concyclic.

Proposed by Anton Trygub, Ukraine
129 replies
lminsl
Jul 16, 2019
Aiden-1089
May 27, 2025
Strange angle condition and concyclic points
G H J
G H BBookmark kLocked kLocked NReply
Source: IMO 2019 Problem 2
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
lminsl
544 posts
#1 • 31 Y
Y by e_plus_pi, Hexagrammum16, abdelkrim, cauchyguess, char2539, Mr.Chagol, Seicchi28, a_simple_guy, Carpemath, FadingMoonlight, AlastorMoody, samuel, RAMUGAUSS, Davrbek, Vietjung, JustKeepRunning, poplintos, sabrinamath, OlympusHero, k12byda5h, bariboaa, math31415926535, HWenslawski, megarnie, TFIRSTMGMEDALIST, ImSh95, SADAT, GeoKing, Adventure10, deplasmanyollari, Rounak_iitr
In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$.

Prove that points $P,Q,P_1$, and $Q_1$ are concyclic.

Proposed by Anton Trygub, Ukraine
This post has been edited 1 time. Last edited by djmathman, Jul 18, 2019, 4:40 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
lminsl
544 posts
#4 • 20 Y
Y by TUGUMEANDREW, Assassino9931, jeteagle, Wizard_32, pavel kozlov, JustKeepRunning, HolyMath, Gumnaami_1945, jnzm02, myh2910, Kagebaka, k.ata.tkm, AllanTian, Illuzion, ImSh95, rayfish, Adventure10, Mango247, ihatemath123, Rounak_iitr
Let $R$ be the intersection of $AA_1$, $BB_1$. Let $\Omega$ denote the circumcircle of triangle $ABC$, and let $A_0=AA_1 \cap \Omega, B_0=BB_1 \cap \Omega$.

Step 1. The points $P,Q, A_0, B_0$ are concyclic.
Since $PQ \parallel BC$, $\angle PQR=\angle ABB_0=\angle PA_0B_0$. The conclusion follows.

Step 2. The points $P,Q, B_0, P_1$ are concyclic, hence $P_1$ and $Q_1$ lies in $\odot(PQA_0B_0)$.

Note that $\angle B_1B_0C=\angle BAC=\angle B_1P_1C$, or $(B_0B_1P_1C)$ are concyclic. Hence $$\angle B_0P_1P=\angle B_1CB_0=\angle ABB_0=\angle PQB_0,$$or $P, Q, B_0, P_1$ are cyclic. This concludes the proof. $\square$
This post has been edited 3 times. Last edited by lminsl, Jul 16, 2019, 3:51 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
mofumofu
179 posts
#5 • 14 Y
Y by Flash_Sloth, zhangruichong, jeteagle, Wizard_32, JustKeepRunning, HolyMath, Not_real_name, ImSh95, ike.chen, Quidditch, CyclicISLscelesTrapezoid, SADAT, Adventure10, Rounak_iitr
[asy]
 /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(13cm); 
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ 
pen dotstyle = black; /* point style */ 
real xmin = -7.63900015685, xmax = 14.779588377, ymin = -2.38665690592, ymax = 8.57946862605;  /* image dimensions */

