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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. 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)!

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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
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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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MO _ |_ YZ if O is circumcenter of ABD, <BAD =<CAD
parmenides51   2
N 9 minutes ago by ihategeo_1969
Source: 2019 Geo Mock - Olympiad by Tovi Wen #2 https://artofproblemsolving.com/community/c594864h1787237p11805928
Let $ABC$ be a triangle, and let $D$ be on $\overline{BC}$ so that $\angle BAD = \angle CAD$. Let $\overline{AD}$ intersect the circumcircle of $\triangle ABC$ again at $M$. Let $Y$ be the midpoint of $\overline{MB}$ and let the perpendicular bisector of $\overline{AM}$ intersect $\overline{MC}$ at $Z$. If $O$ is the circumcenter of $\triangle ABD$, prove that $\overline{MO} \perp \overline{YZ}$.
2 replies
parmenides51
Nov 26, 2023
ihategeo_1969
9 minutes ago
J
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IMO ShortList 1998, number theory problem 8
orl   20
N 11 minutes ago by Maximilian113
Source: IMO ShortList 1998, number theory problem 8
Let $a_{0},a_{1},a_{2},\ldots $ be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form $a_{i}+2a_{j}+4a_{k}$, where $i,j$ and $k$ are not necessarily distinct. Determine $a_{1998}$.
20 replies
orl
Oct 22, 2004
Maximilian113
11 minutes ago
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Concurrence in Cyclic Quadrilateral
GrantStar   38
N an hour ago by wu2481632
Source: IMO Shortlist 2023 G3
Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$.

Prove that lines $AD, PM$, and $BC$ are concurrent.
38 replies
GrantStar
Jul 17, 2024
wu2481632
an hour ago
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IMO Shortlist 2014 C9
hajimbrak   13
N an hour ago by One_piece_fan
There are $n$ circles drawn on a piece of paper in such a way that any two circles intersect in two points, and no three circles pass through the same point. Turbo the snail slides along the circles in the following fashion. Initially he moves on one of the circles in clockwise direction. Turbo always keeps sliding along the current circle until he reaches an intersection with another circle. Then he continues his journey on this new circle and also changes the direction of moving, i.e. from clockwise to anticlockwise or $\textit{vice versa}$.
Suppose that Turbo’s path entirely covers all circles. Prove that $n$ must be odd.

Proposed by Tejaswi Navilarekallu, India
13 replies
hajimbrak
Jul 11, 2015
One_piece_fan
an hour ago
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Angle and a point
joybangla   0
May 15, 2014
Source: Tournament of Towns, Fall 2002, Junior O Level, P5
An angle and a point $A$ inside it is given. Is it possible to draw through $A$ three straight lines so that on either side of the angle one of three points of intersection of these lines be the midpoint of two other points of intersection with that side?
0 replies
joybangla
May 15, 2014
0 replies
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Angle and a point
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Source: Tournament of Towns, Fall 2002, Junior O Level, P5
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joybangla
836 posts
joybangla
#1 h  May 15, 2014, 4:42 PM • 2 Y
Y by Adventure10, Mango247
An angle and a point $A$ inside it is given. Is it possible to draw through $A$ three straight lines so that on either side of the angle one of three points of intersection of these lines be the midpoint of two other points of intersection with that side?
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