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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.

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[*]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
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USAMO solutions
ABCD1728   0
31 minutes ago
Do USAMO USATSTST and USATST have OFFICIAL solutions? Thanks!
0 replies
ABCD1728
31 minutes ago
0 replies
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Distance between any two points is irrational
orl   21
N an hour ago by cursed_tangent1434
Source: IMO 1987, Day 2, Problem 5
Let $n\ge3$ be an integer. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
21 replies
orl
Nov 11, 2005
cursed_tangent1434
an hour ago
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Symmetric inequality
mrrobotbcmc   5
N an hour ago by youochange
Let a,b,c,d belong to positive real numbers such that a+b+c+d=1. Prove that a^3/(b+c)+b^3/(c+d)+c^3/(d+a)+d^3/(a+b)>=1/8
5 replies
mrrobotbcmc
3 hours ago
youochange
an hour ago
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Two perpendiculars
jayme   0
an hour ago
Source: Own?
Dear Mathlinkers,

1. ABC a triangle
2. 0 the circumcircle
3. D the pole of BC wrt 0
4. B', C' the symmetrics of B, C wrt AC, AB
5. 1b, 1c the circumcircles of the triangles BB'D, CC'D
6. J the center of 1b
7. V the second point of intersection of DJ and 1c.

Prove : CV is perpendicular to BC.

Sincerely
Jean-Louis
0 replies
jayme
an hour ago
0 replies
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Diophantine Equation (cousin of Mordell)
urfinalopp   4
N Yesterday at 10:54 PM by FoeverResentful
Find pairs of integers $(x;y)$ such that:

$x^2=y^5+32$
4 replies
urfinalopp
Yesterday at 6:38 PM
FoeverResentful
Yesterday at 10:54 PM
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p+2^p-3=n^2
tom-nowy   1
N Yesterday at 6:51 PM 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
Yesterday at 11:16 AM
urfinalopp
Yesterday at 6:51 PM
J
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Perfect cubes
Entrepreneur   6
N Yesterday at 6:23 PM 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
Yesterday at 6:04 PM
NamelyOrange
Yesterday at 6:23 PM
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Ez comb proposed by ME
IEatProblemsForBreakfast   1
N Yesterday at 3:09 PM by n1g3r14n
A and B play a game on two table:
1.At first one table got $n$ different coloured marbles on it and another one is empty
2.At each move player choose set of marbles that hadn't choose either players before and all chosen marbles from same table, and move all the marbles in that set to another table
3.Player who can not move lose
If A starts and they move alternatily who got the winning strategy?
1 reply
IEatProblemsForBreakfast
Yesterday at 9:02 AM
n1g3r14n
Yesterday at 3:09 PM
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geometry
luckvoltia.112   0
Yesterday at 3:04 PM
ChGiven an acute triangle ABC inscribed in circle $(O)$ The altitudes $BE, CF$ , intersect
each other at $H$. The tangents at $B$ and $C $of $(O)$ intersect at $S$. Let $M $be the midpoint of $BC$. $EM$ intersects $SC$
at $I$, $FM$ intersects $SB$ at $J.$
a) Prove that the points $I, S, M, J$ lie on the same circle.
b) The circle with diameter $AH$ intersects the circle $(O)$ at the second point $T.$ The line $AH$ intersects
$(O)$ at the second point $K$. Prove that $S,K,T$ are collinear.
0 replies
luckvoltia.112
Yesterday at 3:04 PM
0 replies
J
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Exponents of integer question
Dheckob   4
N Yesterday at 2:45 PM 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
Yesterday at 2:45 PM
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ISI 2025
Zeroin   0
Yesterday at 2:29 PM
Let $\mathbb{N}$ denote the set of natural numbers and let $(a_i,b_i),1 \leq i \leq 9$ denote $9$ ordered pairs in $\mathbb{N} \times \mathbb{N}$. Prove that there exist $3$ distinct elements in the set $2^{a_i}3^{b_i}$ for $1 \leq i \leq 9$ whose product is a perfect cube.
0 replies
Zeroin
Yesterday at 2:29 PM
0 replies
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Inequalities
sqing   3
N Yesterday at 1:49 PM by sqing
Let $ a,b>0 $ . Prove that
$$ \frac{a}{a^2+a +2b+1}+ \frac{b}{b^2+2a +b+1} \leq \frac{2}{5} $$$$ \frac{a}{a^2+2a +b+1}+ \frac{b}{b^2+a +2b+1} \leq \frac{2}{5} $$
3 replies
sqing
May 13, 2025
sqing
Yesterday at 1:49 PM
J
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Max and min of ab+bc+ca-abc
Tiira   5
N Yesterday at 1:01 PM by sqing
a, b and c are three non-negative reel numbers such that a+b+c=1.
What are the extremums of
ab+bc+ca-abc
?
5 replies
Tiira
Jan 29, 2021
sqing
Yesterday at 1:01 PM
J
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2017 DMI Individual Round - Downtown Mathematics Invitational
parmenides51   14
N Yesterday at 11:39 AM 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
Yesterday at 11:39 AM
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Combinatorics or algebra?
persamaankuadrat   1
N Apr 14, 2025 by removablesingularity
Source: OSNK 2024
How many sequences of $\left(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6} \right)$ where the $a_{i}$ are positive integers such that each $1 \le a_{i} \le 4$ and none of two consecutive terms when summed are equal to $4$.
1 reply
persamaankuadrat
Apr 13, 2025
removablesingularity
Apr 14, 2025
J
Combinatorics or algebra?
G H J
Reveal topic tags
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algebra
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G H BBookmark kLocked kLocked NReply
Source: OSNK 2024
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persamaankuadrat
156 posts
persamaankuadrat
#1 h  Apr 13, 2025, 6:34 AM • 1 Y
Y by cubres
How many sequences of $\left(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6} \right)$ where the $a_{i}$ are positive integers such that each $1 \le a_{i} \le 4$ and none of two consecutive terms when summed are equal to $4$.
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removablesingularity
569 posts
removablesingularity
#2 h  Apr 14, 2025, 4:02 AM
Y by
Define the sequence $f_k\left(n\right)$ as the number of sequence $a_1,a_2,\cdots,a_6$ such that $a_1 = k$. Define also Click to reveal hidden text Conclude first the recurence Click to reveal hidden text
We can conclude that Click to reveal hidden text with Click to reveal hidden text and find Click to reveal hidden text as the desired answer.
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