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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)!

Be sure to mark your calendars for the following events:
[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
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]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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one nice!
teomihai   0
9 minutes ago
3 girls and 4 boys must be seated at a round table. In how many distinct ways can they be seated so that the 3 girls do not sit next to each other and there can be a maximum of 2 girls next to each other. (The table is round so the seats are not numbered.)
0 replies
teomihai
9 minutes ago
0 replies
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Least common multiple from $n+1$ to $n+2016$.
EmersonSoriano   0
11 minutes ago
Source: 2017 Peru Southern Cone TST P7
For each positive integer $n$, define $P_n = (n+1)(n+2)(n+3)\dots(n+2016)$ and $Q_n = \text{lcm}(n+1, n+2, n+3, \dots, n+2016)$, meaning $Q_n$ is the least common multiple of the numbers $n+1, n+2, n+3, \dots, n+2016$. Determine whether or not there exists a constant $C$ such that
$$ \frac{P_n}{Q_n} < C, $$for every positive integer $n$.
0 replies
EmersonSoriano
11 minutes ago
0 replies
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Moving tokens from the leftmost end to the rightmost end.
EmersonSoriano   0
16 minutes ago
Source: 2017 Peru Southern Cone TST P6
Let $n$ and $\ell$ be positive integers with $\ell > 7$. There are $n$ tokens placed initially in the leftmost cell of a horizontal row consisting of $\ell$ cells. A move consists of shifting any token $1$, $2$, $3$, $4$, $5$, or $6$ positions to the right. Andrés and Beto alternate turns, starting with Andrés. The winner is the player who moves a token into the rightmost cell. Determine who has a winning strategy in terms of $n$ and $\ell$.
0 replies
EmersonSoriano
16 minutes ago
0 replies
J
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A point on the midline of BC.
EmersonSoriano   0
20 minutes ago
Source: 2017 Peru Southern Cone TST P5
Let $ABC$ be an acute triangle with circumcenter $O$. Draw altitude $BQ$, with $Q$ on side $AC$. The parallel line to $OC$ passing through $Q$ intersects line $BO$ at point $X$. Prove that point $X$ and the midpoints of sides $AB$ and $AC$ are collinear.
0 replies
EmersonSoriano
20 minutes ago
0 replies
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Problem 5 of Second round
Pinko   0
Oct 10, 2019
Source: V International Festival of Young Mathematicians Sozopol 2014, Theme for 10-12 grade
Let $ABCD$ be a convex quadrilateral. The rays $AB$ and $DC$ intersect in point $E$. Rays $AD$ and $BC$ intersect in point $F$. The angle bisector of $\angle DCF$ intersects $EF$ in point $K$. Let $I_1$ and $I_2$ be the centers of the inscribed circles in $\Delta ECB$ and $\Delta FCD$. $M$ is the projection of $I_2$ on line $CF$ and $N$ is the projection of $I_1$ on line $BC$. Let $P$ be the reflection of $N$ in $I_1$. If $P,M,K$ are colinear, prove that $ABCD$ is tangential.
0 replies
Pinko
Oct 10, 2019
0 replies
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Problem 5 of Second round
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Source: V International Festival of Young Mathematicians Sozopol 2014, Theme for 10-12 grade
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Pinko
437 posts
Pinko
#1 h  Oct 10, 2019, 2:52 PM • 2 Y
Y by Adventure10, Mango247
Let $ABCD$ be a convex quadrilateral. The rays $AB$ and $DC$ intersect in point $E$. Rays $AD$ and $BC$ intersect in point $F$. The angle bisector of $\angle DCF$ intersects $EF$ in point $K$. Let $I_1$ and $I_2$ be the centers of the inscribed circles in $\Delta ECB$ and $\Delta FCD$. $M$ is the projection of $I_2$ on line $CF$ and $N$ is the projection of $I_1$ on line $BC$. Let $P$ be the reflection of $N$ in $I_1$. If $P,M,K$ are colinear, prove that $ABCD$ is tangential.
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