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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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ABC + DE 2024 TMC AIME Mock #14
parmenides51   4
N 15 minutes ago by vincentwant
Let $P_n$ be the product of all positive integers less than $n$ not divisible by $5$. Let $N$ be the remainder when $P_{3749}$ is divided by $15625 = 5^6$. If $N =\overline{ABCDE}_{10}$, find $\overline{ABC} + \overline{DE}$.
4 replies
parmenides51
Yesterday at 8:18 PM
vincentwant
15 minutes ago
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Polynomial with integer ciefficient
girishpimoli   11
N 25 minutes ago by Shan3t
Let $P(x)$ be a polynomial with integer coefficient, It is known that $P(x)$ takes the values $2015$ for $4$ distinct integers , Then the number of integer values of $x$ for which $ P(x)=2022$
11 replies
+1 w
girishpimoli
Yesterday at 12:57 AM
Shan3t
25 minutes ago
J
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equilateral geo 2024 TMC AIME Mock #12
parmenides51   3
N 25 minutes ago by Sedro
Let $ABC$ be an equilateral triangle. Let $D$ be a point on $AB$, $E$ be a point on $BC$, and $F$ be a point on $AC$ such that $\angle DEF = 120^o$ and̸ $\angle DFE = 30^o$. If $AB = 108$ and $CF = 38$, find $AD$.
3 replies
parmenides51
Yesterday at 8:15 PM
Sedro
25 minutes ago
J
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Another binomial coefficients sum
aether01   8
N an hour ago by soryn
Prove that $$\sum_{k=0}^{n}{{\left(-1\right)}^{k}\binom{n}{k} \binom{2n-k}{n-k}} = 1$$
8 replies
aether01
Mar 3, 2022
soryn
an hour ago
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Double Sum
P162008   2
N an hour ago by soryn
Compute the value of $\Omega = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{\left(\frac{1}{4}\right)^{m+n}}{(2m + 1)(m + n + 1)}.$
2 replies
P162008
Yesterday at 9:31 AM
soryn
an hour ago
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Combination
aria123   1
N 2 hours ago by aria123
Prove that three squares of side length $4$ cannot completely cover a square of side length $5$, if the three smaller squares do not overlap in their interiors (i.e., they may touch at edges or corners, but no part of one lies over another).
1 reply
aria123
Apr 15, 2025
aria123
2 hours ago
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Inequalities
sqing   0
3 hours ago
Let \( x, y \geq \frac{3}{2} \). Prove that
$$ \frac{2}{1+xy} + x + y \geq \frac{x}{y} + \frac{y}{x}+ \frac{21}{13}$$Let \( x, y \geq 2 \). Prove that
$$ \frac{2}{1+xy} + x + y \geq \frac{x}{y} + \frac{y}{x}+ \frac{12}{5}$$Let \(0< x, y \leq \frac{3}{2} \). Prove that
$$ \frac{2}{1+xy} + x + y \leq \frac{x}{y} + \frac{y}{x}+\frac{21}{13}$$Let \(0< x, y \leq 2 \). Prove that
$$ \frac{2}{1+xy} + x + y \leq \frac{x}{y} + \frac{y}{x}+\frac{12}{5}$$
0 replies
sqing
3 hours ago
0 replies
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Number theory
MathsII-enjoy   2
N 3 hours ago by MathsII-enjoy
$Find$ $all$ $integers$ $n$ $such$ $that$ $n-1$ $and$ $\frac{n(n+1)}{2}$ $is$ $a$ $perfect$ $number$.
2 replies
MathsII-enjoy
Today at 5:21 AM
MathsII-enjoy
3 hours ago
J
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Sequence
lgx57   5
N 3 hours ago by lgx57
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
5 replies
lgx57
5 hours ago
lgx57
3 hours ago
J
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set of sum of three or fewer powers of 2, 2024 TMC AIME Mock #13
parmenides51   5
N 3 hours ago by vincentwant
Let $S$ denote the set of all positive integers that can be expressed as a sum of three or fewer powers of $2$. Let $N$ be the smallest positive integer that cannot be expressed in the form $a-b$, where $a, b \in S$. Find the remainder when $N$ is divided by $1000$.
5 replies
parmenides51
Yesterday at 8:16 PM
vincentwant
3 hours ago
J
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trigonogeometry 2024 TMC AIME Mock #15
parmenides51   3
N 3 hours ago by franklin2013
Let $\vartriangle ABC$ have angles $ \alpha, \beta$ and $\gamma$ such that $\cos (\alpha) = \frac1 3$ and $\cos (\beta) = \frac{1}{17}$ . Moreover, suppose that the product of the side lengths of the triangle is equal to its area. Let $(ABC)$ denote the circumcircle of $ABC$. Let $AO$ intersect $(BOC)$ at $D$, $BO$ intersect $(COA)$ at $ E$, and $CO$ intersect $(AOB)$ at $F$. If the area of $DEF$ can be written as $\frac{p\sqrt{r}}{q}$ for relatively prime integers $p$ and $q$ and squarefree $r$, find the sum of all prime factors of $q$, counting multiplicities (so the sum of prime factors of $48$ is $2 + 2 + 2 + 2 + 3 = 11$), given that $q$ has $30$ divisors.
3 replies
parmenides51
Yesterday at 8:22 PM
franklin2013
3 hours ago
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Max and min of a/5+b/4+c/3+d/2 under a/4+b/3+c/2+d=1/2 and a^2/7+b^2/5+c^2/3+...
tom-nowy   1
N 4 hours ago by steve4916
Let $a, b, c, d$ be real numbers satisfying the following two conditions:
\begin{align*} &\frac{a}{4}+\frac{b}{3}+\frac{c}{2}+d=\frac{1}{2}, \\ &\frac{a^2}{7}+\frac{b^2}{5}+\frac{c^2}{3}+d^2 + \frac{ab}{3}+\frac{2ac}{5}+\frac{ad}{2} +\frac{bc}{2}+\frac{2bd}{3}+\frac{cd}{2}=\frac{1}{3} . \end{align*}Find the possible maximum and minimum values of
$$\frac{a}{5}+\frac{b}{4}+\frac{c}{3}+\frac{d}{2}.$$
1 reply
tom-nowy
Today at 11:48 AM
steve4916
4 hours ago
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20 fair coins are flipped, N of them land heads 2024 TMC AIME Mock #6
parmenides51   3
N 5 hours ago by ostriches88
$20$ fair coins are flipped. If $N$ of them land heads, find the expected value of $N^2$.
3 replies
parmenides51
Yesterday at 8:05 PM
ostriches88
5 hours ago
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Inequalities
sqing   10
N 5 hours ago by sqing
Let $ a,b \in [0 ,1] . $ Prove that
$$\frac{a}{ 1-ab+b }+\frac{b }{ 1-ab+a } \leq 2$$$$ \frac{a}{ 1+ab+b^2 }+\frac{b }{ 1+ab+a^2 }+\frac{ab }{2+ab } \leq 1$$$$\frac{a}{ 1-ab+b^2 }+\frac{b }{ 1-ab+a^2 }+\frac{1 }{1+ab }\leq \frac{5}{2}$$$$\frac{a}{ 1-ab+b^2 }+\frac{b }{ 1-ab+a^2 }+\frac{1 }{1+2ab }\leq \frac{7}{3}$$$$\frac{a}{ 1+ab+b^2 }+\frac{b }{ 1+ab+a^2 } +\frac{ab }{1+ab }\leq \frac{7}{6 }$$
10 replies
sqing
Apr 25, 2025
sqing
5 hours ago
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J
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Very Nice equations
steven_zhang123   6
N Feb 27, 2025 by eric201291
Solve this:$\left\{\begin{matrix} x^{2023}+y^{2023}+z^{2023}=2023 \\ x^{2024}+y^{2024}+z^{2024}=2024 \\ x^{2025}+y^{2025}+z^{2025}=2025 \end{matrix}\right.$
6 replies
steven_zhang123
Oct 13, 2024
eric201291
Feb 27, 2025
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Very Nice equations
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Reveal topic tags
algebra
system of equations
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steven_zhang123
411 posts
steven_zhang123
#1 h  Oct 13, 2024, 12:30 PM • 1 Y
Y by William_Mai
Solve this:$\left\{\begin{matrix} x^{2023}+y^{2023}+z^{2023}=2023 \\ x^{2024}+y^{2024}+z^{2024}=2024 \\ x^{2025}+y^{2025}+z^{2025}=2025 \end{matrix}\right.$
This post has been edited 1 time. Last edited by steven_zhang123, Oct 13, 2024, 1:08 PM
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SomeonecoolLovesMaths
3206 posts
SomeonecoolLovesMaths
#2 h  Oct 13, 2024, 1:04 PM
Y by
steven_zhang123 wrote:
Solve this:$\left\{\begin{matrix} x^{2023}+y^{2023}+z^{2023}=2023 \\ x^{2024}+y^{2024}+z^{2024}=2024 \\ x^{2025}+y^{2025}+x^{2025}=2025 \end{matrix}\right.$

