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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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fibonacci number theory
FFA21   1
N 3 hours ago by alexheinis
Source: OSSM Comp'25 P3 (HSE IMC qualification)
$F_n$ fibonacci numbers ($F_1=1, F_2=1$) find all n such that:
$\forall i\in Z$ and $0\leq i\leq F_n$
$C^i_{F_n}\equiv (-1)^i\pmod{F_n+1}$
1 reply
FFA21
May 14, 2025
alexheinis
3 hours ago
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Trapezium problem very nice
manlio   0
Today at 11:00 AM
Given trapezium ABCD with basis AB and CD parallel. Choose a point E on side BC and a point F on side AD such that AE Is parallel to FC . Prove that DE Is parallel to FB.
0 replies
manlio
Today at 11:00 AM
0 replies
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Proving a group is abelian
dragosgamer12   8
N Today at 10:14 AM by dragosgamer12
Source: Florin Stanescu, Gazeta Matematica seria B Nr.2/2025
Let $(G,\cdot)$ be a group, $K$ a subgroup of $G$ and $f : G \rightarrow G$ an endomorphism with the following property:
There exists a nonempty set $H\subset G$ such that for any $k \in G \setminus K$ there exist $h \in H$ with $f(h)=k$ and $z \cdot h= h \cdot z$, for any $z \in H$.

a)Prove that $(G, \cdot)$ is abelian.
b)If, additionally, $H$ is a subgroup of $G$, prove that $H=G$
8 replies
dragosgamer12
Thursday at 10:06 PM
dragosgamer12
Today at 10:14 AM
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Minimum and Maximum of Complex Numbers
pythagorazz   1
N Today at 8:39 AM by alexheinis
Let $a,b,$ and $c$ be complex numbers. For a complex number $z=p+qi$ where $i=\sqrt(-1)$, define the norm $|z|$ to be the distance of $z$ from the origin, or $|z|=\sqrt(p^2+q^2 )$. Let $m$ be the minimum value and $M$ be the maximum value of $\frac{(|a+b|+|b+c|+|c+a|)}{(|a|+|b|+|c| )}$ for all complex numbers $a,b,c$ where $|a|+|b|+|c|\ne 0$. Find $M+m$.
1 reply
pythagorazz
Apr 14, 2025
alexheinis
Today at 8:39 AM
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Folklore
Osim_09   2
N Today at 8:36 AM by pigeon123
Let ABCD be a circumscribed quadrilateral, which is also cyclic. Let I be the incenter, O the circumcenter, and E the intersection point of the diagonals of the quadrilateral. Prove that the points O, I, and E are collinear.
2 replies
Osim_09
Jan 21, 2025
pigeon123
Today at 8:36 AM
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A challenging sum
Polymethical_   1
N Today at 4:16 AM by Polymethical_
I tried to integrate series of log(1-x) / x
1 reply
Polymethical_
Today at 4:09 AM
Polymethical_
Today at 4:16 AM
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Bounding With Powers
Shreyasharma   5
N Today at 2:20 AM by jacosheebay
Is this a valid solution for the following problem (St. Petersburg 1996):

