Prove that $n$ is a prime number or the square of a prime number.

by kyotaro, Mar 23, 2025, 10:12 AM

Let $n$ be an odd positive integer satisfying $2^n-1$ with exactly 2 distinct prime factors. Prove that $n$ is a prime number or the square of a prime number.
number theory
prime numbers
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that statement is true

by pennypc123456789, Mar 23, 2025, 9:27 AM

we have $a^3+b^3 = 2$ and $3(a^4+b^4)+2a^4b^4 \le 8 $ , then we can deduce $a^2+b^2$ \le 2 $ ?
This post has been edited 1 time. Last edited by pennypc123456789, 4 hours ago
inqualities
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Help me please

by ntu0301, Mar 23, 2025, 7:37 AM

Determine all integers $n>1$ that satisfy the following condition: For every integer k such that $0\le k<n$ there always exists a positive integer $A$ that is divisible by n and $S(n)\equiv k (mod n) $. $S(n)$: sum of elements of $A$
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Geometry Problem

by JetFire008, Mar 23, 2025, 5:47 AM

Equilateral $\triangle ADC$ is drawn externally on side $AC$ of $\triangle ABC$. Point $P$ is taken on $BD$. Find $\angle APC$ if $BD=PA+PB+PC$.
geometry
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what actually happens after the usamo

by bubby617, Mar 23, 2025, 2:47 AM

i keep getting different answers for how the selection process gets down from the usamo winners to the IMO team so can someone set the record straight for me
AMC
USA(J)MO
USAMO
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USAMO question

by bubby617, Mar 23, 2025, 2:32 AM

if i had qualified for the usa(j)mo (i wish), would i have been flown out for free like mathcounts nationals or do you have to plan your own trip for going to the usamo
AMC
USA(J)MO
USAMO
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A hard inequality

by JK1603JK, Mar 23, 2025, 1:40 AM

Let a,b,c\ge 0: a+b+c=3. Prove \frac{1}{abc}+\frac{12}{a^2b+b^2c+c^2a}\ge 5.
inequalities
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Number theory national Olympiad

by LoRD2022, Mar 22, 2025, 9:09 PM

Find all polynomials with integer coefficients such that, $a^2+b^2-c^2|P(a)+P(b)-P(c)$ for all $a,b,c \in \mathbb{Z}$.
This post has been edited 1 time. Last edited by LoRD2022, Yesterday at 9:10 PM
number theory unsolved
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Number theory question with many (confusing) variables

by urfinalopp, Mar 22, 2025, 4:06 PM

Given m,n,p,q \in \mathbb{N+}, find all solutions to 2^{m}3^{n}+5^{p}=7^{q}$

One of the paths I've found is to boil it down to solving two non-simultaneous equations 2^{m_1}+5^{n_1}=7^{q_1} and
7^{m_1}+5^{n_1}=2^{q_1} but its too hard. Any other approaches/solutions or a continuation of this path?
number theory
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Discord Server

by mathprodigy2011, Mar 21, 2025, 11:00 PM

Theres a server where we are all like discussing problems+helping each other practice. Hopefully you guys can join.

https://discord.gg/6hN3w4eK
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Click here if you did not make white MOP (Fake copy of pythag's blog)

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