Corrected version:

Solve this:$\left\{\begin{matrix} x^{2023}+y^{2023}+z^{2023}=2023 \\ x^{2024}+y^{2024}+z^{2024}=2024 \\ x^{2025}+y^{2025}+z^{2025}=2025 \end{matrix}\right.$
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steven_zhang123
411 posts
steven_zhang123
#4 h  Oct 13, 2024, 1:08 PM
Y by
SomeonecoolLovesMaths wrote:
steven_zhang123 wrote:
Solve this:$\left\{\begin{matrix} x^{2023}+y^{2023}+z^{2023}=2023 \\ x^{2024}+y^{2024}+z^{2024}=2024 \\ x^{2025}+y^{2025}+x^{2025}=2025 \end{matrix}\right.$

Corrected version:

Solve this:$\left\{\begin{matrix} x^{2023}+y^{2023}+z^{2023}=2023 \\ x^{2024}+y^{2024}+z^{2024}=2024 \\ x^{2025}+y^{2025}+z^{2025}=2025 \end{matrix}\right.$
Oh, thanks! I'll fixed it.
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persamaankuadrat
152 posts
persamaankuadrat
#6 h  Oct 21, 2024, 11:52 PM
Y by
Does anyone have any ideas?
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k2i_forever
33 posts
k2i_forever
#7 h  Oct 22, 2024, 1:36 AM
Y by
$x^{2025}+y^{2025}=(x^{2014}+y^{2024})(x+y)-xy(x^{2013}+y^{2013})$
$2025-z^{2025}=(x+y)(2024-z^{2024})-xy(2023-z^{2023})$
This is the farthest I can get

Or maybe
There are three roots for n to the equation below:
$x^{n}+y^{n}+z^{n}-n=0$
This post has been edited 2 times. Last edited by k2i_forever, Oct 22, 2024, 1:39 AM
Reason: LaTeX problem
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steven_zhang123
411 posts
steven_zhang123
#8 h  Oct 27, 2024, 7:18 AM
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Any thoughts?
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eric201291
208 posts
eric201291
#9 h  Feb 27, 2025, 12:54 PM
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Use Newton
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