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Middle School Math Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

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Middle School Math Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

3 M G

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a My Retirement & New Leadership at AoPS

rrusczyk 1481

N 5 minutes ago by ZuzabKoit

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

1481 replies

+5 w

rrusczyk

Mar 24, 2025

ZuzabKoit

5 minutes ago

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]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

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0 replies

jlacosta

Mar 2, 2025

0 replies

Newton Sums

mithu542 5

N 9 minutes ago by xHypotenuse

Should I memorize Newton's Sums? I already know the first two, but do you think it is worthy to memorize some more (especially 3)?

Website about Newton Sums

5 replies

mithu542

Today at 1:59 AM

xHypotenuse

9 minutes ago

Construct problem

makurorin 1

N 20 minutes ago by vincentwant

Construct angle 3 degree.

1 reply

makurorin

Today at 6:20 AM

vincentwant

20 minutes ago

Anyone who knows MATHCOUNTS well?

ysn613 19

N 44 minutes ago by ysn613

I missed the signup for MATHCOUNTS this year, and I think I could have made it to nats, based on my scores on the tests(I took them after MATHCOUNTS posted them on their website). Anyone have any study tips for next year?

P.S. Does anybody know if you can sign up for MATHCOUNTS individually, or do I need to try to convince 3 other kids from my school to compete?

19 replies

ysn613

Yesterday at 11:33 PM

ysn613

44 minutes ago

Bogus Proof Marathon

pifinity 7519

N an hour ago by vincentwant

Hi!
I'd like to introduce the Bogus Proof Marathon.

In this marathon, simply post a bogus proof that is middle-school level and the next person will find the error. You don't have to post the real solution :P

Use classic Marathon format:

[hide=P#]a1b2c3[/hide]
[hide=S#]a1b2c3[/hide]

Example posts:

P(x)
-----
S(x)
P(x+1)
-----
Let's go!! Just don't make it too hard!

7519 replies

pifinity

Mar 12, 2018

vincentwant

an hour ago

Girls in Math at Yale 2025 Team P12:tangent circles

Bluesoul 3

N 4 hours ago by ehuseyinyigit

Consider triangle $ABC$ with $AB=5, BC=7, AC=8$ . Denote the incircle of $\triangle{ABC}$ as $\omega$ and let $\omega$ meet $AB,AC,BC$ at $D,E,F$ respectively. Draw circle $\gamma$ such that $\gamma$ passes through $A$ and tangents to $BC$ at $F$ . Denote the intersections of line $DE$ and $\gamma$ as $X$ and $Y$ , the length of $XY$ can be written in the simplest form of $\frac{p\sqrt{q}}{r}$ , compute $p+q+r$ .

3 replies

Bluesoul

Today at 8:49 AM

ehuseyinyigit

4 hours ago

3D Geometry Problem

ReticulatedPython 1

N 5 hours ago by ReticulatedPython

Three mutually tangent non-degenerate spheres rest on a plane. Let their centers be $C_1, C_2$ , and $C_3$ . The spheres with centers $C_1, C_2$ , and $C_3$ touch the plane at $P_1, P_2$ , and $P_3$ , respectively. Prove that $\frac{(P_1P_2)(P_2P_3)(P_1P_3)}{(C_1P_1)(C_2P_2)(C_3P_3)}=8.$
Source: Own

1 reply

ReticulatedPython

Yesterday at 8:12 PM

ReticulatedPython

5 hours ago

Inequalities

sqing 13

N Today at 12:18 PM by sqing

Let $a,b,c\geq 0$ and $a+b+c=1$ . Prove that $a^3b+b^3c+c^3a+\frac{473}{256}abc\le\frac{27}{256}$ Equality holds when $a=b=c=\frac{1}{3}$ or $a=0,b=\frac{3}{4},c=\frac{1}{4}$ or $a=\frac{1}{4} ,b=0,c=\frac{3}{4}$
or $a=\frac{3}{4} ,b=\frac{1}{4},c=0.$

13 replies

sqing

Mar 22, 2025

sqing

Today at 12:18 PM

Euclid 2022 Question 10

mockingjay11 1

N Today at 12:08 PM by grey_blue_sky7

10. At Pizza by Alex, toppings are put on circular pizzas in a random way. Every topping is placed on a randomly chosen semicircular half of the pizza and each topping’s semi-circle is chosen independently. For each topping, Alex starts by drawing a diameter whose angle with the horizontal is selected uniformly at random. This divides the pizza into two semi-circles. One of the two halves is then chosen at random to be covered by the topping.