 /* draw figures */
draw((1.,4.96)--(-0.798898966671,-0.708008219337)); 
draw((1.,4.96)--(7.68847429913,-0.833514663193)); 
draw((7.68847429913,-0.833514663193)--(-0.798898966671,-0.708008219337)); 
draw(circle((5.34391751537,3.21734977311), 4.68043258616), green); 
draw(circle((0.279066234699,2.06933899636), 2.97920568808), green); 
draw(circle((1.75406921626,4.18630229154), 4.07948863865), blue); 
draw(circle((3.47180131167,1.0560363319), 4.62068545199), red); 
draw((3.88543194857,7.66473878747)--(2.88060830443,-0.762418678059)); 
draw((-2.28184506812,3.59163983096)--(1.27443840444,-0.738667551996)); 
draw((-1.14173984034,1.31288664799)--(7.68847429913,-0.833514663193)); 
draw((-0.798898966671,-0.708008219337)--(5.73321976419,5.08551776613)); 
draw((0.386107411678,0.343008384736)--(3.00779285736,0.304240392932)); 
 /* dots and labels */
dot((1.,4.96),dotstyle); 
label("$C$", (1.0680093857,5.04959989258), NE * labelscalefactor); 
dot((7.68847429913,-0.833514663193),dotstyle); 
label("$B$", (7.75122752106,-0.739384830301), NE * labelscalefactor); 
dot((-0.798898966671,-0.708008219337),dotstyle); 
label("$A$", (-0.736145744742,-0.613878386444), NE * labelscalefactor); 
dot((3.3230046287,2.94782789959),dotstyle); 
label("$A_1$", (3.38987859704,3.04149679088), NE * labelscalefactor); 
dot((-0.22800366956,1.09078053776),dotstyle); 
label("$B_1$", (-0.171366747388,1.19027674399), NE * labelscalefactor); 
dot((0.386107411678,0.343008384736),dotstyle); 
label("$P$", (0.456165471895,0.437238080855), NE * labelscalefactor); 
dot((3.00779285736,0.304240392932),dotstyle); 
label("$Q$", (3.0761124874,0.405861469891), NE * labelscalefactor); 
dot((1.27443840444,-0.738667551996),dotstyle); 
label("$R$", (1.33471057889,-0.645254997408), NE * labelscalefactor); 
dot((2.88060830443,-0.762418678059),dotstyle); 
label("$S$", (2.95060604354,-0.66094330289), NE * labelscalefactor); 
dot((-2.28184506812,3.59163983096),dotstyle); 
label("$P_1$", (-2.22653476554,3.68471731564), NE * labelscalefactor); 
dot((3.88543194857,7.66473878747),dotstyle); 
label("$Q_1$", (3.9546575944,7.76367674098), NE * labelscalefactor); 
dot((-1.14173984034,1.31288664799),dotstyle); 
label("$E$", (-1.08128846535,1.40991302074), NE * labelscalefactor); 
dot((5.73321976419,5.08551776613),dotstyle); 
label("$D$", (5.7901893358,5.17510633644), NE * labelscalefactor); 
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); 
 /* end of picture */
[/asy]

Let $AA_1,BB_1$ intersect the circumcircle of $\triangle ABC$ again at $D,E$ respectively, and $PB_1,QA_1$ meet $BC$ again at $R,S$ respectively. It is clear that $A,R,C,P_1$ and $B,S,C,Q_1$ are cyclic. By Reim's theorem, we know that $P,Q,D,E$ are cyclic. Taking power of a point at $B_1$, we get $B_1B\cdot B_1E=B_1A\cdot B_1C=B_1R\cdot B_1P_1$. Also, since $\triangle B_1PQ\sim \triangle B_1RB$, we get $B_1B\cdot B_1P=B_1R\cdot B_1Q$. Combining the two gives us $B_1P_1\cdot B_1P= B_1Q\cdot B_1E$, whence $P_1\in (PQDE)$. Similarly $Q_1\in (PQDE)$ are we are done.
This post has been edited 4 times. Last edited by mofumofu, Jul 17, 2019, 12:39 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
nguyenhaan2209
111 posts
#6 • 5 Y
Y by top1csp2020, jeteagle, ImSh95, Pranav1056, Adventure10
$PB_1$ cuts $CB$,$AB$ at $E$,$D$,$QA_1$ cuts $CA$,$AB$ at $G$,$F$. Apply Pappus for ${B_1GA}$ and ${A_1BE}$ we have $B_1B$-$A_1G$=$Q$, $B_1E$-$A_1A$=$P$, $GE$-$AB$ collinear or parallel but $PQ$//$AB$ thus $GE//AB$. It's obvious that $P_1$,$Q_1$ also lies on $(AGE)$ thus by Reim we have the conclusion!
This post has been edited 1 time. Last edited by nguyenhaan2209, Jul 16, 2019, 12:48 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Seicchi28
252 posts
#7 • 6 Y
Y by a_simple_guy, jeteagle, Wizard_32, CrazyMathMan, ImSh95, Adventure10
Nice problem :)

Let $A_1Q \cap AC = X$, $B_1P \cap BC = Y$, and extension of $PQ$ meets $AC,BC$ at $M,N$ respectively.
From the condition of the problem, $MN||AB$.

Lemma $XY||AB||MN$.
Proof

Construct the circumcircle $\omega$ of $\bigtriangleup CXY$. Denote $P'$ as the second intersection of $YP$ and $\omega$. Since \[ \angle PP'C = \angle YP'C = \angle YXC = \angle BAC, \]therefore $P'=P_1$. Similarly, $Q'=Q_1$. So, $C,X,Y,P_1,Q_1$ are concyclic and
\[  \angle Q_1P_1P = \angle Q_1P_1Y = \angle Q_1XY = \angle Q_1QP.  \]
Done. $\blacksquare$
This post has been edited 2 times. Last edited by Seicchi28, Jul 16, 2019, 1:00 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
JustKeepRunning
2958 posts
#9 • 2 Y
Y by ImSh95, Adventure10
mofumofu wrote:
[asy]
 /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(15cm); 
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ 
pen dotstyle = black; /* point style */ 
real xmin = -7.63900015685, xmax = 14.779588377, ymin = -2.38665690592, ymax = 8.57946862605;  /* image dimensions */

