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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
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0 replies
jlacosta
Mar 2, 2025
0 replies
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quadratics
luciazhu1105   15
N an hour ago by KF329
I really need help on quadratics and I don't know why I also kinda need a bit of help on graphing functions and finding the domain and range of them.
15 replies
luciazhu1105
Feb 14, 2025
KF329
an hour ago
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Combi counting
Caasi_Gnow   0
2 hours ago
Find the number of different ways to arrange seven people around a circular meeting table if A and B must sit together and C and D cannot sit next to each other. (Note: the order for A and B might be A,B or B,A)
0 replies
Caasi_Gnow
2 hours ago
0 replies
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Mathcounts state iowa
iwillregretthisnamelater   1
N 4 hours ago by Dutchb
Ok I’m a 6th grader in Iowa who got 38 in chapter which was first, so what are the chances of me getting in nats? I should feel confident but I don’t. I have a week until states and I’m getting really anxious. What should I do? And also does the cdr count in Iowa? Because I heard that some states do cdr for fun or something and that it doesn’t count to final standings.
1 reply
iwillregretthisnamelater
5 hours ago
Dutchb
4 hours ago
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I need help!-tpulak
tpulak   16
N 5 hours ago by mdk2013
Hello peoples:
I need help on a problem. If you know the solution, could you please post it, and tell me how to do it? Here is the problem:
"The product (66)(9)(22)(39) has a prime factorization of the form (2A)(3B)(11C)(13D). What is the value of ac-bd? "

{Note: 2a actually represents "2 to the power of a", same goes for 3b,11c, & 13D}

PLEASE HELP ME! :?: :!: :!: :!:
16 replies
tpulak
Nov 30, 2007
mdk2013
5 hours ago
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Inequalities
sqing   2
N Today at 2:19 AM by sqing
Let $ a,b,c\geq 2 . $ Prove that
$$(a-1)(b^2-2)(c-1) - \frac{1}{2}abc\geq -2$$$$(a^2-2)(b-1)(c^2-2) - \frac{3}{2}abc\geq -6$$
2 replies
sqing
Yesterday at 1:42 PM
sqing
Today at 2:19 AM
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2019 Chile Classification / Qualifying NMO Juniors XXXI
parmenides51   5
N Today at 1:14 AM by liyufish
p1. Consider the sequence of positive integers $2, 3, 5, 6, 7, 8, 10, 11 ...$. which are not perfect squares. Calculate the $2019$-th term of the sequence.


p2. In a triangle $ABC$, let $D$ be the midpoint of side $BC$ and $E$ be the midpoint of segment $AD$. Lines $AC$ and $BE$ intersect at $F$. Show that $3AF = AC$.


p3. Find all positive integers $n$ such that $n! + 2019$ is a square perfect.


p4. In a party, there is a certain group of people, none of whom has more than $3$ friends in this. However, if two people are not friends at least they have a friend in this party. What is the largest possible number of people in the party?
5 replies
parmenides51
Oct 11, 2021
liyufish
Today at 1:14 AM
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2^n = p+3^p
reeh_haan   8
N Today at 12:20 AM by MajesticCheese
Find all pairs $(p, n)$ of positive integers which satisfy the equation $$2^n = p+3^p$$
8 replies
reeh_haan
Dec 28, 2021
MajesticCheese
Today at 12:20 AM
J
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2017 Mock ARML Team Round #7 Revenge of the incenters
parmenides51   1
N Yesterday at 10:16 PM by Giant_PT
Let $ABC$ be a triangle with side lengths $AB = 20$, $BC = 17$, $AC = 13$, incenter $I$ and circumcircle $\Gamma$, and let $\omega_A$ be the circle that is tangent to $AB$, $AC$, $\Gamma$ at $D$ ,$E$, $F$, respectively. If $I_F$ ,$ I_I$, $I_C$ denote the incenters of $CFI$, $FIE$, $ECF$, respectively, find the measure, in degrees, of the largest angle in $\vartriangle I_F I_II_C$.
