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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
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0 replies
jlacosta
May 1, 2025
0 replies
Angles made with the median
BBNoDollar   1
N 13 minutes ago by Ianis
Determine the measures of the angles of triangle \(ABC\), knowing that the median \(BM\) makes an angle of \(30^\circ\) with side \(AB\) and an angle of \(15^\circ\) with side \(BC\).
1 reply
BBNoDollar
2 hours ago
Ianis
13 minutes ago
Find all rationals s.t..
Rushil   12
N 20 minutes ago by SomeonecoolLovesMaths
Source: Indian RMO 1994 Problem 7
Find the number of rationals $\frac{m}{n}$ such that

(i) $0 < \frac{m}{n} < 1$;

(ii) $m$ and $n$ are relatively prime;

(iii) $mn = 25!$.
12 replies
Rushil
Oct 25, 2005
SomeonecoolLovesMaths
20 minutes ago
An inequality
Rushil   11
N 25 minutes ago by SomeonecoolLovesMaths
Source: Indian RMO 1994 Problem 8
If $a,b,c$ are positive real numbers such that $a+b+c = 1$, prove that \[ (1+a)(1+b)(1+c) \geq 8 (1-a)(1-b)(1-c) . \]
11 replies
Rushil
Oct 25, 2005
SomeonecoolLovesMaths
25 minutes ago
Minimum moves to reach 25
lkason   0
44 minutes ago
Source: Final of the XXI Polish Championship in Mathematical and Logical Games
Mateusz plays a game of erasing-writing on a large board. The board is initially empty.

In each move, he can either:
-- Write two numbers equal to $1$ on the board.
-- Erase two numbers equal to $n$ and write instead the numbers $n-1$ and $n+1$.

What is the minimal number of moves Mateusz needs to make for the number 25 to appear on the board?

Note: Numbers on the board retain their values; their digits cannot be combined or split.

Spoiler, answer:
Click to reveal hidden text
0 replies
lkason
44 minutes ago
0 replies
Might be the first equation marathon
steven_zhang123   35
N 4 hours ago by lightsbug
As far as I know, it seems that no one on HSM has organized an equation marathon before. Click to reveal hidden textSo why not give it a try? Click to reveal hidden text Let's start one!
Some basic rules need to be clarified:
$\cdot$ If a problem has not been solved within $5$ days, then others are eligible to post a new probkem.
$\cdot$ Not only simple one-variable equations, but also systems of equations are allowed.
$\cdot$ The difficulty of these equations should be no less than that of typical quadratic one-variable equations. If the problem involves higher degrees or more variables, please ensure that the problem is solvable (i.e., has a definite solution, rather than an approximate one).
$\cdot$ Please indicate the domain of the solution to the equation (e.g., solve in $\mathbb{R}$, solve in $\mathbb{C}$).
Here's an simple yet fun problem, hope you enjoy it :P :
P1
35 replies
steven_zhang123
Jan 20, 2025
lightsbug
4 hours ago
Maximum number of empty squares
Ecrin_eren   0
5 hours ago


There are 16 kangaroos on a giant 4×4 chessboard, with exactly one kangaroo on each square. In each round, every kangaroo jumps to a neighboring square (up, down, left, or right — but not diagonally). All kangaroos stay on the board. More than one kangaroo can occupy the same square. What is the maximum number of empty squares that can exist after 100 rounds?



0 replies
Ecrin_eren
5 hours ago
0 replies
THREE People Meet at the SAME. TIME.
LilKirb   7
N 6 hours ago by hellohi321
Three people arrive at the same place independently, at a random between $8:00$ and $9:00.$ If each person remains there for $20$ minutes, what's the probability that all three people meet each other?

