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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
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0 replies
jlacosta
Jun 2, 2025
0 replies
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Extended Wilson's?
NamelyOrange   1
N 15 minutes ago by Vivaandax
Let $\mathbb{Z}^*_n$ be the set of positive integers less than $n$ relatively prime to it. Is there a nice pattern for $\left(\prod_{k\in \mathbb{Z}^*_n} k\right) \text{ mod }n$? I know from a Wilson's theorem-style argument that it's either $1$ or $-1$, but when is it which?
1 reply
NamelyOrange
3 hours ago
Vivaandax
15 minutes ago
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Exponents and Logarithms
wpdnjs   1
N 35 minutes ago by RedFireTruck
Suppose A and B are positive real numbers for which log base A of B = log base B of A. If neither A nor B is 1 and A does not equal B, find the value of AB.
1 reply
wpdnjs
2 hours ago
RedFireTruck
35 minutes ago
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Polynomials
P162008   5
N 38 minutes ago by RedFireTruck
P1. Find $p(0) + p(5)$ where $p$ is a monic polynomial of degree $4$ satisfying $p(r) = 2^r ; r = 1,2,3,4.$

P2. Find $p(1), p(-1)$ where $p$ is a polynomial of smallest degree possible satisfying $p(r) = \frac{1}{r^2 - 1}; r = 2,3,4,\cdots, 10.$

P3. Find $k$ and $p(0)$, if polynomial $p$ satisfies $x.p(x) + 1 = k\left(\prod_{i = 1}^{5} (x - i)\right).$

P4. Find $p(0)$ where $p$ is a polynomial of smallest degree satisfying $p(r) = \frac{1}{r}; r = 1,2,3,\cdots,10.$

P5. Find $p(0),p(6),k$ and $\alpha$ if polynomial $p$ satisfies $(x^2 - 6x).p(x) + 1 = k\left(\prod_{i=1}^{5} (x - i)\right)(x - \alpha).$

P6. If $f(x)$ is a polynomial of degree $50$ such that $f(x) = \frac{x}{x + 1}; x = 0,1,2,\cdots,50.$ Evaluate $f(-1).$
5 replies
P162008
Today at 12:17 AM
RedFireTruck
38 minutes ago
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Logarithms
P162008   1
N 39 minutes ago by vanstraelen
Let $a = \frac{(\log_{2} 3 - \log_{5} 7)(\log_{2} 3 - \log_{7} 9)}{(\log_{3} 5 - \log_{5} 7)(\log_{3} 5 - \log_{7} 9)}, b = \frac{(\log_{2} 3 - \log_{3} 5)(\log_{2} 3 - \log_{7} 9)}{(\log_{5} 7 - \log_{3} 5)(\log_{5} 7 - \log_{7} 9)}$ and $c = \frac{(\log_{2} 3 - \log_{3} 5)(\log_{2} 3 - \log_{5} 7)}{(\log_{7} 9 - \log_{3} 5)(\log_{7} 9 - \log_{5} 7)}.$
Find the value of $\lfloor a + b + c \rfloor$ where $\lfloor.\rfloor$ denotes greatest integer function.
1 reply
P162008
May 30, 2025
vanstraelen
39 minutes ago
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No more topics!
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What are vectors?
SomeonecoolLovesMaths   3
N Apr 20, 2025 by programjames1
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?
3 replies
SomeonecoolLovesMaths
Apr 20, 2025
programjames1
Apr 20, 2025
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What are vectors?
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Reveal topic tags
vector
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SomeonecoolLovesMaths
3327 posts
SomeonecoolLovesMaths
#1 h  Apr 20, 2025, 4:19 AM
Y by
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?
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greenturtle3141
3563 posts
greenturtle3141
#2 h  Apr 20, 2025, 5:45 AM
Y by
For the sort of "vectors" you are probably thinking of, they are simply elements of $\mathbb{R}^n$.

For example, if the net force on an object is "10 units of force directed upward", that would mathematically be something like $(0,0,10)$.

(More generally and abstractly, a mathematician would define a vector as an element of a vector space, but that's likely out of the scope of this discussion.)
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SomeonecoolLovesMaths
3327 posts
SomeonecoolLovesMaths
#3 h  Apr 20, 2025, 5:53 AM
Y by
greenturtle3141 wrote:
For the sort of "vectors" you are probably thinking of, they are simply elements of $\mathbb{R}^n$.

For example, if the net force on an object is "10 units of force directed upward", that would mathematically be something like $(0,0,10)$.

(More generally and abstractly, a mathematician would define a vector as an element of a vector space, but that's likely out of the scope of this discussion.)

Okay I get it. It is probably out of my league to grasp the concept of vectors but I just wanted to get an idea of what they truly are. Thanks! :D
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programjames1
3046 posts
programjames1
#4 h  Apr 20, 2025, 7:24 AM
Y by
See this discussion from a few months ago:

https://artofproblemsolving.com/community/c4h3520259p34184343
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