 /* draw figures */
draw((1.,4.96)--(-0.798898966671,-0.708008219337)); 
draw((1.,4.96)--(7.68847429913,-0.833514663193)); 
draw((7.68847429913,-0.833514663193)--(-0.798898966671,-0.708008219337)); 
draw(circle((5.34391751537,3.21734977311), 4.68043258616), green); 
draw(circle((0.279066234699,2.06933899636), 2.97920568808), green); 
draw(circle((1.75406921626,4.18630229154), 4.07948863865), blue); 
draw(circle((3.47180131167,1.0560363319), 4.62068545199), red); 
draw((3.88543194857,7.66473878747)--(2.88060830443,-0.762418678059)); 
draw((-2.28184506812,3.59163983096)--(1.27443840444,-0.738667551996)); 
draw((-1.14173984034,1.31288664799)--(7.68847429913,-0.833514663193)); 
draw((-0.798898966671,-0.708008219337)--(5.73321976419,5.08551776613)); 
draw((0.386107411678,0.343008384736)--(3.00779285736,0.304240392932)); 
 /* dots and labels */
dot((1.,4.96),dotstyle); 
label("$C$", (1.0680093857,5.04959989258), NE * labelscalefactor); 
dot((7.68847429913,-0.833514663193),dotstyle); 
label("$B$", (7.75122752106,-0.739384830301), NE * labelscalefactor); 
dot((-0.798898966671,-0.708008219337),dotstyle); 
label("$A$", (-0.736145744742,-0.613878386444), NE * labelscalefactor); 
dot((3.3230046287,2.94782789959),dotstyle); 
label("$A_1$", (3.38987859704,3.04149679088), NE * labelscalefactor); 
dot((-0.22800366956,1.09078053776),dotstyle); 
label("$B_1$", (-0.171366747388,1.19027674399), NE * labelscalefactor); 
dot((0.386107411678,0.343008384736),dotstyle); 
label("$P$", (0.456165471895,0.437238080855), NE * labelscalefactor); 
dot((3.00779285736,0.304240392932),dotstyle); 
label("$Q$", (3.0761124874,0.405861469891), NE * labelscalefactor); 
dot((1.27443840444,-0.738667551996),dotstyle); 
label("$R$", (1.33471057889,-0.645254997408), NE * labelscalefactor); 
dot((2.88060830443,-0.762418678059),dotstyle); 
label("$S$", (2.95060604354,-0.66094330289), NE * labelscalefactor); 
dot((-2.28184506812,3.59163983096),dotstyle); 
label("$P_1$", (-2.22653476554,3.68471731564), NE * labelscalefactor); 
dot((3.88543194857,7.66473878747),dotstyle); 
label("$Q_1$", (3.9546575944,7.76367674098), NE * labelscalefactor); 
dot((-1.14173984034,1.31288664799),dotstyle); 
label("$E$", (-1.08128846535,1.40991302074), NE * labelscalefactor); 
dot((5.73321976419,5.08551776613),dotstyle); 
label("$D$", (5.7901893358,5.17510633644), NE * labelscalefactor); 
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); 
 /* end of picture */
[/asy]

Let $AA_1,BB_1$ intersect the circumcircle of $\triangle ABC$ again at $D,E$ respectively, and $PB_1,QA_1$ meet $BC$ again at $R,S$ respectively. It is clear that $A,R,C,P_1$ and $B,S,C,Q_1$ are cyclic. By Reim's theorem, we know that $P,Q,D,E$ are cyclic. Taking power of a point at $B_1$, we get $B_1B\cdot B_1E=B_1A\cdot B_1C=B_1R\cdot B_1P_1$. Also, since $\triangle B_1PQ\sim \triangle B_1RB$, we get $B_1B\cdot B_1P=B_1R\cdot B_1Q$. Combining the two gives us $B_1P_1\cdot B_1P= B_1Q\cdot B_1E$, whence $P_1\in (PQDE)$. Similarly $Q_1\in (BQDE)$ are we are done.

How do you import a graph form geogebra?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Wictro
119 posts
#10 • 3 Y
Y by ImSh95, Adventure10, Mango247
Seicchi28 wrote:
Nice problem :)
Let $A_1Q$ meet $AC$ at $X$, $B_1P$ meet $BC$ at $Y$, extension of $PQ$ meets $AC,BC$ at $M,N$ respectively.
From the question, $MN||AB$.