I'm already familiar with the variant where there are only two people, where you Click to reveal hidden text It was an AIME problem from the 90s I believe. However, I don't know how one could visualize this in a Click to reveal hidden text Help on what to do?
7 replies
LilKirb
Yesterday at 1:06 PM
hellohi321
6 hours ago
Quite straightforward
steven_zhang123   1
N Today at 3:16 PM by Mathzeus1024
Given that the sequence $\left \{ a_{n} \right \} $ is an arithmetic sequence, $a_{1}=1$, $a_{2}+a_{3}+\dots+a_{10}=144$. Let the general term $b_{n}$ of the sequence $\left \{ b_{n} \right \}$ be $\log_{a}{(1+\frac{1}{a_{n}} )} ( a > 0  \text{and}  a \ne  1)$, and let $S_{n}$ be the sum of the $n$ terms of the sequence $\left \{ b_{n} \right \}$. Compare the size of $S_{n}$ with $\frac{1}{3} \log_{a}{(1+\frac{1}{a_{n}} )} $.
1 reply
steven_zhang123
Jan 11, 2025
Mathzeus1024
Today at 3:16 PM
Inequalities
sqing   0
Today at 2:23 PM
Let $ a,b,c>0. $ Prove that$$a^2+b^2+c^2+abc-k(a+b+c)\geq 3k+2-2(k+1)\sqrt{k+1}$$Where $7\geq k \in N^+.$
$$a^2+b^2+c^2+abc-3(a+b+c)\geq-5$$
0 replies
sqing
Today at 2:23 PM
0 replies
Function and Quadratic equations help help help
Ocean_MathGod   1
N Today at 11:26 AM by Mathzeus1024
Consider this parabola: y = x^2 + (2m + 1)x + m(m - 3) where m is constant and -1 ≤ m ≤ 4. A(-m-1, y1), B(m/2, y2), C(-m, y3) are three different points on the parabola. Now rotate the axis of symmetry of the parabola 90 degrees counterclockwise around the origin O to obtain line a. Draw a line from the vertex P of the parabola perpendicular to line a, meeting at point H.

1) express the vertex of the quadratic equation using an expression with m.
2) If, regardless of the value of m, the parabola and the line y=x−km (where k is a constant) have exactly one point of intersection, find the value of k.

3) (where I'm struggling the most) When 1 < PH ≤ 6, compare the values of y1, y2, and y3.
1 reply
Ocean_MathGod
Aug 26, 2024
Mathzeus1024
Today at 11:26 AM
System of Equations
P162008   1
N Today at 10:30 AM by alexheinis
If $a,b$ and $c$ are complex numbers such that

$\frac{ab}{b + c} + \frac{bc}{c + a} + \frac{ca}{a + b} = -9$

$\frac{ab}{c + a} + \frac{bc}{a + b} + \frac{ca}{b + c} = 10$

Compute $\frac{a}{c + a} + \frac{b}{a + b} + \frac{c}{b + c}.$
1 reply
P162008
Yesterday at 10:34 AM
alexheinis
Today at 10:30 AM
Inequalities
sqing   19
N Today at 8:40 AM by sqing
Let $ a,b,c>0 , a+b+c +abc=4$. Prove that
$$ \frac {a}{a^2+2}+\frac {b}{b^2+2}+\frac {c}{c^2+2} \leq 1$$Let $ a,b,c>0 , ab+bc+ca+abc=4$. Prove that
$$ \frac {a}{a^2+2}+\frac {b}{b^2+2}+\frac {c}{c^2+2} \leq 1$$
19 replies
sqing
May 15, 2025
sqing
Today at 8:40 AM
System of Equations
P162008   1
N Today at 6:33 AM by lbh_qys
If $a,b$ and $c$ are real numbers such that

