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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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NEPAL TST 2025 DAY 2
Tony_stark0094   8
N 5 minutes ago by cursed_tangent1434
Consider an acute triangle $\Delta ABC$. Let $D$ and $E$ be the feet of the altitudes from $A$ to $BC$ and from $B$ to $AC$ respectively.

Define $D_1$ and $D_2$ as the reflections of $D$ across lines $AB$ and $AC$, respectively. Let $\Gamma$ be the circumcircle of $\Delta AD_1D_2$. Denote by $P$ the second intersection of line $D_1B$ with $\Gamma$, and by $Q$ the intersection of ray $EB$ with $\Gamma$.

If $O$ is the circumcenter of $\Delta ABC$, prove that $O$, $D$, and $Q$ are collinear if and only if quadrilateral $BCQP$ can be inscribed within a circle.

$\textbf{Proposed by Kritesh Dhakal, Nepal.}$
8 replies
Tony_stark0094
Apr 12, 2025
cursed_tangent1434
5 minutes ago
J
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Interesting inequalities
sqing   4
N 11 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $ ab+bc+ca+abc=4$ . Prove that
$$k(a+b+c) -ab-bc\geq 4\sqrt{k(k+1)}-(k+4)$$Where $ k\geq \frac{16}{9}. $
$$ \frac{16}{9}(a+b+c) -ab-bc\geq \frac{28}{9}$$
4 replies
sqing
3 hours ago
sqing
11 minutes ago
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NEPAL TST DAY 2 PROBLEM 2
Tony_stark0094   6
N 23 minutes ago by cursed_tangent1434
Kritesh manages traffic on a $45 \times 45$ grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the $45 \times 45$ grid.

Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove $k$ cars from the $45 \times 45$ grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid.

What is the minimum value of $k$ that guarantees that Kritesh's job is possible?

$\textbf{Proposed by Shining Sun. USA}$
6 replies
Tony_stark0094
Apr 12, 2025
cursed_tangent1434
23 minutes ago
J
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NEPAL TST DAY-2 PROBLEM 1
Tony_stark0094   9
N 42 minutes ago by cursed_tangent1434
Let the sequence $\{a_n\}_{n \geq 1}$ be defined by
\[ a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N} \]Prove that
\[ a_n^{2025} >n^{2024} \]for all positive integers $n \geq 2$.

$\textbf{Proposed by Prajit Adhikari, Nepal.}$
9 replies
Tony_stark0094
Apr 12, 2025
cursed_tangent1434
42 minutes ago
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line segments among 5 intersecting lines
parmenides51   0
Dec 5, 2020
Source: 2001 Belarusian MO D 8.4
The five straight lines intersect as shown. Can the segments obtained when crossing, have the lengths indicated in the figure?
IMAGE
0 replies
parmenides51
Dec 5, 2020
0 replies
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line segments among 5 intersecting lines
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Source: 2001 Belarusian MO D 8.4
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parmenides51
30629 posts
parmenides51
#1 h  Dec 5, 2020, 1:01 PM • 2 Y
Y by Mango247, Mango247
The five straight lines intersect as shown. Can the segments obtained when crossing, have the lengths indicated in the figure?
//cdn.artofproblemsolving.com/images/0/3/f/03fd27a63b29160f497b6a61075d4f42a9daced1.jpg
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