Find all positive integers $n$ such that,

$$ 3^{n-1} + 5^{n-1} | 3^n + 5^n$$
Solution
5 replies
Shreyasharma
Jul 11, 2023
jacosheebay
Today at 2:20 AM
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Tough integral
Martin.s   3
N Yesterday at 9:42 PM by GreenKeeper
$$\int_0^{\pi/2}\ln(\tan(\theta/2)) \;\frac{4\cos\theta\cos(2\theta)}{4\sin^4\theta+1}\,d\theta.$$
3 replies
Martin.s
May 12, 2025
GreenKeeper
Yesterday at 9:42 PM
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2021 SMT Guts Round 5 p17-20 - Stanford Math Tournament
parmenides51   7
N Yesterday at 8:05 PM 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
Yesterday at 8:05 PM
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2024 Mock AIME 1 ** p15 (cheaters' trap) - 128 | n^{\sigma (n)} - \sigma(n^n)
parmenides51   6
N Yesterday at 7:32 PM by NamelyOrange
Let $N$ be the number of positive integers $n$ such that $n$ divides $2024^{2024}$ and $128$ divides
$$n^{\sigma (n)} - \sigma(n^n)$$where $\sigma (n)$ denotes the number of positive integers that divide $n$, including $1$ and $n$. Find the remainder when $N$ is divided by $1000$.
6 replies
parmenides51
Jan 29, 2025
NamelyOrange
Yesterday at 7:32 PM
J
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Minimum number of points
Ecrin_eren   6
N Yesterday at 6:01 PM by Ecrin_eren
There are 18 teams in a football league. Each team plays against every other team twice in a season—once at home and once away. A win gives 3 points, a draw gives 1 point, and a loss gives 0 points. One team became the champion by earning more points than every other team. What is the minimum number of points this team could have?

6 replies
Ecrin_eren
May 15, 2025
Ecrin_eren
Yesterday at 6:01 PM
J
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b+c <=a/sin(A/2)
lgx57   4
N Yesterday at 4:27 PM by cosinesine
Prove that: In $\triangle ABC$,$b+c \le \dfrac{a}{\sin \frac{A}{2}}$
4 replies
lgx57
Yesterday at 1:11 PM
cosinesine
Yesterday at 4:27 PM
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2014 preRMO p10, computational with ratios and areas
parmenides51   11
N Yesterday at 3:16 PM by MATHS_ENTUSIAST
In a triangle $ABC, X$ and $Y$ are points on the segments $AB$ and $AC$, respectively, such that $AX : XB = 1 : 2$ and $AY :YC = 2:1$. If the area of triangle $AXY$ is $10$, then what is the area of triangle $ABC$?
11 replies
parmenides51
Aug 9, 2019
MATHS_ENTUSIAST
Yesterday at 3:16 PM
J
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Graphs and Trig
Math1331Math   7
N Yesterday at 2:43 PM by BlackOctopus23
The graph of the function $f(x)=\sin^{-1}(2\sin{x})$ consists of the union of disjoint pieces. Compute the distance between the endpoints of any one piece
7 replies
Math1331Math
Jun 19, 2016
BlackOctopus23
Yesterday at 2:43 PM
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Investigating functions
mikejoe   1
N May 14, 2025 by Mathzeus1024
Source: Edwards and Penney
Investigate the function $f(x) = (x-2) \sqrt{x+1}$
Also determine its domain and range.
1 reply
mikejoe
Nov 2, 2012
Mathzeus1024
May 14, 2025
J
Investigating functions
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Reveal topic tags
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calculus computations
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Source: Edwards and Penney
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mikejoe
8 posts
mikejoe
#1 h  Nov 2, 2012, 8:02 PM • 2 Y
Y by Adventure10, Mango247
Investigate the function $f(x) = (x-2) \sqrt{x+1}$
Also determine its domain and range.
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The post below has been deleted. Click to close.
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Mathzeus1024
895 posts
Mathzeus1024
#2 h  May 14, 2025, 1:08 PM
Y by
The domain is clearly $x \in [-1, \infty)$ due to the function's radicand. As for the range, taking the derivative of $f$ equal to zero yields the critical value:

$f'(x) = \sqrt{x+1}+\frac{x-2}{2\sqrt{x+1}}=0 \Rightarrow 2(x+1) = 2-x \Rightarrow x = 0$;

and second derivative check of $f$ against this critical value yields:

$f''(x) = \frac{1}{2\sqrt{x+1}} + \frac{x+4}{4(x+1)^{3/2}} \Rightarrow f''(0) = \frac{3}{2} > 0$ (a global minimum).

Thus, the range of $f$ is $[-2,\infty)$.
This post has been edited 1 time. Last edited by Mathzeus1024, May 14, 2025, 1:11 PM
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