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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Thursday at 11:16 PM
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
Thursday at 11:16 PM
0 replies
Equivalent condition of the uniformly continuous fo a function
Alphaamss   0
4 hours ago
Source: Personal
Let $f_{a,b}(x)=x^a\cos(x^b),x\in(0,\infty)$. Get all the $(a,b)\in\mathbb R^2$ such that $f_{a,b}$ is uniformly continuous on $(0,\infty)$.
0 replies
Alphaamss
4 hours ago
0 replies
Find all continuous functions
bakkune   2
N 4 hours ago by bakkune
Source: Own
Find all continuous function $f, g\colon\mathbb{R}\to\mathbb{R}$ satisfied
$$
(x - k)f(x) = \int_k^x g(y)\mathrm{d}y 
$$for all $x\in\mathbb{R}$ and all $k\in\mathbb{Z}$.
2 replies
bakkune
6 hours ago
bakkune
4 hours ago
ISI 2019 : Problem #2
integrated_JRC   40
N 5 hours ago by Sammy27
Source: I.S.I. 2019
Let $f:(0,\infty)\to\mathbb{R}$ be defined by $$f(x)=\lim_{n\to\infty}\cos^n\bigg(\frac{1}{n^x}\bigg)$$(a) Show that $f$ has exactly one point of discontinuity.
(b) Evaluate $f$ at its point of discontinuity.
40 replies
integrated_JRC
May 5, 2019
Sammy27
5 hours ago
Cube Colouring Problems
Saucepan_man02   0
5 hours ago
Could anyone kindly post some problems (and hopefully along the solution thread/final answer) related to combinatorial colouring of cube?
0 replies
Saucepan_man02
5 hours ago
0 replies
Floor function
Ro.Is.Te.   4
N 5 hours ago by aidan0626
Find all the real solution for this equation $$\left\lfloor \frac{x}{2} \right\rfloor + \left\lfloor \frac{2x}{3} \right\rfloor = x$$
4 replies
Ro.Is.Te.
Today at 1:53 AM
aidan0626
5 hours ago
A twist on a classic
happypi31415   13
N 5 hours ago by brownbear.bb
Rank from smallest to largest: $\sqrt[2]{2}$, $\sqrt[3]{3}$, and $\sqrt[5]{5}$.

