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k a My Retirement & New Leadership at AoPS
rrusczyk   1571
N Mar 26, 2025 by SmartGroot
I write today to announce my retirement as CEO from Art of Problem Solving. When I founded AoPS 22 years ago, I never imagined that we would reach so many students and families, or that we would find so many channels through which we discover, inspire, and train the great problem solvers of the next generation. I am very proud of all we have accomplished and I’m thankful for the many supporters who provided inspiration and encouragement along the way. I'm particularly grateful to all of the wonderful members of the AoPS Community!

I’m delighted to introduce our new leaders - Ben Kornell and Andrew Sutherland. Ben has extensive experience in education and edtech prior to joining AoPS as my successor as CEO, including starting like I did as a classroom teacher. He has a deep understanding of the value of our work because he’s an AoPS parent! Meanwhile, Andrew and I have common roots as founders of education companies; he launched Quizlet at age 15! His journey from founder to MIT to technology and product leader as our Chief Product Officer traces a pathway many of our students will follow in the years to come.

Thank you again for your support for Art of Problem Solving and we look forward to working with millions more wonderful problem solvers in the years to come.

And special thanks to all of the amazing AoPS team members who have helped build AoPS. We’ve come a long way from here:IMAGE
1571 replies
rrusczyk
Mar 24, 2025
SmartGroot
Mar 26, 2025
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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.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

