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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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0 replies
jlacosta
Apr 2, 2025
0 replies
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Calculus
EthanWYX2009   2
N 6 minutes ago by watery
Determine the value of$$\int\limits_{0}^{\infty}\frac{\sin x}{x^2}\sum_{n=1}^{\infty}\frac{\sin (nx)}{n!}\mathrm dx$$
2 replies
EthanWYX2009
4 hours ago
watery
6 minutes ago
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Picking a College
missionsqhc   2
N 2 hours ago by missionsqhc
I applied to college as a math major, and my options are Georgetown, UVA, Stony Brook, and Binghamton. I was waitlisted from CMU, Columbia, Northwestern, Berkeley, Williams, UNC, and UMich.

I’ve done competition math throughout middle school and high school and obviously am currently slotted to study math. But I am also very much interested in politics, government, history, etc. I could easily see myself double majoring or even completely switching to something like political science or history. I don’t have a clear-cut vision for a future career. I used to really want to become a mathematician, but now I think it’s more likely that I’ll do something more “practical,” like finance or law. I also have aspirations of working in government, even possibly running for elected office.

If someone has gone to one of the school’s I’ve been accepted by or has experience in one of the careers I’ve mentioned (or possesses some other characteristics that gives insight into my situation), I would greatly appreciate your thoughts. On one hand, I really like Georgetown because of its strong programs in government, international relations, and other social sciences; its DC location; and its stated goal (which I hope is genuine) of educating students for life and not just work. But the hard sciences, and particularly math, are relatively smaller programs and less of the school’s emphasis. I worry that I may end up sticking mainly with math and would have been better off picking something like UVA or even Stony or Bing.

A related question I have regards how the undergraduate math departments compare at different schools. I wouldn't be surprised if the very top-tier places, like MIT, Caltech, CMU, Harvard, Stanford, and Princeton, were significantly stronger than Georgetown. But how does Georgetown compare to places that are good for math but not necessarily hyper-elite, like a Cornell or a UMich?

Also, Georgetown has a 3 + 2 program with Columbia Engineering, in which you study for three years at Georgetown to get a BA/BS in any major in any school (but preferably in math/science) and then study for two years at Columbia to get a BS in their engineering school. This seems like a way to get the best of both worlds between humanities and STEM (and to gain connections in both DC and NYC). If anyone has done this, please do share your experience.
2 replies
missionsqhc
Today at 1:42 AM
missionsqhc
2 hours ago
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high school math
aothatday   7
N 3 hours ago by ZMB038
Let $x_n$ be a positive root of the equation $x^n=x^2+x+1$. Prove that the following sequence converges: $n^2(x_n-x_{ n+1})$
7 replies
aothatday
Apr 10, 2025
ZMB038
3 hours ago
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Solve an equation
lgx57   2
N 4 hours ago by lgx57
Find all positive integers $n$ and $x$ such that:
$$2^{2n+1}-7=x^2$$
2 replies
lgx57
Mar 12, 2025
lgx57
4 hours ago
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Indonesia Regional MO 2019 Part A
parmenides51   17
N 4 hours ago by Rohit-2006
Indonesia Regional MO
Year 2019 Part A

Time: 90 minutes Rules


p1. In the bag there are $7$ red balls and $8$ white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ...


p2. Given a regular hexagon with a side length of $1$ unit. The area of the hexagon is ...


p3. It is known that $r, s$ and $1$ are the roots of the cubic equation $x^3 - 2x + c = 0$. The value of $(r-s)^2$ is ...


p4. The number of pairs of natural numbers $(m, n)$ so that $GCD(n,m) = 2$ and $LCM(m,n) = 1000$ is ...


p5. A data with four real numbers $2n-4$, $2n-6$, $n^2-8$, $3n^2-6$ has an average of $0$ and a median of $9/2$. The largest number of such data is ...


p6. Suppose $a, b, c, d$ are integers greater than $2019$ which are four consecutive quarters of an arithmetic row with $a <b <c <d$. If $a$ and $d$ are squares of two consecutive natural numbers, then the smallest value of $c-b$ is ...


p7. Given a triangle $ABC$, with $AB = 6$, $AC = 8$ and $BC = 10$. The points $D$ and $E$ lies on the line segment $BC$. with $BD = 2$ and $CE = 4$. The measure of the angle $\angle DAE$ is ...


p8. Sequqnce of real numbers $a_1,a_2,a_3,...$ meet $\frac{na_1+(n-1)a_2+...+2a_{n-1}+a_n}{n^2}=1$ for each natural number $n$. The value of $a_1a_2a_3...a_{2019}$ is ....


p9. The number of ways to select four numbers from $\{1,2,3, ..., 15\}$ provided that the difference of any two numbers at least $3$ is ...


p10. Pairs of natural numbers $(m , n)$ which satisfies $$m^2n+mn^2 +m^2+2mn = 2018m + 2019n + 2019$$are as many as ...


p11. Given a triangle $ABC$ with $\angle ABC =135^o$ and $BC> AB$. Point $D$ lies on the side $BC$ so that $AB=CD$. Suppose $F$ is a point on the side extension $AB$ so that $DF$ is perpendicular to $AB$. The point $E$ lies on the ray $DF$ such that $DE> DF$ and $\angle ACE = 45^o$. The large angle $\angle AEC$ is ...


