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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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Floor function for polynomials
kred9   0
4 minutes ago
Source: 2025 Utah Math Olympiad #2
Given polynomials $f(x)$ and $g(x)$, where $g(x)$ is not the zero polynomial, we define $\left \lfloor \frac{f(x)}{g(x)} \right \rfloor$ to be the unique polynomial $q(x)$ such that we can write $f(x)=g(x)\cdot q(x) + r(x)$, where $r(x)$ is a polynomial such that either $r(x)=0$ or the degree of $r(x)$ is less than the degree of $g(x)$. Find all polynomials $p(x)$ with real coefficients such that $$\left \lfloor \frac{p(x)}{x} \right \rfloor + \left \lfloor \frac{p(x)}{x+1} \right \rfloor =x^2.$$
0 replies
kred9
4 minutes ago
0 replies
J
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Squares in an Octagon
kred9   0
7 minutes ago
Source: 2025 Utah Math Olympiad #1
A regular octagon and all of its diagonals are drawn. Find, with proof, the number of squares that appear in the resulting diagram. (The side of each square must lie along one of the edges or diagonals of the octagon.)
0 replies
kred9
7 minutes ago
0 replies
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equal angles
jhz   4
N 23 minutes ago by sami1618
Source: 2025 CTST P16
In convex quadrilateral $ABCD, AB \perp AD, AD = DC$. Let $ E$ be a point on side $BC$, and $F$ be a point on the extension of $DE$ such that $\angle ABF = \angle DEC>90^{\circ}$. Let $O$ be the circumcenter of $\triangle CDE$, and $P$ be a point on the side extension of $FO$ satisfying $FB =FP$. Line BP intersects AC at point Q. Prove that $\angle AQB =\angle DPF.$
4 replies
jhz
Mar 26, 2025
sami1618
23 minutes ago
J
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Divisibility involving the product and sum of two numbers
EmersonSoriano   1
N 29 minutes ago by mathprodigy2011
Source: 2017 Peru Southern Cone TST P8
Determine the smallest positive integer $n$ for which the following statement is true: If positive integers $a$ and $b$ are such that $a + b$ is a multiple of $100$ and $ab$ is a multiple of $n$, then each of the numbers $a$ and $b$ is a multiple of $100$.

1 reply
EmersonSoriano
4 hours ago
mathprodigy2011
29 minutes ago
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triangle of medians of ABC similar to ABC criterion
parmenides51   0
Jan 1, 2022
Source: 2011 Saudi Arabia Pre-TST February 2.4 https://artofproblemsolving.com/community/c2745403_2011_
Let $ABC$ be a triangle with medians $m_a$ , $m_b$, $m_c$. Prove that:
(a) There is a triangle with side lengths $m_a$ ,$m_b$, $m_c$.
(b) This triangle is similar to $ABC$ if and only if the squares of the side lengths of triangle $ABC$ form an arithmetical sequence.
0 replies
parmenides51
Jan 1, 2022
0 replies
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triangle of medians of ABC similar to ABC criterion
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Source: 2011 Saudi Arabia Pre-TST February 2.4 https://artofproblemsolving.com/community/c2745403_2011_
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parmenides51
30629 posts
parmenides51
#1 h  Jan 1, 2022, 10:11 PM
Y by
Let $ABC$ be a triangle with medians $m_a$ , $m_b$, $m_c$. Prove that:
(a) There is a triangle with side lengths $m_a$ ,$m_b$, $m_c$.
(b) This triangle is similar to $ABC$ if and only if the squares of the side lengths of triangle $ABC$ form an arithmetical sequence.
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