1 reply
parmenides51
Jan 12, 2024
Giant_PT
Yesterday at 10:16 PM
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Don Sirloin Bowel's Algorithm
BadAtCompetitionMath21420   1
N Yesterday at 10:07 PM by ohiorizzler1434
Can you sum
\begin{align*} \sum_{k=1}^n F^m_k? \end{align*}
I coined a strategy Don Sirloin Bowel's algorithm because I found it and didn't see it anywhere else. Please tell me if this looks familiar because it can have absolutely scrumptious applications. (I hope that this stuff can help you sum that.) Starting with the "golden quadratic" or whatever the sigma it's called $x^2=x+1$. Multiplying by $x$ on both sides and resubstituting $x^2$, we find $x^3=x^2+x=(x+1)+x=2x+1$. Continuing this process, we find $x^4=3x+2$, $x^5=5x+3$, and so forth. We claim that $x^n=F_nx+F_{n-1}.$ We already have our base case. $x^n=F_nx+F_{n-1}$. Multiplying by $x$, $x^{n+1}=F_nx^2+F_{n-1}x=F_n(x+1)+F_{n-1}x=F_{n+1}x+F_{n}$. Hence, we have proved our claim by induction. Now, raising both sides to the $m$th power,
\begin{align*} x^{mn}&=\sum_{k=0}^m \binom{m}{k}F_{n}^{m-k}F_{n-1}^kx^m\\ &=\sum_{k=0}^m \binom{m}{k}F_{n}^{m-k}F_{n-1}^k(F_mx+F_{m-1})\\ &=x\sum_{k=0}^m \binom{m}{k}F_{n}^{m-k}F_{n-1}^kF_m+\sum_{k=0}^mF_{n}^{m-k}F_{n-1}^kF_{m-1}\\ &=F_{mn}x+F_{mn-1} \end{align*}So,
\begin{align*} F_{mn}=\sum_{k=0}^m \binom{m}{k}F_{n}^{m-k}F_{n-1}^kF_m \end{align*}and
\begin{align*} F_{mn-1}=\sum_{k=0}^mF_{n}^{m-k}F_{n-1}^kF_{m-1} \end{align*}Are these identities useful? And can you use them to compute the sum?
1 reply
BadAtCompetitionMath21420
Yesterday at 9:50 PM
ohiorizzler1434
Yesterday at 10:07 PM
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Chessboard
Ecrin_eren   5
N Yesterday at 8:44 PM by TrendCrusher
On an 8×8 checkerboard, what is the minimum number of squares that must be marked (including the marked ones) so that every square has exactly one marked neighbor? (We define neighbors as squares that share a common edge, and a square is not considered a neighbor of itself.)

5 replies
Ecrin_eren
Tuesday at 8:55 PM
TrendCrusher
Yesterday at 8:44 PM
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Why does the combined equation have two negative solutions?
Luking   1
N Yesterday at 7:30 PM by vanstraelen
It is known that the moving point $G(x,y)$ is on the curve $C_1: y^2 - x^2 = 1$. There is a parabola $C_2: x^2 = 4y$ with focus $F$. Two tangent lines to$C_2$ are drawn through a point $P$ on $C1$, and the tangent points are $A$ and $B$ respectively. The line $l$ parallel to the line $AB$ is tangent to $C_2$ at point $Q$. Question: When the line $l$ and $C_1$ have two intersection points, find the range of $|QF|$.\
This may make some of the information in the question useless, because I deleted the first two questions of this big question in order to avoid making the question too long and get straight to the point.\
According to the calculation, I get the analytical expression of the line $l$ as $y=\frac{x_0}{2} x - \frac{x_0^2}{4}$.\
At first, I thought it only needed to be not parallel to the parabola asymptotes.\
That is, the slope of the straight line $k \neq \frac{a}{b}$ , then $\frac{x_0}{2} \neq \frac{a}{b}$ , so $x_0 \neq \pm \frac{2a}{b} = \pm2$ ,and $x_0^2 \neq 4$.\
$|QF| = y_0 +1 \neq 5$.\
But when I checked the answer, it was wrong.\
It combines the straight line $l$ with the curve $C_1$ to get an equation and then uses Vieta's theorem.
