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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

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[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
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[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]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
Holomorphic function
Sifan.C.Maths   0
12 minutes ago
Source: m exercise
Is there a complex function $f$ such that $f$ satisfies two following statements?
(i) f is holomorphic on a domain $\Omega$ which contains $z=0$.
(ii) $f(\dfrac{1}{n})=0$ if $n$ is an odd natural number, $f(\dfrac{1}{n})=2$ if $n$ is an even natural number ($n$ is different from 0).
0 replies
Sifan.C.Maths
12 minutes ago
0 replies
Putnam 1952 B1
centslordm   4
N 3 hours ago by Gauler
A mathematical moron is given two sides and the included angle of a triangle and attempts to use the Law of Cosines: $a^2 = b^2 + c^2 - 2bc \cos A,$ to find the third side $a.$ He uses logarithms as follows. He finds $\log b$ and doubles it; adds to that the double of $\log c;$ subtracts the sum of the logarithms of $2, b, c,$ and $\cos A;$ divides the result by $2;$ and takes the anti-logarithm. Although his method may be open to suspicion his computation is accurate. What are the necessary and sufficient conditions on the triangle that this method should yield the correct result?
4 replies
centslordm
May 30, 2022
Gauler
3 hours ago
Miklós Schweitzer 1956- Problem 8
Coulbert   2
N 3 hours ago by izzystar
8. Let $(a_n)_{n=1}^{\infty}$ be a sequence of positive numbers and suppose that $\sum_{n=1}^{\infty} a_n^2$ is divergent. Let further $0<\epsilon<\frac{1}{2}$. Show that there exists a sequence $(b_n)_{n=1}^{\infty}$ of positive numbers such that $\sum_{n=1}^{\infty}b_n^2$ is convergent and

$\sum_{n=1}^{N}a_n b_n >(\sum_{n=1}^{N}a_n^2)^{\frac{1}{2}-\epsilon}$

for every positive integer $N$. (S. 8)
2 replies
Coulbert
Oct 11, 2015
izzystar
3 hours ago
Very straightforward linear recurrence
Assassino9931   1
N 3 hours ago by Etkan
Source: Vojtech Jarnik IMC 2025, Category II, P1
Let $x_0=a, x_1= b, x_2 = c$ be given real numbers and let $x_{n+2} = \frac{x_n + x_{n-1}}{2}$ for all $n\geq 1$. Show that the sequence $(x_n)_{n\geq 0}$ converges and find its limit.
1 reply
Assassino9931
3 hours ago
Etkan
3 hours ago
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
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
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
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
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
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
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
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
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
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
Inverse of absolute value function
MetaphysicalWukong   2
N Apr 2, 2025 by paxtonw
how does the function have an inverse for k= 101, 203, 305, 509, 611 and 713?

how do we deduce this without graphing software?
2 replies
MetaphysicalWukong
Apr 2, 2025
paxtonw
Apr 2, 2025
Inverse of absolute value function
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MetaphysicalWukong
260 posts
#1
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how does the function have an inverse for k= 101, 203, 305, 509, 611 and 713?

how do we deduce this without graphing software?
Attachments:
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MetaphysicalWukong
260 posts
#2
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Addendum: Show the following statements are not true.
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paxtonw
29 posts
#3
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Claim: The function f_k(x) = x^k * |x| has an inverse if k + 1 is even.

First, note that f_k(x) = x^k * |x| behaves differently for positive and negative x:

If x > 0, then |x| = x, so f_k(x) = x^(k+1)
If x < 0, then |x| = -x, so f_k(x) = x^k * (-x) = -x^(k+1)

So the function is:

f_k(x) = x^(k+1) when x > 0
f_k(x) = -(-x)^(k+1) when x < 0

This function is odd (symmetric about the origin), and is continuous for all x.
We now check if the function is one-to-one. If it is, then it has an inverse.
We use calculus: take the derivative of f_k(x).

For x > 0: derivative is (k+1) * x^k, which is always positive.
For x < 0: after simplification, the derivative is also positive if k+1 is even.

So if k+1 is even, then the function is increasing everywhere.
A function that is strictly increasing everywhere is one-to-one, which means it has an inverse.

For the values k = 101, 203, 305, 509, 611, and 713:
k + 1 = 102, 204, 306, 510, 612, and 714 — all even

Conclusion: Since k + 1 is even for all given values of k, the function f_k(x) has an inverse for each of them.
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