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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Inequalities
nhathhuyyp5c   0
14 minutes ago
Let $a, b, c$ be non-negative real numbers such that $a^2 + b^2 + c^2 = 3$. Find the maximum and minimum values of the expression
\[ P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}. \]
0 replies
nhathhuyyp5c
14 minutes ago
0 replies
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What are vectors?
SomeonecoolLovesMaths   2
N an hour ago by SomeonecoolLovesMaths
How does one define vectors with mathematical rigority? Most of the time it is stated as an object with a magnitude and a direction, but it in itself is very vague in my opinion. Like how do you define direction and magnitude itself?
2 replies
SomeonecoolLovesMaths
3 hours ago
SomeonecoolLovesMaths
an hour ago
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Inequalities
sqing   10
N an hour ago by DAVROS
Let $ a,b,c> 0 $ and $ \frac{a}{a^2+ab+c}+\frac{b}{b^2+bc+a}+\frac{c}{c^2+ca+b} \geq 1$. Prove that
$$ a+b+c\leq 3 $$
10 replies
sqing
Apr 4, 2025
DAVROS
an hour ago
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Recursion
Sid-darth-vater   1
N 2 hours ago by aidan0626
Help, I can't characterize ts and I dunno what to do
1 reply
Sid-darth-vater
4 hours ago
aidan0626
2 hours ago
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Inequalities
sqing   19
N 4 hours ago by sqing
Let $ a,b $ be reals such that $ a^2+ab+b^2 =3$ . Prove that
$$ \frac{4}{ 3}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{11}{4 }$$$$ \frac{13}{ 4}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{2}{3 }$$$$ \frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq -\frac{17}{6 }$$$$ \frac{19}{ 6}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2}$$Let $ a,b $ be reals such that $ a^2-ab+b^2 =1 $ . Prove that
$$ \frac{3}{ 2}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }+ab \geq \frac{4}{15 }$$$$ \frac{14}{ 15}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }-ab \geq -\frac{1}{2 }$$$$ \frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq \frac{13}{42 }$$$$ \frac{41}{ 42}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2 }$$
19 replies
sqing
Apr 16, 2025
sqing
4 hours ago
J
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weird permutation problem
Sedro   0
5 hours ago
Let $\sigma$ be a permutation of $1,2,3,4,5,6,7$ such that there are exactly $7$ ordered pairs of integers $(a,b)$ satisfying $1\le a < b \le 7$ and $\sigma(a) < \sigma(b)$. How many possible $\sigma$ exist?
0 replies
Sedro
5 hours ago
0 replies
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Calculus BC help
needcalculusasap45   7
N 5 hours ago by ehz2701
So basically, I have the AP Calculus BC exam in less than a month, and I have only covered until Unit 6 or 7 of the cirriculum. I am self studying this course (no teacher) and have not had much time to study bc of 6 other APs. I need to finish 8, 9, and 10 in less than 2 weeks. What can I do ? I would appreciate any help or resources anyone could provide. Could I just learn everything from barrons and princeton? Also, I have not taken AP Calculus AB before.

7 replies
needcalculusasap45
Yesterday at 1:51 PM
ehz2701
5 hours ago
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Some problems
hashbrown2009   2
N Today at 12:12 AM by UberPiggy
1. Real numbers a,b,c are satisfy a+1/b = b+1/c = c+1/a =x. If a,b,c are distinct, what is the value of x?
2. If x^2+y^2=1, then what is the value of : root(x^2-2x+1) + root(xy-2x+y-2) ?
3. Find the value of the sequence 2^2 + (3^2+1) + (4^2+2) + … + (97^2+95) + (98^2+96).
4. If x^2+x-1=0, then evaluate (1-x^2-x^3-x^4-…-x^2022-x^2023)/x^2022 .
5. If triangle XYZ has 3 sides that are all whole numbers, and the perimeter of XYZ is 24, what is the probability XYZ is a right triangle?