p12. The set of $S$ consists of $n$ integers with the following properties: For every three different members of $S$ there are two of them whose sum is a member of $S$. The largest value of $n$ is ....


p13. The minimum value of $\frac{a^2+2b^2+\sqrt2}{\sqrt{ab}}$ with $a, b$ positive reals is ....


p14. The polynomial P satisfies the equation $P (x^2) = x^{2019} (x+ 1) P (x)$ with $P (1/2)= -1$ is ....


p15. Look at a chessboard measuring $19 \times 19$ square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of $k$ coins on the chessboard where each coin is only loaded exactly on one square and each square can contain coins or blanks. At each turn. You must select exactly one plot that holds the minimum number of coins in the number of neighbors of the plot and then you must give exactly one coin to each neighbor of the selected plot. The game ends if you are no longer able to select squares with the intended conditions. The smallest number of $k$ so that the game never ends for any initial square selection is ....
17 replies
parmenides51
Nov 11, 2021
Rohit-2006
4 hours ago
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How to prove one-one function
Vulch   6
N 4 hours ago by Vulch
Hello everyone,
I am learning functional equations.
To prove the below problem one -one function,I have taken two non-negative real numbers $ (1,2)$ from the domain $\Bbb R_{*},$ and put those numbers into the given function f(x)=1/x.It gives us 1=1/2.But it's not true.So ,it can't be one-one function.But in the answer,it is one-one function.Would anyone enlighten me where is my fault? Thank you!
6 replies
Vulch
Apr 11, 2025
Vulch
4 hours ago
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Inequalities
sqing   6
N 4 hours ago by sqing
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
$$ a+b+c\leq 3 $$
6 replies
sqing
Apr 4, 2025
sqing
4 hours ago
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hard number theory
eric201291   0
4 hours ago
Prove:There are no integers x, y, that y^2+9998587980=x^3.
0 replies
eric201291
4 hours ago
0 replies
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Amc 10 mock
Mathsboy100   3
N 5 hours ago by iwastedmyusername
let \[\lfloor x \rfloor\]denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which \[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]
3 replies
Mathsboy100
Oct 9, 2024
iwastedmyusername
5 hours ago
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Inequalities
lgx57   4
N 5 hours ago by pooh123
Let $0 < a,b,c < 1$. Prove that

$$a(1-b)+b(1-c)+c(1-a)<1$$
4 replies
lgx57
Mar 19, 2025
pooh123
5 hours ago
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Putnam 1958 November B1
sqrtX   10
N 5 hours ago by StarLex1
Source: Putnam 1958 November
Given
$$b_n = \sum_{k=0}^{n} \binom{n}{k}^{-1}, \;\; n\geq 1,$$prove that
$$b_n = \frac{n+1}{2n} b_{n-1} +1, \;\; n \geq 2.$$Hence, as a corollary, show
$$ \lim_{n \to \infty} b_n =2.$$
10 replies
sqrtX
Jul 19, 2022
StarLex1
5 hours ago
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Let x,y,z be non-zero reals
Purple_Planet   3
N 5 hours ago by sqing
Let $x,y,z$ be non-zero real numbers. Define $E=\frac{|x+y|}{|x|+|y|}+\frac{|x+z|}{|x|+|z|}+\frac{|y+z|}{|y|+|z|}$, then the number of all integers which lies in the range of $E$ is equal to.
3 replies
Purple_Planet
Jul 16, 2019
sqing
5 hours ago
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Identity Proof
jjsunpu   2
N 6 hours ago by fruitmonster97
Hi this is my identity I name it Excalibur

I proved it already using induction what other ways?
2 replies
jjsunpu
Today at 10:35 AM
fruitmonster97
6 hours ago
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Three 3-digit numbers
miiirz30   5
N 6 hours ago by fruitmonster97
Leonard wrote three 3-digit numbers on the board whose sum is $1000$. All of the nine digits are different. Determine which digit does not appear on the board.

Proposed by Giorgi Arabidze, Georgia
5 replies
miiirz30
Mar 31, 2025
fruitmonster97
6 hours ago
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On coefficients of a polynomial over a finite field
Ciobi_   0
Apr 2, 2025
Source: Romania NMO 2025 12.4
Let $p$ be an odd prime number, and $k$ be an odd number not divisible by $p$. Consider a field $K$ be a field with $kp+1$ elements, and $A = \{x_1,x_2, \dots, x_t\}$ be the set of elements of $K^*$, whose order is not $k$ in the multiplicative group $(K^*,\cdot)$. Prove that the polynomial $P(X)=(X+x_1)(X+x_2)\dots(X+x_t)$ has at least $p$ coefficients equal to $1$.
0 replies
Ciobi_
Apr 2, 2025
0 replies
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On coefficients of a polynomial over a finite field
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Reveal topic tags
number theory
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Ciobi_
25 posts
Ciobi_
#1 h  Apr 2, 2025, 2:59 PM
Y by
Let $p$ be an odd prime number, and $k$ be an odd number not divisible by $p$. Consider a field $K$ be a field with $kp+1$ elements, and $A = \{x_1,x_2, \dots, x_t\}$ be the set of elements of $K^*$, whose order is not $k$ in the multiplicative group $(K^*,\cdot)$. Prove that the polynomial $P(X)=(X+x_1)(X+x_2)\dots(X+x_t)$ has at least $p$ coefficients equal to $1$.
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