Lemma

Now by the lemma, create the circumcircle $\omega$ of $\bigtriangleup CXY$. Denote $P'$ as the second intersection of $YP$ and $\omega$. Since \[ \angle PP'C = \angle YP'C = \angle YXC = \angle BAC, \]therefore $P'=P_1$. Similarly, $Q'=Q_1$. So, $C,X,Y,P_1,Q_1$ are concyclic and
\[  \angle Q_1P_1P = \angle Q_1P_1Y = \angle Q_1XY = \angle Q_1QP.\]Done.

Exactly how I did it. Though I proved the lemma with Pappus and then Desargues on perspective triangles.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
TUGUMEANDREW
5 posts
#13 • 2 Y
Y by ImSh95, Adventure10
Since Problem 2 was geometry and problem 2 is medium level, is there a chance that Problem 4 won't be geometry?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
richrow12
411 posts
#14 • 2 Y
Y by ImSh95, Adventure10
Seicchi28 wrote:
Nice problem :)

Let $A_1Q \cap AC = X$, $B_1P \cap BC = Y$, and extension of $PQ$ meets $AC,BC$ at $M,N$ respectively.
From the condition of the problem, $MN||AB$.

Lemma $XY||AB||MN$.
Proof

Construct the circumcircle $\omega$ of $\bigtriangleup CXY$. Denote $P'$ as the second intersection of $YP$ and $\omega$. Since \[ \angle PP'C = \angle YP'C = \angle YXC = \angle BAC, \]therefore $P'=P_1$. Similarly, $Q'=Q_1$. So, $C,X,Y,P_1,Q_1$ are concyclic and
\[  \angle Q_1P_1P = \angle Q_1P_1Y = \angle Q_1XY = \angle Q_1QP.  \]
Done. $\blacksquare$

Just the same but my proof of the lemma as follows:

Consider two lines $(P, B_1, Y)$ and $(Q, A_1, X)$. By Pappus's theorem points $S=PA_1\cap QB_1$, $T=PX\cap QY$ and $C=B_1Y\cap A_1X$ are collinear. Now, consider trinagles $APX$ and $BQY$. We claim that they are perspective. Indeed, $AP\cap BQ = S$, $AX\cap BY=C$ and $PX\cap QY=T$ are colinear. Hence, by Desargues's theorem lines $AB$, $PQ$ and $XY$ are concurrent. Since $AB\parallel PQ$ we have $AB\parallel PQ\parallel XY$.
TUGUMEANDREW wrote:
Since Problem 2 was geometry and problem 2 is medium level, is there a chance that Problem 4 won't be geometry?

Usually (at least in recent years) problems 1,2,4,5 have different topics (ACGN).
This post has been edited 1 time. Last edited by richrow12, Jul 16, 2019, 1:56 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
-[]-
97 posts
#15 • 5 Y
Y by richrow12, ImSh95, carefully, Adventure10, Mango247
Finally, the first medium geometry problem since the introduction of the new protocol! So surprised that they didn't vote for G1 this time.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Math-wiz
6107 posts
#16 • 3 Y
Y by ImSh95, Adventure10, Mango247
TUGUMEANDREW wrote:
Since Problem 2 was geometry and problem 2 is medium level, is there a chance that Problem 4 won't be geometry?

P6 can be a Geo. P4,5 would be NT and Combi
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
MS_Kekas
275 posts
#17 • 65 Y
Y by a_simple_guy, Wictro, richrow12, MathPassionForever, Mr.Chagol, rmtf1111, RudraRockstar, SHREYAS333, FadingMoonlight, AlastorMoody, UlanKZ, 62861, MNJ2357, GourmetSalad, Illuzion, Systematicworker, anantmudgal09, juckter, niyu, samuel, Anzoteh, Kayak, AlbertEinstein1905, jeteagle, Euler1728, Wizard_32, myh2910, Gumnaami_1945, Bassiskicking, Geronimo_1501, Abbas11235, aa1024, MarkBcc168, Aryan-23, amar_04, Atpar, k12byda5h, CHLORG1, CrazyMathMan, blackbluecar, 554183, tigerzhang, Bubu-Droid, Seicchi28, Wizard0001, rayfish, ImSh95, ike.chen, Pranav1056, Quidditch, parmenides51, IMUKAT, sabkx, wenwenma, CyclicISLscelesTrapezoid, kamatadu, Adventure10, Mango247, LoloChen, EpicBird08, Kingsbane2139, khina, ihatemath123, farhad.fritl, Sedro
Lol, this is my problem! Didn't expect to see this at P2
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
AngleChasingXD
109 posts
#18 • 5 Y
Y by jeteagle, ImSh95, DashTheSup, Adventure10, Mango247
Beautiful problem!!! :)
My solution is kinda long, especially because I used Moving Points during it.