(a) For a 2-topping pizza, determine the probability that at least $\frac{1}{4}$ of the pizza is covered by both toppings.

(b) For a 3-topping pizza, determine the probability that some region of the pizza with non-zero area is covered by all 3 toppings.

(c) Suppose that $N$ is a positive integer. For an $N$ -topping pizza, determine the probability, in terms of $N$ , that some region of the pizza with non-zero area is covered by all $N$ toppings.

I already solved (a), but I don't seem to be able to do (b). Is there any trick to tackle this problem?

1 reply

mockingjay11

Sep 3, 2022

grey_blue_sky7

Today at 12:08 PM

Inequalities

sqing 1

N Today at 11:58 AM by sqing

Let $a,b,c\geq 0$ and $a+b+c=1$ . Prove that
$a(b+c+ 5bc +1)\leq\frac{676}{675}$ $a(b+c+6bc +1)\leq\frac{245}{243}$

1 reply

sqing

Today at 11:34 AM

sqing

Today at 11:58 AM

Serving some good tea

smartvong 1

N Today at 10:43 AM by Chanome

Consider a teapot that holds $1$ liter and is initially filled with tea of $m$ % concentration. I want to serve tea to my guests, but they insist that the tea must have at least a $n$ % tea concentration to be considered good, such that $0<n<m\le100$ . I can add as much water as needed, as long as the teapot’s capacity isn’t exceeded, and I can pour out tea in arbitrarily small amounts—allowing me to continuously adjust the concentration by topping it off with water. Using an optimal strategy, what is the maximum total volume of good tea (i.e. tea with at least $n$ % concentration) that I can serve to my guests?

1 reply

smartvong

Today at 8:42 AM

Chanome

Today at 10:43 AM

8 question contest for fun :)