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.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
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0 replies
jlacosta
Mar 2, 2025
0 replies
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2025 USC Math Comp (SCMC) individual round
Bluesoul   0
an hour ago
1. For an integer $x,$ we define a \textit{step} as either doubling the value of the integer or subtracting 3 from it. What is the minimum number of steps required to obtain 25 from 11?
2. Find the sum of all integer values of $n$ that satisfy the inequality chain \[n^3<2025<3^n.\]3. A rectangle with length 20 units and height 16 units is divided into 10 smaller congruent rectangles. Let $P$ be the largest possible perimeter of one of these small rectangles; compute the value of $10P.$
4. Positive integers $x,y,z$ satisfy the system
\[\left\{\begin{array}{l} x^2+y^2=z^2+22\\ y^2+z^2=x^2+76\\ x^2+z^2=y^2-4. \end{array} \right.\]What is the value of $xyz?$
5. \begin{problem}
Four husband-wife couples go ballroom dancing one evening. The husbands' names are Henry, Peter, Steve, and Roger, while the wives' names are Elizabeth, Keira, Mary, and Anne. At a given moment, Henry's wife is dancing with Elizabeth's husband, who is not Henry; Roger and Anne are not dancing; Peter is playing the trumpet; and Mary is playing the piano. Given that Anne's husband is not Peter, how many different letters are in the name of Roger's wife?
6. A laser is fired from vertex $A$ into the interior of regular hexagon $ABCDEF,$ whose sides are mirrors, and hits side $\overline{CD}$ at $G.$ It then reflects and hits $\overline{AF}$ at $H,$ and finally reflects and hits $\overline{DE}$ at $I.$ If $\angle BAG=45^\circ,$ then how many degrees are in $\angle HIE?$
7. What is the value of the expression
\[\frac{\log_2(\log_2 3)}{\log_4(\log_4 9)}?\]8. The area of equiangular octagon $ABCDEFGH,$ with $AB=EF=2,$ $BC=FG=3,$ $CD=GH=4,$ and $DE=AH=5,$ can be written in the form $a+b\sqrt{2}$, find $a+b$.
9. A toe-wrestling tournament between Don and Kam consists of three matches. In each match, the winner is the first person to reach five points. After the three matches, each person’s score is the number of matches they won, plus the sum of the points they earned during all of their matches. Let $d$ and $k$ denote Don and Kam’s final scores, respectively. How many ordered pairs $(d, k)$ are possible?
10. For each positive integer $n$, let $s(n)$ denote the sum of the remainders when $n$ is divided by $2,3,4,5,$ and $6.$ For example, when $n=93,$ we have $s(93)=1+0+1+3+3=8.$ Compute the integer $N$ for which \[\sum_{n=1}^{N}s(n)=2025.\]11. For complex numbers $z,$ we define the function \[f(z)=\frac{z+3}{z-2i}.\]Over all values of $z$ for which $f(z)$ is real, the minimum possible value of $|z|^2$ can be written in the form $\dfrac{m}{n}$ for positive integers $m$ and $n.$ Compute the value of $100m+n.$
12. In convex quadrilateral $ABCD,$ $AB=6, BC=10,$ and $\angle{ABC}=90^{\circ}$. Let $M$ and $N$ be the midpoints of $\overline{AD}$ and $\overline{CD},$ respectively. Compute the area of $\triangle{BMN},$ given that the area of $ABCD$ is $50$.
13. What is the remainder when $20^{25}$ is divided by 2025?
14. Your friend plays a prank on you by changing your phone's password. Your friend chooses a password consisting of 4 decimal digits $\overline{abcd}$ uniformly at random and tells you the sum of its digits. (Leading zeros are allowed, so your friend can choose any password from 0000, 0001, and so on to 9999.) Then, you select a digit $e;$ your friend tells you the password if and only if $e$ is the median of the set $\{a,b,c,d,e\}.$
\null
Now, your friend picks a password whose digits sum to 20; let $S$ be the set of all such passwords. Suppose you select $e$ such that the probability that your friend tells you the password, given this information, is maximized. Compute the number of passwords in $S$ for which this would not occur, given your choice of $e.$
15. For real numbers $x,$ we define the function \[f(x)=\lceil{1+\sqrt{x+1}}\rceil+\lfloor{1-\sqrt{x-1}}\rfloor.\]Compute the $100^\text{th}-$smallest integer $x$ for which $f(x)=2$.
16. How many ordered pairs $(x,y),$ with $1\le x,y\le 100,$ satisfy the congruence \[2^{2^x+2^y}\equiv 1\pmod{101}?\]17. $\triangle ABC$ has circumcircle $\omega$ and incenter $I.$ $\overline{AI}$ is extended to intersect $\omega$ at a point $P\ne A,$ and $\overline{BI}$ is extended to intersect $\overline{AC}$ at $Q.$ If $AB=5,$ $BC=8,$ and $IPCQ$ is a cyclic quadrilateral, then compute $AC^2.$
18. The $\textbf{Cantor set}$ is constructed as follows:
i) Start with the closed interval $[0, 1]$.
ii)Remove the open middle third of the interval, so we remove $\left(\frac{1}{3}, \frac{2}{3}\right)$ at first and leave $\left[0, \frac{1}{3}\right]$ and $\left[\frac{2}{3}, 1\right]$.
iii) Remove the open middle third from each of the remaining closed intervals, and repeat this step infinitely.
For how many integer values of $i$, where $0 \leq i \leq 10$, is $\frac{i}{10}$ an element of the Cantor set?
19. For complex numbers $a,b,c$ satisfying $|a|^2+|b|^2+|c|^2=1$, the maximum value of $|ab(a^2-b^2)+ca(c^2-a^2)+bc(b^2-c^2)|$ can be expressed in the simplest form of $\frac{p}{q}, \gcd(p,q)=1$, find $p+q$.
20. Consider cyclic quadrilateral $ABCD$ with all integer side lengths and $AB=AD=6$. Let $AC$ meet $BD$ at $F$, $AF=3,CF=9$. Denote the centers of the circumcircles of polygons $CBF, ABCD, DCF$ as $H,I,J$ respectively. Compute the area of $\triangle{HIJ}$. The answer is in the simplest form of $\frac{p\sqrt{q}}{r},\gcd(p,r)=1$ and $q$ is square-free, compute $p+q+r$.
0 replies
Bluesoul
an hour ago
0 replies
J
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Inequalities
sqing   10
N 2 hours ago by sqing
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +21abc\leq\frac{512}{441}$$Equality holds when $a=b=\frac{38}{21},c=\frac{5}{214}.$
$$a^2+b^2+ ab +19abc\leq\frac{10648}{9747}$$Equality holds when $a=b=\frac{22}{57},c=\frac{13}{57}.$
$$a^2+b^2+ ab +22abc\leq\frac{15625}{13068}$$Equality holds when $a=b=\frac{25}{66},c=\frac{8}{33}.$
10 replies
sqing
Mar 26, 2025
sqing
2 hours ago
J
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Calculating combinatorial numbers
lgx57   5
N 2 hours ago by generatingFraction
Try to simplify this expression:

$$\sum_{i=1}^n \sum_{j=1}^i C_{n}^i C_{n}^j$$
5 replies
lgx57
3 hours ago
generatingFraction
2 hours ago
J
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Proper subsets of R
lgx57   0
3 hours ago
Let $S_1,S_2 \cdots S_n$ are proper subsets of $\mathbb{R}$ and they are closed for addition and subtraction. Try to prove that:

$$\displaystyle\bigcup_{i=1}^n S_i \ne \mathbb{R}$$
0 replies
lgx57
3 hours ago
0 replies
J
No more topics!
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This a better problem than that FE: Geometry
First   6
N Sep 3, 2016 by henry24816
Let ABCD be a quadrilateral. If a circle can be inscribed in it, prove that AB +CD= BC+DA. (The Pitot theorem)
6 replies
First
Sep 2, 2016
henry24816
Sep 3, 2016
J
This a better problem than that FE: Geometry
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Reveal topic tags
geometry
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First
2352 posts
First
#1 h  Sep 2, 2016, 9:51 PM • 2 Y
Y by Adventure10, Mango247
Let ABCD be a quadrilateral. If a circle can be inscribed in it, prove that AB +CD= BC+DA. (The Pitot theorem)
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checkmatetang
3454 posts
checkmatetang
#2 h  Sep 2, 2016, 9:52 PM • 2 Y
Y by Adventure10, Mango247
You put in the easy version. The better statement is quadrilateral $ABCD$ is circumscriptable if and only if $AB+CD=AD+BC.$

First, we prove the "if". Let the incircle meet $AB, BC, CD, DA$ at $E,F,G,H$ respectively. Then, $AE=AH, BE=BF, CF=CG, DG=DH.$ Summing then combining gives $AB+CD=AD+BC.$

Now, we assume $AB+CD=AD+BC$ and prove $ABCD$ is crcumscriptable. Construct circle $O$ tangent to $AD, DC, BC.$ We generate point $B'$ on $BC$ such that $AB'$ is tangent to circle $O$ as well. Thus, by the "if" statement, $AB'+CD=AD+B'C.$ By hypothesis, $AB+CD=AD+BC.$ Thus, $AB-AB'=BC-B'C.$ Note that since $BC-B'C=BB',$ we have $AB-AB'=BB'.$ By the triangle inequality, this means $B'$ lies on $AB.$ However, $B'$ also lies on $BC,$ so $B'=B$ as we wished.
This post has been edited 1 time. Last edited by checkmatetang, Sep 2, 2016, 9:53 PM
Reason: more info!
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First
2352 posts
First
#3 h  Sep 2, 2016, 9:56 PM • 2 Y
Y by Adventure10, Mango247
What does circumscriptable mean? Do you mean circumscribable? I knew, how to prove it, it was mainly redemption for my ridiculous problem I posted earlier.
This post has been edited 1 time. Last edited by First, Sep 2, 2016, 9:58 PM
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checkmatetang
3454 posts
checkmatetang
#4 h  Sep 2, 2016, 9:58 PM • 2 Y
Y by Adventure10, Mango247
No, circumscriptable is the same as tangential, which means a circle can be inscribed in the figure.
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tree3
641 posts
tree3
#5 h  Sep 3, 2016, 1:05 AM • 2 Y
Y by Adventure10, Mango247
Consider the intersections of the quadrilateral and circle. For distinct intersections each by two tangents implies that there are four sets of tangent lines. Let them be w,x,y, and z. Now w and x, say are on one side, x and y on another, y and z on another, and z and w on the last. Now w+x+y+z, summing up two opposite sides equals itself. Q.E.D.
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jayme
9772 posts
jayme
#6 h  Sep 3, 2016, 6:54 AM • 1 Y
Y by Adventure10
Dear Mathlinkers,

see
http://jl.ayme.pagesperso-orange.fr/Docs/Urquhart.pdf p. 6...
and also

http://jl.ayme.pagesperso-orange.fr/Docs/Demir.pdf p. 7-11.

Sincerely
Jean-Louis
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henry24816
268 posts
henry24816
#7 h  Sep 3, 2016, 1:30 PM • 1 Y
Y by Adventure10
I just used a bunch of congruent tangents, added up their measures and matched up numbers.
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