God do bosses have a hard job

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

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.
[*]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

Polynomials and their shift with all real roots and in common

Assassino9931 1

N 3 minutes ago by joeym2011

Source: Bulgaria Spring Mathematical Competition 2025 11.4

We call two non-constant polynomials friendly if each of them has only real roots, and every root of one polynomial is also a root of the other. For two friendly polynomials $P(x), Q(x)$ and a constant $C \in \mathbb{R}, C \neq 0$ , it is given that $P(x) + C$ and $Q(x) + C$ are also friendly polynomials. Prove that $P(x) \equiv Q(x)$ .

1 reply

Assassino9931

an hour ago

joeym2011

3 minutes ago

When is this well known sequence periodic?

Assassino9931 1

N 5 minutes ago by RagvaloD

Source: Bulgaria Spring Mathematical Competition 2025 12.2

Determine all values of $a_0$ for which the sequence of real numbers with $a_{n+1}=3a_n - 4a_n^3$ for all $n\geq 0$ is periodic from the beginning.

1 reply

Assassino9931

an hour ago

RagvaloD

5 minutes ago

$$ac=bd$$

sqing 0

7 minutes ago

Source: Own

Let $a,b,c,d$ be reals such that $a^2+b^2=4,c^2+d^2=9$ and $abcd\ge 9.$ Prove that $ac=bd$ Let $a,b,c,d$ be reals such that $a^2+b^2=4,c^2+d^2=9$ and $ad+bc \ge 6.$ Prove that $ac=bd$ Let $a,b,c,d$ be reals such that $a^2+b^2=4,c^2+d^2=9$ and $ab+cd \geq \frac{13}{2}.$ Prove that $ac=bd$

0 replies

1 viewing

sqing

7 minutes ago

0 replies

nice problem

hanzo.ei 1

N 11 minutes ago by hanzo.ei

Source: I forgot

Let triangle $ABC$ be inscribed in the circumcircle $(O)$ and circumscribed about the incircle $(I)$ , with $AB < AC$ . The incircle $(I)$ touches the sides $BC$ , $CA$ , and $AB$ at $D$ , $E$ , and $F$ , respectively. A line through $I$ , perpendicular to $AI$ , intersects $BC$ , $CA$ , and $AB$ at $X$ , $Y$ , and $Z$ , respectively. The line $AI$ meets $(O)$ at $M$ (distinct from $A$ ). The circumcircle of triangle $AYZ$ intersects $(O)$ at $N$ (distinct from $A$ ). Let $P$ be the midpoint of the arc $BAC$ of $(O)$ . The line $AI$ cuts segments $DF$ and $DE$ at $K$ and $L$ , respectively, and the tangents to the circle $(DKL)$ at $K$ and $L$ intersect at $T$ . Prove that $AT \perp BC$ .

1 reply

hanzo.ei

Yesterday at 5:58 PM

hanzo.ei

11 minutes ago

No more topics!

God do bosses have a hard job

AshAuktober 2

N Mar 26, 2025 by Mathdreams

Source: 05JPN5 (couldn't find on search function)

The boss has to assign ten job positions to ten candidates, considering two parameters: preference and ability. If candidate A prefers job $v$ to job $u$ and has a better ability in job $v$ than candidate B, but A is assigned job $u$ and B is assigned job $v$ , then A will complain. Also, if it is possible to assign each job to a candidate with a higher ability, the director will complain. Show that the boss can assign the jobs so as to avoid any complaints.

2 replies

AshAuktober

Mar 25, 2025

Mathdreams

Mar 26, 2025

God do bosses have a hard job

G H J

Reveal topic tags

combinatorics

G H BBookmark kLocked kLocked NReply

Source: 05JPN5 (couldn't find on search function)

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AshAuktober

944 posts

AshAuktober

#1 h Mar 25, 2025, 4:36 PM

Y by

Z K Y

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tyrantfire4

1096 posts

tyrantfire4

#2 h Mar 25, 2025, 4:55 PM

Y by

Ummm err unable to under stand ,comprehend, read,do and help with problem but to answer if bosses have hard jobs yes

Z K Y

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Mathdreams

1443 posts

Mathdreams

#4 h Mar 26, 2025, 1:24 PM

Y by

Solution

For each job, rank the performance of each candidate.

Then, we repeat the following until each job has a candidate assigned to it:

For each job, assign the candidate out of the remaining ones with the highest ranking, and tiebreak between possible jobs for each candidate by assigning the one they have the highest preference for.

This process clearly terminates, since at least one person is assigned a job after tiebreaks each time this process is performed.

Now, we show that the resulting assignment of candidates to jobs satisfies the desired properties.

\textbf{Lemma 1:} No candidate will complain under this system.

Consider any job and the candidate that ends up being assigned to it.

Any candidates with a lower ability in the job cannot complain.

Any candidate with a higher ability that does not end up being assigned the job must have preferred another job, meaning they will not complain.

Hence, no candidate will complain about any of the jobs.

\textbf{Lemma 2:} The director will not complain under this system.

This is clear, as some job has the person with the highest ranking assigned to it in the first occurrence of this process.

From Lemma $1$ and Lemma $2$ it follows that the resulting assignment of candidates to jobs satisfies the desired properties.

Z K Y

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