Let $B_1P_1$ cut $AB$ at $U$ and $Q_1A_1$ cut $AB$ at $V$. Moreover, let $L$ be the intersection point of lines $B_1P_1$ and $BC$ and $K$ the intersection point of lines $Q_1A_1$ and $AC$.

Since $\angle PP_1C\equiv \angle BAC$, we conclude that $AP_1CU$ is cyclic. Similarly, since $\angle QQ_1C\equiv \angle ABC$, $BQ_1CV$ is cyclic, too. Keep these in mind as (1).

The conclusion is equivalent to $\angle QPP_1\equiv \angle QQ_1P_1$. However, since $PQ\parallel AB$, we have $\angle QPP_1\equiv \angle AUP_1$, so it suffices to prove $\angle AUP_1\equiv \angle QQ_1P_1$, which is equivalent to $UVP_1Q_1$ being cyclic.

$\textbf{Claim.}$ We have that $KL\parallel AB$.

$\textbf{Proof:}$ We show this by the Moving Points method. Fix $\Delta ABC$, points $A_1$, $B_1$ and animate $P$ on $AA_1$. Consider $L'$ on $AB$ such that $KL'\parallel AB$. We prove that the map $$L\to P\to Q\to K\to L'~~by~~\overline{BC}\to \overline{AA_1}\to \overline{BB_1}\to \overline{AC}\to \overline{BC}$$is projective, thus it would ve enough to check $L=L'$ for $3$ distinct positions of $P$.
$\textbf{(1)}$ $L\to P$ by $\overline{BC}\to \overline{AA_1}$ is projective since it is a perspectivity through $B_1$ from $BC$ to $AA_1$.
$\textbf{(2)}$ $P\to Q$ by $\overline{AA_1}\to \overline{BB_1}$ is projective since it preserves linear motion.
$\textbf{(3)}$ $Q\to K$ by $\overline{BB_1}\to \overline{AC}$ is projective since it is a perspectivity through $A_1$ from $BB_1$ to $AC$.
$\textbf{(4)}$ $K\to L'$ by $ \overline{AC}\to \overline{BC}$ is projective since it preserves linear motion.

Now it suffices to check for $3$ distinct positions of $P$ the desired parallelism.
$\textbf{(1)}$ When $P\in AA_1\cap BB_1$, we actually have $Q=P$, $L=B$ and $K=A$ so there is nothing to prove.
$\textbf{(2)}$ When $P=A$, $Q=B$ so $C=K=L$, again nothing to prove.
$\textbf{(3)}$ When $P$ is the point at infinity on $\overline{AA_1}$, $Q$ is the point at infinity on $\overline{BB_1}$ so $B_1L\parallel AA_1$ and $A_1K\parallel BB_1$. Then apply Thales' Lemma to obtain $CB_1\cdot CA_1=CK\cdot CB$ and $CA_1\cdot CB_1=CL\cdot CA$, hence $CK\cdot CB=CL\cdot CA$ so indeed $KL\parallel BC$.$\blacksquare$

Now we are ready to finish. From (1) and Power of a Point Theorem we get $PB_1\cdot B_1U=AB_1\cdot CB_1$. By our lemma and Thales' Lemma, $B_1K\cdot B_1U=B_1L\cdot B_1A$. Divide these two last relations in order to get $PB_1\cdot B_1L=CB_1\cdot B_1K$, which again by Power of a Point leads to $CP_1KL$ being cyclic. In a similar manner one proves the cyclicity of $CQ_1KL$, so the pentagon $CP_1KLQ_1$ is cyclic. But then $\angle KQ_1P_1\equiv \angle KP_1C$, and since by (1) we have $\angle KCP_1\equiv \angle AUP_1$, we arrive at $\angle AUP_1\equiv \angle VQ_1P_1$, hence $UVQ_1P_1$ is cyclic and done.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
prague123
230 posts
#19 • 2 Y
Y by ImSh95, Adventure10
-[]- wrote:
Finally, the first medium geometry problem since the introduction of the new protocol! So surprised that they didn't vote for G1 this time.

This probably just means that G1 and G2 got eliminated as known problems, and that G3 contained traces of combinatorial geometry.
This post has been edited 1 time. Last edited by prague123, Jul 16, 2019, 2:13 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Math-wiz
6107 posts
#20 • 2 Y
Y by ImSh95, Adventure10
MS_Kekas wrote:
Lol, this is my problem! Didn't expect to see this at P2

Wowww. Where are you from?
Z K Y
G
H
=
a