Click to reveal hidden text
13 replies
happypi31415
Mar 17, 2025
brownbear.bb
5 hours ago
AM-GM Inequalitie with square root
NiltonCesar   3
N 5 hours ago by brownbear.bb
For $x \geq 0$, show that
$2x+\frac{3}{8} \geq 4\sqrt{x}$.
3 replies
NiltonCesar
Apr 27, 2025
brownbear.bb
5 hours ago
Putnam 1956 A4
sqrtX   2
N Today at 5:22 AM by sangsidhya
Source: Putnam 1956
Suppose that the $n$ times differentiable real function $f(x)$ has at least $n+1$ distinct zeros in the closed interval $[a,b]$ and that the polynomial $P(z)=z^n +c_{n-1}z^{n-1}+\ldots+c_1 x +c_0$ has only real zeroes. Show that
$f^{(n)}(x)+ c_{n-1} f^{(n-1)}(x) +\ldots +c_1 f'(x)+ c_0 f(x)$ has at least one zero in $[a,b]$, where $f^{(n)}$ denotes the $n$-th derivative of $f.$
2 replies
sqrtX
Jul 5, 2022
sangsidhya
Today at 5:22 AM
How to get a 300+ on the NWEA MAP MATH test (URGENT)
nmlikesmath   11
N Today at 4:17 AM by nmlikesmath
I have 4 days till this test, I'm wondering how do I get a 300+ and what do I need to know, thank you.
11 replies
nmlikesmath
Today at 1:57 AM
nmlikesmath
Today at 4:17 AM
A problem in point set topology
tobylong   0
Today at 3:14 AM
Source: Basic Topology, Armstrong
Let $f:X\to Y$ be a closed map with the property that the inverse image of each point in $Y$ is a compact subset of $X$. Prove that $f^{-1}(K)$ is compact whenever $K$ is compact in $Y$.
0 replies
tobylong
Today at 3:14 AM
0 replies
Putnnam 1954 B2
sqrtX   3
N Yesterday at 9:13 PM by centslordm
Source: Putnam 1954
Let $s$ denote the sum of the alternating harmonic series. Rearrange this series as follows
$$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} +\frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \ldots$$Assume as known that this series converges as well and denote its sum by $S$. Denote by $s_k, S_k$ respectively the $k$-th partial sums of both series. Prove that
$$ \!\!\!\! \text{i})\; S_{3n} = s_{4n} +\frac{1}{2} s_{2n}.$$$$ \text{ii}) \; S\ne s.$$
3 replies
sqrtX
Jul 17, 2022
centslordm
Yesterday at 9:13 PM
Putnam 1954 B1
sqrtX   5
N Yesterday at 8:56 PM by centslordm
Source: Putnam 1954
Show that the equation $x^2 -y^2 =a^3$ has always integral solutions for $x$ and $y$ whenever $a$ is a positive integer.
5 replies
sqrtX
Jul 17, 2022
centslordm
Yesterday at 8:56 PM
Putnam 1954 A6
sqrtX   1
N Yesterday at 8:52 PM by centslordm
Source: Putnam 1954
Suppose that $u_0 , u_1 ,\ldots$ is a sequence of real numbers such that
$$u_n =  \sum_{k=1}^{\infty} u_{n+k}^{2}\;\;\; \text{for} \; n=0,1,2,\ldots$$Prove that if $\sum u_n$ converges, then $u_k=0$ for all $k$.
1 reply
sqrtX
Jul 17, 2022
centslordm
Yesterday at 8:52 PM
Putnam 1954 A3
sqrtX   2
N Yesterday at 8:49 PM by centslordm
Source: Putnam 1954
Prove that if the family of integral curves of the differential equation
$$ \frac{dy}{dx} +p(x) y= q(x),$$where $p(x) q(x) \ne 0$, is cut by the line $x=k$ the tangents at the points of intersection are concurrent.
2 replies
sqrtX
Jul 17, 2022
centslordm
Yesterday at 8:49 PM
easy olympiad problem
kjhgyuio   7
N Apr 23, 2025 by Charizard_637
Find all positive integer values of \( x \) such that
\[
\sqrt{x - 2011} + \sqrt{2011 - x} + 10
\]is an integer.
7 replies
kjhgyuio
Apr 17, 2025
Charizard_637
Apr 23, 2025
easy olympiad problem
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kjhgyuio
57 posts
#1
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Find all positive integer values of \( x \) such that
\[
\sqrt{x - 2011} + \sqrt{2011 - x} + 10
\]is an integer.
This post has been edited 2 times. Last edited by kjhgyuio, Apr 17, 2025, 2:01 PM
Reason: nil
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Mathdreams
1470 posts
#2
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Solution
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Charizard_637
111 posts
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$x-2011$ and $2011-x$ are each other's negatives, and you can't take the square root of a negative number while having an integer solution. Therefore the only solution that would work is if they were both non-negative, and based on this they must be zero because one positive number will lead to one negative number. Since 0 is neither negative or positive it's what's under both square roots. Therefore, $\sqrt{x-2011} = 0$. Squaring both sides gives $x-2011 = 0$, hence $x = 2011$. This is the only solution.

Edit: I know it's verbose but it's an "olympiad problem"
This post has been edited 2 times. Last edited by Charizard_637, Apr 21, 2025, 5:44 PM
Reason: e
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vincentwant
1370 posts
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Charizard_637 wrote:
$x-2011$ and $2011-x$ are each other's negatives, and you can't take the square root of a negative number while having an integer solution. Therefore the only solution that would work is if they were both non-negative, and based on this they must be zero because one positive number will lead to one negative number. Since 0 is neither negative or positive it's what's under both square roots. Therefore, $\sqrt{x-2011} = 0$. Squaring both sides gives $x-2011 = 0$, hence $x = 2011$. This is the only solution.

Edit: I know it's verbose but it's an olympiad problem

you dont have to do this, this is enough

Notice that for the expression to be real, $x-2011\geq0$ and $2011-x\geq 0$, otherwise the imaginary part of the expression would be positive. Thus no solutions other than $x=2011$ exist, and inspection gives that $x=2011$ works. Thus the answer is $x=2011$.
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deduck
220 posts
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this is a past amc8 problem
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Roger.Moore
5 posts
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The both the roots there are reals only if x=2011
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Maxklark
6 posts
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Charizard_637
111 posts
#8
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deduck wrote:
this is a past amc8 problem

oh
my
:rotfl:
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