\begin{cases}
y=\frac{x_0}{2} x - \frac{x_0^2}{4}
x^2=4y
\end{cases}
$$( 4 x_{0}^{2}-1 6 ) y^{2}-8 x_{0}^{2} y-x_{0}^{4}-4 x_{0}^{2}=0 $$The answer is that according to the question, the equation has two negative roots. I can't understand this, and this is exactly where my problem lies.\
Then it gets the following system of equations:
\begin{cases}
4x_0-16 \neq 0
\Delta > 0
x_1+x_2 <0
x_1\cdot x_2>0
\end{cases}
Solve, $2\sqrt{5}-2<x_0<4$.
So we get $|QF|=\frac {x_0^2}{4} + 1 \in (\frac {\sqrt{5}+1}{2},2)$.
I hope you can help me figure out why both roots of that equation are negative.
IMAGE
IMAGE
1 reply
Luking
Yesterday at 4:02 PM
vanstraelen
Yesterday at 7:30 PM
J
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Rolling a die
Ecrin_eren   2
N Yesterday at 6:16 PM by tofubear
John has a standard die with values from 1 to 6. He plays a game where, after each roll, the value on the top face permanently increases by 1. After rolling the die three times, the probability that at least one face has a value of at least 7 is given as a/b , where a and b are coprime. What is the value of a/b ?
2 replies
Ecrin_eren
Yesterday at 5:05 PM
tofubear
Yesterday at 6:16 PM
J
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Travel group
Ecrin_eren   0
Yesterday at 5:12 PM
A group of 5 students will form a travel group of 3 students. Each student is asked to select two other students they would like to travel with. Given that the probability of forming a 3-person group where each member has selected the other two is a/b , where a and b are coprime positive integers, what is the value of a/b ?

0 replies
Ecrin_eren
Yesterday at 5:12 PM
0 replies
J
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Area problem
MTA_2024   1
N Yesterday at 4:46 PM by vanstraelen
Let $\omega$ be a circle inscribed inside a rhombus $ABCD$. Let $P$ and $Q$ be variable points on $AB$ and $AD$ respectively, such as $PQ$ is always the tangent line to $\omega$.
Prove that for any position of $P$ and $Q$ the area of triangle $\triangle CPQ$ is the same.
1 reply
MTA_2024
Mar 16, 2025
vanstraelen
Yesterday at 4:46 PM
J
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J
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MATHCOUNTS Chapter Score Thread
apex304   103
N Mar 17, 2025 by captainnobody
$\begin{tabular}{c|c|c|c|c}Username & Grade & Score \\ \hline apex304 & 8 & 46 \\ \end{tabular}$
103 replies
apex304
Mar 1, 2025
captainnobody
Mar 17, 2025
J
MATHCOUNTS Chapter Score Thread
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Reveal topic tags
MATHCOUNTS
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apex304
519 posts
apex304
#1 h  Mar 1, 2025, 5:20 PM • 2 Y
Y by GlitchyBoy, BigOrca
$\begin{tabular}{c|c|c|c|c}Username & Grade & Score \\ \hline apex304 & 8 & 46 \\ \end{tabular}$
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Jaxman8
95 posts
Jaxman8
#2 h  Mar 1, 2025, 5:23 PM
Y by