Chanome 0

Today at 8:49 AM

$\begin{aligned} &\text{Each question is worth 10 marks. If you just provide the answer, you get 5 marks. If you provide sufficient workings, you receive up to 5 marks.} \\[10pt] &\textbf{Q1.} \text{ Alice and Bob are playing a game where Alice starts first. There is a common positive integer } x \text{ given, and on their turn,} \\ &\text{each player subtracts an integer } n \text{ where } 1 \leq n \leq 9, \text{ such that the common number becomes } (x-n). \text{ Given a target } y, \\ &\text{the player wins when their turn ends with } (x-n) = y. \\[10pt] &\text{E.g. } x = 25, y = 1: \\ &\text{On Alice's turn, she chooses to subtract 9, so the common number is now 14.} \\ &\text{On Bob's turn, he chooses to subtract 3, so the common number is now 11.} \\ &\text{On Alice's turn, she chooses to subtract 2, so the common number is now 9.} \\ &\text{On Bob's turn, he chooses to subtract 8, so the common number is now 1. Bob wins.} \\[10pt] &(i) \text{ Assuming } (x, y) \text{ is a pair of integers such that Alice will have a strategy to guarantee a win, find that strategy.} \\ &(ii) \text{ Find all } (x, y) \text{ where Bob will have a strategy to guarantee a win.} \\ &\text{[Modified Intermediate Mathematical Olympiad Maclaurin paper Q2]} \\[20pt] &\textbf{Q2.} \text{ Given a fair } n\text{-sided die, where the sides are } 1, 2, 3, \ldots, n-1, n, \text{ find the probability of rolling } n \\ &\text{at least once in } m \text{ rolls.} \\ &\text{[Original question]} \\[20pt] &\textbf{Q3.} \text{ Determine the smallest natural number } n \text{ such that } n^n \text{ is not a divisor of } 2025!. \\ &\text{[Modified Flanders Math Olympiad 2016 Q2]} \\[20pt] &\textbf{Q4.} (n+1)^{n-1} = (n-1)^{n+1}. \text{ Find all real } n. \\ &\text{[Original Question]} \\[20pt] &\textbf{Q5.} a, b, c, d, x \text{ are integers. } 0 \leq a, b, c, d \leq 9. \text{ Find the number of possible } (a, b, c, d) \text{ such that} \\ &7^a + 7^b + 7^c + 7^d = 100x. \\ &\text{Note: } (2, 0, 2, 4) \text{ and } (2, 0, 4, 2) \text{ are 2 separate solutions.} \\ &\text{[Intermediate Mathematical Olympiad Maclaurin paper Q3]} \\[20pt] &\textbf{Q6.} \text{A sequence is defined as } a_1 = 2025, \text{ and for all } n \geq 2: \\ &a_n = \frac{a_{n-1} + 1}{n}. \\ &\text{Determine the smallest } k \text{ such that } a_k < \frac{1}{2025}. \\ &\text{[Malaysian APMO Camp Selection Test for APMO 2025 Q1]} \\[20pt] &\textbf{Q7.} \text{There are } n \geq 3 \text{ students in a classroom. Every day, the teacher splits the students into exactly 2 non-empty groups,} \\ &\text{and each pair of students from the same group will shake hands once. Suppose after } k \text{ days, each pair of students} \\ &\text{have shaken hands exactly once, and } k \text{ is as minimal as possible.} \\ &\text{Prove that } \sqrt{n} \leq k - 1 \leq 2\sqrt{n}. \\ &\text{[Malaysian APMO Camp Selection Test for APMO 2025 Q2]} \\[20pt] &\textbf{Q8.} \text{Given a fair } n\text{-sided die with sides } 1, 2, \ldots, n: \\ &\text{Roll the die. If you roll } n, \text{ you win. Else, roll again.} \\ &\text{HOWEVER, if your roll is not greater than your previous roll, you lose.} \\[10pt] &\text{E.g. } n = 4: \\ &\text{134: win, } \quad 31: \text{ lose, } \quad 122: \text{ lose, } \quad 24: \text{ win.} \\ &\text{Find the probability that you win for any given } n \text{ without using summation.} \\ &\text{[Original Question]} \end{aligned}$

0 replies

Chanome

Today at 8:49 AM

0 replies

computational in an isosceles triangle (Singapore Junior 2018)

parmenides51 2

N Today at 3:46 AM by lightsynth123

In $\vartriangle ABC, AB=AC=14 \sqrt2 , D$ is the midpoint of $CA$ and $E$ is the midpoint of $BD$ . Suppose $\vartriangle CDE$ is similar to $\vartriangle ABC$ . Find the length of $BD$ .

2 replies

parmenides51

Jul 10, 2019

lightsynth123

Today at 3:46 AM

a+b+c=3 inequality

JK1603JK 2

N Today at 3:42 AM by lbh_qys

Let a,b,c\ge 0: a+b+c=3 then prove \frac{a+bc}{b^{2}+c^{2}+2}+\frac{b+ca}{c^{2}+a^{2}+2}+\frac{c+ab}{a^{2}+b^{2}+2}\le \frac{3}{2}
When does equality hold?

2 replies

JK1603JK

Yesterday at 2:04 PM

lbh_qys

Today at 3:42 AM

digit reversing and divisibility

roundtablepizza 6

N Yesterday at 11:47 PM by roundtablepizza

an interesting problem i thought of:

for what integers k will the following statement be true: if k divides a number, then it will also divide that number reversed.

for example, since 3 divides 321, it also divides 123.

i know this applies for 3, 9, and 11(maybe??) but are there infinitely many more values of k?

6 replies

roundtablepizza

Mar 24, 2025

roundtablepizza

Yesterday at 11:47 PM

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