$\prod_{cyc} (a + b) = abc$

$\prod_{cyc} (a^3 + b^3) = (abc)^3$

Compute the value of $abc.$
1 reply
P162008
Yesterday at 10:43 AM
lbh_qys
Today at 6:33 AM
Max min in geometry
son2007vn   0
Today at 5:33 AM
Given a triangle ABC and positive real numbers m, n, p, find the point M in the plane of the triangle such that m \cdot MA + n \cdot MB + p \cdot MC is minimized.
0 replies
son2007vn
Today at 5:33 AM
0 replies
Sequence with infinite primes which we see again and again and again
Assassino9931   4
N May 2, 2025 by SimplisticFormulas
Source: Balkan MO Shortlist 2024 N6
Let $c$ be a positive integer. Prove that there are infinitely many primes, each of which divides at least one term of the sequence $a_1 = c$, $a_{n+1} = a_n^3 + c$.
4 replies
Assassino9931
Apr 27, 2025
SimplisticFormulas
May 2, 2025
Sequence with infinite primes which we see again and again and again
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Source: Balkan MO Shortlist 2024 N6
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Assassino9931
1361 posts
#1
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Let $c$ be a positive integer. Prove that there are infinitely many primes, each of which divides at least one term of the sequence $a_1 = c$, $a_{n+1} = a_n^3 + c$.
This post has been edited 1 time. Last edited by Assassino9931, Apr 27, 2025, 1:08 PM
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Parsia--
79 posts
#2
Y by
We prove the generalization (I'm not quite sure which contest this problem is from):
Let $P(x)$ be a non-constant monic polynomial with non-negative integer coefficients such that $P'(0) = 0$. Let ${a_n}$ be a sequence such that $a_0=0$ and $P(a_n)=a_{n+1}$. Prove that for every $n$, there exists a prime $r$ such that $r|a_n$ but $r \not | a_1 \cdots a_{n-1}$.
Claim: for all prime $q$ and integers $m,n$, $v_q(a_n)=v_q(a_{mn})$
Proof: Let $v_q(a_n) = \alpha$. Since $P'(0)=0$, we get $$a_{n+i} = P^i(a_n) \equiv P^i(0) = a_i \mod q^{2\alpha} \Rightarrow a_{mn} \equiv a_n \mod q^{2\alpha} \Rightarrow v_q(a_{mn})=v_q(a_n)$$Suppose that for some $m$, and each $r$ dividing $a_m$, there exists $i$ such that $r|a_i$. Then from the claim we get $$v_r(a_m) = v_r(a_{mi}) = v_r(a_i) \Rightarrow a_m | a_1 \cdots a_{m-1}$$But we have $a_m > a_{m-1}^2 > a_{m-1}a_{m-2}^2 > \cdots > a_{m-1}\cdots a_1$ which is a contradiction.
And so for every $n$, there exists a prime $r$ such that $r|a_n$ but $r \not | a_1 \cdots a_{n-1}$.

The problem is simply letting $P(x)= x^3+c$.
This post has been edited 1 time. Last edited by Parsia--, Apr 27, 2025, 2:31 PM
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Assassino9931
1361 posts
#3 • 1 Y
Y by ehuseyinyigit
Parsia-- wrote:
We prove the generalization (I'm not quite sure which contest this problem is from)

The thing you proved: Bulgaria National Round 2018

The given problem posted: here

See also: China IMO TST 2016 , IMO Shortlist 2014 N7, Silk Road 2023
This post has been edited 1 time. Last edited by Assassino9931, Apr 27, 2025, 2:44 PM
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grupyorum
1427 posts
#4
Y by
Easier (though less general) solution: It's well-known that $x\mapsto x^3$ is injective modulo $p$ for any $p\equiv 5\pmod{p}$ prime (e.g., see my solution to a 1999 Balkan problem).

We now prove any such prime works. Note that there exists $(i,j)$ such that $a_i\equiv a_j\pmod{p}$. Take such a pair $j>i\ge 1$ with the smallest coordinate sum and assume $i>1$. Then, $a_j=a_{j-1}^3+c\equiv a_{i-1}^3+c\pmod{p}$ forces $a_{j-1}\equiv a_{i-1}$, contradicting with minimality of $(i,j)$. So, $i=1$. But then, there is a $j>2$ such that $a_j=a_{j-1}^3+c\equiv c\pmod{p}$, forcing $p\mid a_{j-1}$.
This post has been edited 1 time. Last edited by grupyorum, Apr 30, 2025, 10:12 PM
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SimplisticFormulas
118 posts
#5 • 1 Y
Y by L13832
sol
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N Quick Reply
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