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SOLVE: CDR style problem quick algebra
ryfighter   6
N an hour ago by EthanNg6
It takes 3 people 10 minutes to mow 2 lawns. How many minutes will it take for 2 people to mow 10 lawns? Express your answer in hours as a decimal.

$(A)$ $1.25$
$(B)$ $75$
$(C)$ $01.025$
$(D)$ $1.5$
$(E)$ $15.25$
6 replies
ryfighter
Today at 3:19 AM
EthanNg6
an hour ago
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Fun challange problem :)
TigerSenju   32
N an hour ago by maxamc
Scenario:

Master Alchemist Aurelius is renowned for his mastery of elemental fusion. He works with seven fundamental, yet mysterious, elements: Ignis (Fire), Aqua (Water), Terra (Earth), Aer (Air), Lux (Light), Umbra (Shadow), and Aether (Spirit). Each element possesses a unique 'potency' value, a positive integer crucial for his most complex fusions

Aurelius has lost his master log of these potencies. All he has left are seven cryptic scrolls, each containing a precise relationship between the potencies of various elements. He needs these values to complete his Grand Device. Can you help him deduce the exact potency of each element?

The Elements and Their Potencies:

Let I represent the potency of Ignis (Fire).
Let A represent the potency of Aqua (Water).
Let T represent the potency of Terra (Earth).
Let R represent the potency of Aer (Air).
Let L represent the potency of Lux (Light).
Let U represent the potency of Umbra (Shadow).
Let E represent the potency of Aether (Spirit).
The Cryptic Scrolls (System of Equations):

Aurelius's scrolls reveal the following relationships:

The combined potency of Ignis, Aqua, and Terra is equal to the potency of Aer plus Lux, plus a constant of two.

If you sum the potencies of Aqua and Umbra, it precisely equals the sum of Lux and Aether, minus one.

The sum of Terra and Aer potencies is the same as the sum of Ignis, Aqua, and Aether potencies, minus one.

Three times the potency of Ignis, plus the potency of Aer, is equal to the sum of Aqua, Terra, and Aether potencies, plus five.

The difference between Lux and Ignis potencies is identical to the difference between Umbra and Aqua potencies.

The sum of Umbra and Aether potencies, when decreased by the potency of Terra, results in twice the potency of Aqua.

The potency of Ignis added to Lux, minus the potency of Aer, is equivalent to the potency of Aether minus Umbra, plus one.

The Grand Challenge:

Using only the information from the cryptic scrolls, set up and solve the system of seven linear equations to determine the unique positive integer potency value for each of the seven elements: I,A,T,R,L,U,E.

good luck, and whoever finds the potencies first, gets a title of The SYSTEMS OF EQUATIONS MASTER

p.s. Yes, I did just come up with a whole story of words to make a ridiculously long problem, but hey, you're reading this, so you probably have nothing better to be doing. ;)
32 replies
TigerSenju
May 18, 2025
maxamc
an hour ago
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Warning!
VivaanKam   40
N 2 hours ago by ayeshaaq
This problem will try to trick you! :!:

40 replies
VivaanKam
May 5, 2025
ayeshaaq
2 hours ago
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MathDash help
Spacepandamath13   8
N 5 hours ago by Yiyj
AkshajK ORZ by the way invited me to do MathDash a few months ago and I did try it one day but haven't done it much after (Sorry). Now, I'm getting back into it and finding the format kind of weird. When selecting certain problem type sometimes it lets me pick immediately, other times not. Any fixes?
8 replies
Spacepandamath13
May 29, 2025
Yiyj
5 hours ago
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MIT PRIMES STEP
pingpongmerrily   5
N Today at 4:56 PM by pingpongmerrily
Anyone else applying? How cooked am I for the placement test... (106.5 AMC 10, 5 AIME, 36/27 States/Nationals)
5 replies
pingpongmerrily
Yesterday at 9:01 PM
pingpongmerrily
Today at 4:56 PM
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Combi counting
Caasi_Gnow   4
N Today at 3:49 PM by Rabbit47
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)
4 replies
Caasi_Gnow
Mar 20, 2025
Rabbit47
Today at 3:49 PM
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Math with Connect4 Boards
Math-lover1   12
N Today at 2:47 PM by Math-lover1
Hi! So I was playing Connect4 with my friends the other day and I wondered: how many "legal" arrangements of Connect4 can be reached at the ending position?

We assume that we do not stop the game when there is a four in a row, and we have 21 red pieces and 21 yellow pieces. We also drop the pieces one by one into a standard 7 by 6 board. We can start the game with any color piece.

https://en.wikipedia.org/wiki/Connect_Four

Initial Thoughts
Attempt to use one-to-one correspondences
12 replies
Math-lover1
May 1, 2025
Math-lover1
Today at 2:47 PM
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Challenge: Make every number to 100 using 4 fours
CJB19   272
N Today at 2:42 PM by bbojy
I've seen this attempted a lot but I want to see if the AoPS community can actually do it. Using ONLY 4 fours and math operations, make as many numbers as you can. Try to go in order. I'll start:
$$(4-4)*4*4=0$$$$4-4+4/4=1$$$$4/4+4/4=2$$$$(4+4+4)/4=3$$$$4+(4-4)*4=4$$$$4+4^{4-4}=5$$$$4!/4+4-4=6$$$$4+4-4/4=7$$$$4+4+4-4=8$$
272 replies
CJB19
May 15, 2025
bbojy
Today at 2:42 PM
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The daily problem!
Leeoz   216
N Today at 1:42 PM by kjhgyuio
Every day, I will try to post a new problem for you all to solve! If you want to post a daily problem, you can! :)

Please hide solutions and answers, hints are fine though! :)

Problems usually get harder throughout the week, so Sunday is the easiest and Saturday is the hardest!

Past Problems!
216 replies
Leeoz
Mar 21, 2025
kjhgyuio
Today at 1:42 PM
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Geometry question !
kjhgyuio   1
N Today at 11:13 AM by whwlqkd
........
1 reply
kjhgyuio
Today at 10:13 AM
whwlqkd
Today at 11:13 AM
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Polynomial
Z_.   1
N Apr 23, 2025 by rchokler
Let \( m \) be an integer greater than zero. Then, the value of the sum of the reciprocals of the cubes of the roots of the equation
\[ mx^4 + 8x^3 - 139x^2 - 18x + 9 = 0 \]is equal to:
1 reply
Z_.
Apr 23, 2025
rchokler
Apr 23, 2025
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Z_.
28 posts
Z_.
#1 h  Apr 23, 2025, 9:21 PM
Y by
Let \( m \) be an integer greater than zero. Then, the value of the sum of the reciprocals of the cubes of the roots of the equation
\[ mx^4 + 8x^3 - 139x^2 - 18x + 9 = 0 \]is equal to:
This post has been edited 1 time. Last edited by Z_., Apr 23, 2025, 9:21 PM
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rchokler
2975 posts
rchokler
#2 h  Apr 23, 2025, 10:22 PM
Y by
Let $a,b,c,d$ be the reciprocals of the roots. Then they solve $9x^4-18x^3-139x^2+8x+m=0$.

Then by Newton's identities, where $p_n=a^n+b^n+c^n+d^n$ and $e_n$ are elementary symmetric polynomials give:

$p_3=e_1p_2-e_2p_1+3e_3=e_1(e_1p_1-2e_2)-e_2p_1+3e_3=e_1(e_1^2-2e_2)-e_1e_2+3e_3=e_1^3-3e_1e_2+3e_3=2^3+3\cdot 2\cdot\frac{139}{9}-3\cdot\frac{8}{9}=98$
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