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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
5 hours ago
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
5 hours ago
0 replies
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Inequality
Ecrin_eren   1
N 3 hours ago by sqing


Let a, b, c be positive real numbers. Prove the inequality:

sqrt(a² - ab + b²) + sqrt(b² - bc + c²) ≥ sqrt(a² + ac + c²)



1 reply
Ecrin_eren
Yesterday at 8:47 PM
sqing
3 hours ago
J
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Number of dodecagons (OTIS MOCK AIME 2025 I #4)
dolphinday   2
N 3 hours ago by lpieleanu
In the Cartesian plane, let $A = (0, 10+12\sqrt{3})$, $B = (8, 10+12\sqrt{3})$, $G = (8, 0)$ and $H = (0, 0)$.
Compute the number of ways to draw an equiangular dodecagon $\mathcal P$ in the Cartesian plane such that all side lengths of $\mathcal P$ are positive integers and line segments $AB$ and $GH$ are both sides of $\mathcal P$.


Alansha Jiang
2 replies
dolphinday
Jan 22, 2025
lpieleanu
3 hours ago
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Find the minimum
Ecrin_eren   1
N 3 hours ago by imbadatmath1233
The polynomial is given by P(x) = x^4 + ax^3 + bx^2 + cx + d, and its roots are x1, x2, x3, x4. Additionally, it is stated that d ≥ 5.Find the minimum value of the product:

(x1^2 + 1)(x2^2 + 1)(x3^2 + 1)(x4^2 + 1).

1 reply
Ecrin_eren
Yesterday at 9:03 PM
imbadatmath1233
3 hours ago
J
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trigonometric functions
VivaanKam   10
N 3 hours ago by aok
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
10 replies
VivaanKam
Apr 29, 2025
aok
3 hours ago
J
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Geometry
BBNoDollar   0
5 hours ago
Let ABCD be a convex quadrilateral with angles BAD and BCD obtuse, and let the points E, F ∈ BD, such that AE ⊥ BD and CF ⊥ BD.
Prove that 1/(AE*CF) ≥ 1/(AB*BC) + 1/(AD*CD) .
0 replies
BBNoDollar
5 hours ago
0 replies
J
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Coprime sequence
Ecrin_eren   1
N 6 hours ago by revol_ufiaw


"Let N be a natural number. Show that any two numbers from the following sequence are coprime:

2^1 + 1, 2^2 + 1, 2^3 + 1, ..., 2^N + 1."



1 reply
Ecrin_eren
Yesterday at 8:53 PM
revol_ufiaw
6 hours ago
J
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Find the functions
Ecrin_eren   1
N Yesterday at 10:02 PM by undefined-NaN


"Find all differentiable functions f that satisfy the condition f(x) + f(y) = f((x + y) / (1 - xy)) for all x, y ∈ R, where x ≠ 1 and y ≠ 1."





1 reply
Ecrin_eren
Yesterday at 8:58 PM
undefined-NaN
Yesterday at 10:02 PM
J
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If it is an integer then perfect square
Ecrin_eren   0
Yesterday at 8:55 PM


"Let a, b, c, d be non-zero digits, and let abcd and dcba represent four-digit numbers.

Show that if the number abcd / dcba is an integer, then that integer is a perfect square."



0 replies
Ecrin_eren
Yesterday at 8:55 PM
0 replies
J
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Sum of arctan
Ecrin_eren   1
N Yesterday at 8:53 PM by Shan3t


Find the value of the sum:
sum from n = 0 to infinity of arctan(k / (n² + kn + 1))


1 reply
Ecrin_eren
Yesterday at 8:49 PM
Shan3t
Yesterday at 8:53 PM
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Cool vieta sum
Kempu33334   6
N Yesterday at 6:29 PM by Lankou
Let the roots of \[\mathcal{P}(x) = x^{108}+x^{102}+x^{96}+2x^{54}+3x^{36}+4x^{24}+5x^{18}+6\]be $r_1, r_2, \dots, r_{108}$. Find \[\dfrac{r_1^6+r_2^6+\dots+r_{108}^6}{r_1^6r_2^6+r_1^6r_3^6+\dots+r_{107}^6r_{108}^6}\]without Newton Sums.
6 replies
Kempu33334
Wednesday at 11:44 PM
Lankou
Yesterday at 6:29 PM
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đề hsg toán
akquysimpgenyabikho   3
N Yesterday at 5:50 PM by Lankou
làm ơn giúp tôi giải đề hsg

