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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]
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0 replies
jlacosta
May 1, 2025
0 replies
GCD of terms in a sequence
BBNoDollar   0
15 minutes ago
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
0 replies
BBNoDollar
15 minutes ago
0 replies
Number Theory
fasttrust_12-mn   13
N 22 minutes ago by KTYC
Source: Pan African Mathematics Olympiad P1
Find all positive intgers $a,b$ and $c$ such that $\frac{a+b}{a+c}=\frac{b+c}{b+a}$ and $ab+bc+ca$ is a prime number
13 replies
fasttrust_12-mn
Aug 15, 2024
KTYC
22 minutes ago
GCD of terms in a sequence
BBNoDollar   0
25 minutes ago
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
0 replies
BBNoDollar
25 minutes ago
0 replies
Aime type Geo
ehuseyinyigit   3
N 32 minutes ago by sami1618
Source: Turkish First Round 2024
In a scalene triangle $ABC$, let $M$ be the midpoint of side $BC$. Let the line perpendicular to $AC$ at point $C$ intersect $AM$ at $N$. If $(BMN)$ is tangent to $AB$ at $B$, find $AB/MA$.
3 replies
ehuseyinyigit
Yesterday at 9:04 PM
sami1618
32 minutes ago
No more topics!
Robot tries to get to (0,0) or (1,0)
prtQ   3
N Jul 26, 2019 by DVDthe1st
Source: SMO Open 2019 Q3
A robot is placed at point $P$ on the $x$-axis but different from $(0,0)$ and $(1,0)$ and can only move along the axis either to the left or to the right. Two players play the following game. Player $A$ gives a distance and $B$ gives a direction and the robot will move the indicated distance along the indicated direction. Player $A$ aims to move the robot to either $(0,0)$ or $(1,0)$. Player $B$'s aim is to stop $A$ from achieving his aim. For which $P$ can $A$ win?
3 replies
prtQ
Jul 6, 2019
DVDthe1st
Jul 26, 2019
Robot tries to get to (0,0) or (1,0)
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Source: SMO Open 2019 Q3
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prtQ
10 posts
#1 • 1 Y
Y by Adventure10
A robot is placed at point $P$ on the $x$-axis but different from $(0,0)$ and $(1,0)$ and can only move along the axis either to the left or to the right. Two players play the following game. Player $A$ gives a distance and $B$ gives a direction and the robot will move the indicated distance along the indicated direction. Player $A$ aims to move the robot to either $(0,0)$ or $(1,0)$. Player $B$'s aim is to stop $A$ from achieving his aim. For which $P$ can $A$ win?
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NikoIsLife
9657 posts
#2 • 2 Y
Y by Adventure10, Upwgs_2008
Solution
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fattypiggy123
615 posts
#3 • 2 Y
Y by Adventure10, Mango247
Another possible solution goes by setting $S_n$ to be the set of points that are winnable in at most $n$ moves, and noting that $S_n$ consists of the midpoints of $S_0,\ldots, S_{n-1}$, it is clear that $\cup_{n=0}^{\infty} S_n$ is exactly the solution set described above.

However, one cannot conclude that this is every possible $P$ since it is not true in general that if $A$ can win from $P$ in finitely many turns, then there is an $n$ such that $A$ will definitely win in $n$ moves. To illustrate, consider a game where $B$ chooses a positive integer $n$, does nothing afterwards, and every turn $A$ gets to subtract $1$ off it and wins when the number is $0$. Then although $A$ is guaranteed to win from the starting position in finitely many turns, no upper bound for the number of turns exists.

To fix this, consider a binary tree where the vertices are the current point the game is at, and the two descendants are the two possibilities given by the two possible directions $B$ can choose. Assuming $A$ is able to win from $P$ and plays optimally, every path then has finite depth. If there is no upper bound, then there are arbitrary large paths exists and so the graph is infinite. But by Kőnig's lemma there must exist an infinite path --- a contradiction. Hence every point $P$ will exists in $S_n$ for some $n$ and we are done.
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DVDthe1st
341 posts
#4 • 2 Y
Y by Adventure10, Mango247
The following is essentially the above post restated, but it cleared up some confusion for me on why $\cup_{n=0}^{\infty} S_n$ isn't necessarily the set of winnable positions.

Suppose now there's a slightly different game:
modified game wrote:
A robot is placed at point $P$ on [...]. Two players play the following game. Before the game starts, player $B$ gives a positive integer $N$. Player $A$ gives a distance and $B$ gives a direction and the robot will move the indicated distance along the indicated direction. Player $A$ aims to move the robot to either $(0,0)$ or $(1,0)$. Player $B$'s aim is to stop $A$ from achieving his aim, however, after $N$ turns player $B$ runs out of patience and forfeits the game. [...]
It turns out each $S_i$ in the modified game is the same as in the original game. However, all positions are now winnable, so the winnable set is not $\cup_{n=0}^{\infty} S_n$.

Essentially, what must be shown is that in the original game, none of $B$'s decisions (left/right) really have the "same impact" as picking $N$ in the modified game.
This post has been edited 1 time. Last edited by DVDthe1st, Jul 26, 2019, 9:40 AM
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