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A suspcious assumption
NamelyOrange   2
N Yesterday at 1:30 AM by maromex
Let $a,b,c,d$ be positive integers. Maximize $\max(a,b,c,d)$ if $a+b+c+d=a^2-b^2+c^2-d^2=2012$.
2 replies
NamelyOrange
Thursday at 1:53 PM
maromex
Yesterday at 1:30 AM
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Got what it takes to disprove Euler?
4everwise   18
N May 21, 2025 by P162008
One of Euler's conjectures was disproved in then 1960s by three American mathematicians when they showed there was a positive integer $ n$ such that \[133^5 + 110^5 + 84^5 + 27^5 = n^5.\]Find the value of $ n$.
18 replies
4everwise
Feb 20, 2006
P162008
May 21, 2025
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2018 Mock ARML I --7 2^n | \prod^{2048}_{k=0} C(2k , k)
parmenides51   3
N May 21, 2025 by MathIQ.
Find the largest integer $n$ such that $2^n$ divides $\prod^{2048}_{k=0} {2k \choose k}$.
3 replies
parmenides51
Jan 17, 2024
MathIQ.
May 21, 2025
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Pell's Equation
Entrepreneur   0
May 19, 2025
A Pells Equation is defined as follows $$x^2-1=ky^2.$$Where $x,y$ are positive integers and $k$ is a non-square positive integer. If $(x_n,y_n)$ denotes the n-th set of solution to the equation with $(x_0,y_0)=(1,0).$ Then, prove that $$x_{n+1}x_n-ky_{n+1}y_n=x_1,$$$$x_n\pm y_n\sqrt k=(x_1\pm y_1\sqrt k)^n.$$
0 replies
Entrepreneur
May 19, 2025
0 replies
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Diophantine Equation (cousin of Mordell)
urfinalopp   4
N May 18, 2025 by FoeverResentful
Find pairs of integers $(x;y)$ such that:

$x^2=y^5+32$
4 replies
urfinalopp
May 18, 2025
FoeverResentful
May 18, 2025
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p+2^p-3=n^2
tom-nowy   1
N May 18, 2025 by urfinalopp
Let $n$ be a natural number and $p$ be a prime number. How many different pairs $(n, p)$ satisfy the equation:
$$p + 2^p - 3 = n^2 .$$
Inspired by https://artofproblemsolving.com/community/c4h3560823
1 reply
tom-nowy
May 18, 2025
urfinalopp
May 18, 2025
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Perfect cubes
Entrepreneur   6
N May 18, 2025 by NamelyOrange
Find all ordered pairs of positive integers $(a,b,c)$ such that $\overline{abc}$ and $\overline{cab}$ are both perfect cubes.
6 replies
Entrepreneur
May 18, 2025
NamelyOrange
May 18, 2025
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Exponents of integer question
Dheckob   4
N May 18, 2025 by LeYohan
Find the smallest positive integer $m$ such that $5m$ is an exact 5th power, $6m$ is an exact 6th power, and $7m$ is an exact 7th power.
4 replies
Dheckob
Apr 12, 2017
LeYohan
May 18, 2025
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2017 DMI Individual Round - Downtown Mathematics Invitational
parmenides51   14
N May 18, 2025 by SomeonecoolLovesMaths
p1. Compute the smallest positive integer $x$ such that $351x$ is a perfect cube.


p2. A four digit integer is chosen at random. What is the probability all $4$ digits are distinct?


p3. If $$\frac{\sqrt{x + 1}}{\sqrt{x}}+ \frac{\sqrt{x}}{\sqrt{x + 1}} =\frac52.$$Solve for $x$.


p4. In $\vartriangle ABC$, $AB = 13$, $BC = 14$, and $AC = 15$. Let $D$ be the point on $BC$ such that $AD \perp BC$, and let $E$ be the midpoint of $AD$. If $F$ is a point such that $CDEF$ is a rectangle, compute the area of $\vartriangle AEF$.


p5. Square $ABCD$ has a sidelength of $4$. Points $P$, $Q$, $R$, and $S$ are chosen on $AB$, $BC$, $CD$, and $AD$ respectively, such that $AP$, $BQ$, $CR$, and $DS$ are length $1$. Compute the area of quadrilateral $P QRS$.


p6. A sequence $a_n$ satisfies for all integers $n$, $$a_{n+1} = 3a_n - 2a_{n-1}.$$If $a_0 = -30$ and $a_1 = -29$, compute $a_{11}$.


p7. In a class, every child has either red hair, blond hair, or black hair. All but $20$ children have black hair, all but $17$ have red hair, and all but $5$ have blond hair. How many children are there in the class?


p8. An Akash set is a set of integers that does not contain two integers such that one divides the other. Compute the minimum positive integer $n$ such that the set $\{1, 2, 3, ..., 2017\}$ can be partitioned into n Akash subsets.


PS. You should use hide for answers. Collected here.
14 replies
parmenides51
Oct 2, 2023
SomeonecoolLovesMaths
May 18, 2025
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2021 SMT Guts Round 5 p17-20 - Stanford Math Tournament
parmenides51   7
N May 16, 2025 by Rombo
p17. Let the roots of the polynomial $f(x) = 3x^3 + 2x^2 + x + 8 = 0$ be $p, q$, and $r$. What is the sum $\frac{1}{p} +\frac{1}{q} +\frac{1}{r}$ ?


p18. Two students are playing a game. They take a deck of five cards numbered $1$ through $5$, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play?


p19. Compute the sum of all primes $p$ such that $2^p + p^2$ is also prime.


p20. In how many ways can one color the $8$ vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?


PS. You should use hide for answers. Collected here.
7 replies
parmenides51
Feb 11, 2022
Rombo
May 16, 2025
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