3 replies
akquysimpgenyabikho
Apr 27, 2025
Lankou
Yesterday at 5:50 PM
J
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A problem with a rectangle
Raul_S_Baz   13
N Yesterday at 4:38 PM by undefined-NaN
On the sides AB and AD of the rectangle ABCD, points M and N are taken such that MB = ND. Let P be the intersection of BN and CD, and Q be the intersection of DM and CB. How can we prove that PQ || MN?
IMAGE
13 replies
Raul_S_Baz
Apr 26, 2025
undefined-NaN
Yesterday at 4:38 PM
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Find the domain and range of $f(x)=2-|x-5|.$
Vulch   1
N Yesterday at 12:13 PM by Mathzeus1024
Find the domain and range of $f(x)=2-|x-5|.$
1 reply
Vulch
Yesterday at 2:07 AM
Mathzeus1024
Yesterday at 12:13 PM
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nice problem
teomihai   1
N Yesterday at 11:58 AM by Royal_mhyasd
Let set $A =\{0,1,2,3,...,n\}$ , where $n$ it is positiv ,integer number.
How many subsets of A contain at least one odd number?
1 reply
teomihai
Yesterday at 11:46 AM
Royal_mhyasd
Yesterday at 11:58 AM
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Recursion
Sid-darth-vater   6
N Apr 20, 2025 by vanstraelen
Help, I can't characterize ts and I dunno what to do
6 replies
Sid-darth-vater
Apr 20, 2025
vanstraelen
Apr 20, 2025
J
Recursion
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Reveal topic tags
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Sid-darth-vater
42 posts
Sid-darth-vater
#1 h  Apr 20, 2025, 3:02 AM
Y by
Help, I can't characterize ts and I dunno what to do
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aidan0626
1883 posts
aidan0626
#2 h  Apr 20, 2025, 5:14 AM
Y by
well by testing small cases i got the general form to be Click to reveal hidden text, which can probably be proven by induction
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Sid-darth-vater
42 posts
Sid-darth-vater
#3 h  Apr 20, 2025, 3:40 PM
Y by
Ahh yeah, it works out. tyty. also, can you kinda explain how you thought of the numerator part? like I had figured out $a_2 = \frac{7}{24}, a_3 = \frac{13}{96},$ and $a_4 = \frac{25}{384}$ but I couldn't find the $3 \cdot 2^{n-1} + 1$ portion.
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vanstraelen
9004 posts
vanstraelen
#4 h  Apr 20, 2025, 4:22 PM
Y by
$a_{1}=\frac{4}{6},a_{2}=\frac{7}{24},a_{3}=\frac{13}{96},a_{4}=\frac{25}{384},a_{5}=\frac{49}{1536},a_{6}=\frac{97}{6144},\cdots$

$a_{n}=\frac{3 \cdot 2^{n-1} +1}{3 \cdot 2^{2n-1}}$.
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aidan0626
1883 posts
aidan0626
#5 h  Apr 20, 2025, 4:55 PM
Y by
Sid-darth-vater wrote:
Ahh yeah, it works out. tyty. also, can you kinda explain how you thought of the numerator part? like I had figured out $a_2 = \frac{7}{24}, a_3 = \frac{13}{96},$ and $a_4 = \frac{25}{384}$ but I couldn't find the $3 \cdot 2^{n-1} + 1$ portion.
well I noticed that the differences between consecutive numerators was 3, 6, 12, etc.
and so I got a geometric series which resulted in that
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Sid-darth-vater
42 posts
Sid-darth-vater
#6 h  Apr 20, 2025, 5:52 PM
Y by
mmmm i see, thanks!
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vanstraelen
9004 posts
vanstraelen
#7 h  Apr 20, 2025, 5:59 PM
Y by
$a_{k}=\frac{3 \cdot 2^{k-1} +1}{3 \cdot 2^{2k-1}}=2^{-k}+\frac{1}{3} \cdot 2^{1-2k}$.

$S=\lim_{n \to \infty} \sum_{k=1}^{n}a_{k}=\lim_{n \to \infty} \sum_{k=1}^{n}\left[2^{-k}+\frac{1}{3} \cdot 2^{1-2k}\right]$,
$S=\lim_{n \to \infty} \left[1-2^{-n}+\frac{1}{3}(\frac{2}{3}-\frac{2^{1-2n}}{2})\right]=1+\frac{2}{9}=\frac{11}{9}$.
This post has been edited 1 time. Last edited by vanstraelen, Apr 20, 2025, 6:00 PM
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