$\begin{tabular}{c|c|c|c|c}Username & Grade & Score \\ \hline apex304 & 8 & 46 \\ Jaxman8 & 8 & 39(27+12) \\ \end{tabular}$
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mathprodigy2011
226 posts
mathprodigy2011
#3 h  Mar 1, 2025, 5:34 PM
Y by
Jaxman8 wrote:
$\begin{tabular}{c|c|c|c|c}Username & Grade & Score \\ \hline apex304 & 8 & 46 \\ mathprodigy2011 & 8 & 43 \\ Jaxman8 & 8 & 39(27+12) \\ \end{tabular}$
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Existing_Human1
182 posts
Existing_Human1
#4 h  Mar 1, 2025, 5:39 PM
Y by
$\begin{tabular}{c|c|c|c|c}Username & Grade & Target & Sprint & Final Score \\ \hline apex304 & 8 & 16 & 30 & 46\\ Jaxman8 & 8 & 12 & 27 & 39 \\ ExistingHuman1 & 8 & 14 & 29 & 43\\ \end{tabular}$

Added more columns
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jb2015007
1705 posts
jb2015007
#5 h  Mar 1, 2025, 5:39 PM
Y by
$\begin{tabular}{c|c|c|c|c}Username & Grade & Score \\ \hline apex304 & 8 & 46 \\ mathprodigy2011 & 8 & 43 \\ Jaxman8 & 8 & 39(27+12) \\ jb2015007 & 7 & 35 :facepalm: \end{tabular}$
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luckyrabbit11
379 posts
luckyrabbit11
#6 h  Mar 1, 2025, 5:40 PM
Y by
$\begin{tabular}{c|c|c|c|c}Username & Grade & Score \\ \hline apex304 & 8 & 46 \\ mathprodigy2011 & 8 & 43 \\ Jaxman8 & 8 & 39(27+12) \\ luckyrabbit11 & 8 & 41 \\ jb2015007 & 7 & 35 :facepalm: \end{tabular}$
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Soupboy0
158 posts
Soupboy0
#8 h  Mar 1, 2025, 5:58 PM
Y by
$\begin{tabular}{c|c|c|c|c}Username & Grade & Score \\ \hline apex304 & 8 & 46 \\ mathprodigy2011 & 8 & 43 \\ Jaxman8 & 8 & 39(27+12) \\ luckyrabbit11 & 8 & 41 \\ jb2015007 & 7 & 35 :facepalm: \\ Soupboy0 & 7 & 39 \end{tabular}$
4 sillies btw
This post has been edited 1 time. Last edited by Soupboy0, Mar 1, 2025, 5:59 PM
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Existing_Human1
182 posts
Existing_Human1
#9 h  Mar 1, 2025, 6:04 PM
Y by
Can you guys put sprint and target, then I'll add you to the table with more columns
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Soupboy0
158 posts
Soupboy0
#10 h  Mar 1, 2025, 6:05 PM • 1 Y
Y by nmlikesmath
sprint: 25 target: 14 (MASSIVE SELL)
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Existing_Human1
182 posts
Existing_Human1
#11 h  Mar 1, 2025, 6:09 PM
Y by
$\begin{tabular}{c|c|c|c|c}Username & Grade & Target & Sprint & Final Score \\ \hline apex304 & 8 & 16 & 30 & 46\\ Jaxman8 & 8 & 12 & 27 & 39 \\ Mathprodigy2011 & 8 & idk & idk & 43\\ ExistingHuman1 & 8 & 14 & 29 & 43\\ jb2015007 & 7 & idk & idk & 35 :facepalm: \\ luckyrabbit11 & 8 & idk & idk & 41\\ Soupboy0 & 7 & 14 & 25 & 39 \\ \end{tabular}$
This post has been edited 2 times. Last edited by Existing_Human1, Mar 1, 2025, 6:10 PM
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jb2015007
1705 posts
jb2015007
#12 h  Mar 1, 2025, 6:11 PM
Y by