Note: If someone can latex-ify this it would help.
2 replies
hashbrown2009
Yesterday at 11:01 PM
UberPiggy
Today at 12:12 AM
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ez problem....
Cobedangiu   4
N Yesterday at 11:22 PM by iniffur
Let $x,y \in Z$ and $xy \cancel \vdots7$
Find $n \in Z^+$.
$x^2+y^2+xy=7^n$
4 replies
Cobedangiu
Friday at 11:07 AM
iniffur
Yesterday at 11:22 PM
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Twin Primes Digital Root
FerMath10   1
N Yesterday at 8:57 PM by Sedro
Hi,

I noticed something interesting while playing around with twin primes (pairs of primes that differ by 2). Here is what I noticed:
Conjecture: The product of twin primes—excluding the pair (3, 5)—always has a digital root of 8.

Just to clarify, the digital root of a number is the single-digit value you get by repeatedly summing its digits until only one digit remains. For example, the digital root of 77 is 7 + 7 = 14, and then 1 + 4 = 5.

I tested this on several examples, and it seems to hold, but I’m not sure if it’s a well-known result or something that breaks down for larger primes.

Is this an obvious consequence of some known number theory property? Would love to hear your thoughts!
1 reply
FerMath10
Yesterday at 8:51 PM
Sedro
Yesterday at 8:57 PM
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Inscribed Semi-Circle!!!
ehz2701   1
N Yesterday at 7:56 PM by vanstraelen
A right triangle $ABC$ with legs $AB = a$ and $BC = b$ is drawn with a semicircle inscribed into the triangle. What is the smallest possible radius of the semi-circle and the largest possible radius?

1 reply
ehz2701
Sep 11, 2022
vanstraelen
Yesterday at 7:56 PM
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Geometry
AlexCenteno2007   2
N Yesterday at 7:35 PM by AlexCenteno2007
Let A, B, C, and D be four distinct points on a straight line, in that order. The circles with diameters AC and BD intersect at X and Y. The straight line XY intersects BC at Z. Let P be a point on XY distinct from Z. The straight line CP intersects the circle with diameter AC at C and M, and the straight line BP intersects the circle with diameter BD at B and N. Show that AM, DN, and XY are aligned.
2 replies
AlexCenteno2007
Apr 17, 2025
AlexCenteno2007
Yesterday at 7:35 PM
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Geometry
AlexCenteno2007   4
N Yesterday at 6:42 PM by vanstraelen
Let C be a point on a semicircle of diameter AB and let D be the mid-length of arc AC. Let E be the projection of point D on BC and F the intersection of AE with the semicircle. Prove that BF bisects segment DE.
4 replies
AlexCenteno2007
Apr 17, 2025
vanstraelen
Yesterday at 6:42 PM
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Good Problem??
pedronis   0
Yesterday at 6:05 PM
Let \( m \geq 2 \) be a positive integer and \( p \geq 5 \) a prime number. Consider the sequence \( a_1, a_2, \ldots \) defined by
\[ a_n = (p - 2)^n + p^n - m, \quad \text{for } n \geq 1. \]Let \( q \) be the smallest prime such that some term of the sequence is divisible by \( q \). Determine all pairs \( (m, p) \) for which \( q \geq m \).
0 replies
pedronis
Yesterday at 6:05 PM
0 replies
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Basic functional equations
Vulch   1
N Apr 14, 2025 by NamelyOrange
Solve the following problem:
1 reply
Vulch
Apr 14, 2025
NamelyOrange
Apr 14, 2025
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Basic functional equations
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functional equation
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Vulch
2682 posts
Vulch
#1 h  Apr 14, 2025, 8:24 AM
Y by
Solve the following problem:
Attachments:
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NamelyOrange
495 posts
NamelyOrange
#2 h  Apr 14, 2025, 1:36 PM
Y by
The function is not bijective, as $f(1) = f(2) = 1$.

For a function to be bijective, each input must correspond to a unique output.
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