$\begin{tabular}{c|c|c|c|c}Username & Grade & Target & Sprint & Final Score \\ \hline apex304 & 8 & 16 & 30 & 46\\ Jaxman8 & 8 & 12 & 27 & 39 \\ Mathprodigy2011 & 8 & idk & idk & 43\\ ExistingHuman1 & 8 & 14 & 29 & 43\\ jb2015007 & 7 & 14 & 21 & 35 :facepalm: \\ luckyrabbit11 & 8 & idk & idk & 41\\ Soupboy0 & 7 & 14 & 25 & 39 \\ \end{tabular}$
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elizhang101412
1184 posts
elizhang101412
#13 h  Mar 1, 2025, 6:14 PM
Y by
a
$\begin{tabular}{c|c|c|c|c}Username & Grade & Target & Sprint & Final Score \\ \hline apex304 & 8 & 16 & 30 & 46\\ Jaxman8 & 8 & 12 & 27 & 39 \\ Mathprodigy2011 & 8 & idk & idk & 43\\ ExistingHuman1 & 8 & 14 & 29 & 43\\ jb2015007 & 7 & 14 & 21 & 35 :facepalm: \\ luckyrabbit11 & 8 & idk & idk & 41\\ Soupboy0 & 7 & 14 & 25 & 39 \\ elizhang101412 & 6 & 12 & 27 & 39 \\ \end{tabular}$
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derekwang2048
1188 posts
derekwang2048
#14 h  Mar 1, 2025, 6:58 PM
Y by
idk my real score tho (this is what i think i got)
$\begin{tabular}{c|c|c|c|c}Username & Grade & Target & Sprint & Final Score \\ \hline apex304 & 8 & 16 & 30 & 46\\ Jaxman8 & 8 & 12 & 27 & 39 \\ Mathprodigy2011 & 8 & idk & idk & 43\\ ExistingHuman1 & 8 & 14 & 29 & 43\\ jb2015007 & 7 & 14 & 21 & 35 :facepalm: \\ luckyrabbit11 & 8 & idk & idk & 41\\ Soupboy0 & 7 & 14 & 25 & 39 \\ elizhang101412 & 6 & 12 & 27 & 39 \\ derekwang2048 & 6 & 14 & 26-29? & 40-43? \\ \end{tabular}$
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martianrunner
122 posts
martianrunner
#15 h  Mar 1, 2025, 7:31 PM
Y by
$\begin{tabular}{c|c|c|c|c}Username & Grade & Target & Sprint & Final Score \\ \hline apex304 & 8 & 16 & 30 & 46\\ Jaxman8 & 8 & 12 & 27 & 39 \\ Mathprodigy2011 & 8 & idk & idk & 43\\ ExistingHuman1 & 8 & 14 & 29 & 43\\ jb2015007 & 7 & 14 & 21 & 35 :facepalm: \\ luckyrabbit11 & 8 & idk & idk & 41\\ Soupboy0 & 7 & 14 & 25 & 39 \\ elizhang101412 & 6 & 12 & 27 & 39 \\ derekwang2048 & 6 & 14 & 26-29? & 40-43? \\ martianrunner & 8 & 14 & 26-27 & 40-41 \\ \end{tabular}$
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ilikemath247365
220 posts
ilikemath247365
#16 h  Mar 1, 2025, 7:36 PM
Y by
$\begin{tabular}{c|c|c|c|c}Username & Grade & Target & Sprint & Final Score \\ \hline apex304 & 8 & 16 & 30 & 46\\ Jaxman8 & 8 & 12 & 27 & 39 \\ Mathprodigy2011 & 8 & idk & idk & 43\\ ExistingHuman1 & 8 & 14 & 29 & 43\\ jb2015007 & 7 & 14 & 21 & 35 :facepalm: \\ luckyrabbit11 & 8 & idk & idk & 41\\ Soupboy0 & 7 & 14 & 25 & 39 \\ elizhang101412 & 6 & 12 & 27 & 39 \\ derekwang2048 & 6 & 14 & 26-29? & 40-43? \\ martianrunner & 8 & 14 & 26-27 & 40-41 \\ ilikemath247365 & 7 & 14 & 26 & 40 \\ \end{tabular}$
If I didn't silly, I would've gotten a 44 or even a 45(Sillied numbers 27, 28 on sprint, almost solved 29, sillied number 4 on target)
This post has been edited 1 time. Last edited by ilikemath247365, Mar 1, 2025, 7:37 PM
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