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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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0 replies
jlacosta
May 1, 2025
0 replies
J
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INTERSTING
teomihai   23
N 21 minutes ago by ADus
IF $1=3$
$2=3$

$3=4$
$4=5 $
FIND $6=?$
23 replies
teomihai
Today at 7:34 AM
ADus
21 minutes ago
J
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cant understand so dumb
greenplanet2050   13
N an hour ago by CJB19
am i stupid or smth

2001 AMC 10

Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible?

umm why isnt it 3^4
13 replies
greenplanet2050
May 18, 2025
CJB19
an hour ago
J
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Vol 1 enough?
Spacepandamath13   12
N an hour ago by Spacepandamath13
is aops vol 1 book enough for amc10 or is vol 2 required to be studied too?
12 replies
Spacepandamath13
Today at 1:07 AM
Spacepandamath13
an hour ago
J
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Fun challange problem :)
TigerSenju   25
N 2 hours ago by melloncandy
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. ;)
25 replies
TigerSenju
May 18, 2025
melloncandy
2 hours ago
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Inequalities
sqing   3
N 2 hours ago by DAVROS
Let $ a,b,c\geq 0 ,a+b+c\leq 3. $ Prove that
$$a^2+b^2+c^2+ab +2ca+2bc + abc \leq \frac{251}{27}$$$$ a^2+b^2+c^2+ab+2ca+2bc + \frac{2}{5}abc \leq \frac{4861}{540}$$$$ a^2+b^2+c^2+ab+2ca+2bc + \frac{7}{20}abc \leq \frac{2381411}{26460}$$
3 replies
sqing
Today at 12:47 PM
DAVROS
2 hours ago
J
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n is divisible by 5
spiralman   0
3 hours ago
n is an integer. There are n integers such that they are larger or equal to 1, and less or equal to 6. Sum of them is larger or equal to 4n, while sum of their square is less or equal to 22n. Prove n is divisible by 5.
0 replies
spiralman
3 hours ago
0 replies
J
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It is given that $M=1+\frac12+\frac13+\frac14+\cdots+\frac{1}{23}=\frac{n}{23!},
Vulch   2
N 3 hours ago by P162008
It is given that $M=1+\frac12+\frac13+\frac14+\cdots+\frac{1}{23}=\frac{n}{23!},$ where $n$ is a natural number.What is the remainder when $n$ is divided by $13?$
2 replies
Vulch
Apr 9, 2025
P162008
3 hours ago
J
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Symmetric System's Cyclic Sum
worthawholebean   6
N Today at 4:09 PM by kilobyte144
If $ a+1=b+2=c+3=d+4=a+b+c+d+5$, then $ a+b+c+d$ is

$ \text{(A)}\ -5 \qquad \text{(B)}\ -10/3 \qquad \text{(C)}\ -7/3 \qquad \text{(D)}\ 5/3 \qquad \text{(E)}\ 5$
6 replies
worthawholebean
Feb 11, 2008
kilobyte144
Today at 4:09 PM
J
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Inequalities
sqing   5
N Today at 2:45 PM by sqing
Let $ a,b,c>0. $ Prove that$$a^2+b^2+c^2+abc-k(a+b+c)\geq 3k+2-2(k+1)\sqrt{k+1}$$Where $7\geq k \in N^+.$
$$a^2+b^2+c^2+abc-3(a+b+c)\geq-5$$
5 replies
sqing
Yesterday at 2:23 PM
sqing
Today at 2:45 PM
J
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Find the value of k/m
zolfmark   4
N Today at 2:31 PM by Shan3t
If we have
K=1/1+1/3+1/4+......1/2019

m=2018+2017/2+2016/3+......2/2017+1/2018
4 replies
zolfmark
Mar 1, 2024
Shan3t
Today at 2:31 PM
J
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geometry
luckvoltia.112   2
N Today at 1:35 PM by luckvoltia.112
ChGiven an acute triangle ABC inscribed in circle $(O)$ The altitudes $BE, CF$ , intersect
each other at $H$. The tangents at $B$ and $C $of $(O)$ intersect at $S$. Let $M $be the midpoint of $BC$. $EM$ intersects $SC$
at $I$, $FM$ intersects $SB$ at $J.$
a) Prove that the points $I, S, M, J$ lie on the same circle.
b) The circle with diameter $AH$ intersects the circle $(O)$ at the second point $T.$ The line $AH$ intersects
$(O)$ at the second point $K$. Prove that $S,K,T$ are collinear.
2 replies
luckvoltia.112
May 18, 2025
luckvoltia.112
Today at 1:35 PM
J
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positive integer solutions
zolfmark   1
N Today at 1:07 PM by Mathzeus1024
positive integer solutions
x^2+y^2+xy=283
1 reply
zolfmark
Jan 5, 2018
Mathzeus1024
Today at 1:07 PM
J
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Minimize z.
Entrepreneur   1
N Today at 12:18 PM by Lankou
Minimize $z = 6x + 3y.$ Subject to the constraints:
$$\begin{cases} 4x+y\ge80,\\ x+5y \ge115,\\ 3x+2y\le150,\\ x,y\ge0. \end{cases}$$
1 reply
Entrepreneur
Today at 11:30 AM
Lankou
Today at 12:18 PM
J
H
by vectors
zolfmark   1
N Today at 11:54 AM by Mathzeus1024
by vectors shoe that
in ABC .
: cosA + cosB + cosC ≤ 3/2
1 reply
zolfmark
Oct 26, 2023
Mathzeus1024
Today at 11:54 AM
J
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J
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Annoying Probability Math Problem
RYang2   13
N Apr 18, 2025 by FabulousSpider24
I was working in my math textbook(not the AoPS one) when I came across this math problem:

Determine if the events are dependent or independent.
1. Drawing a red and a blue marble at the same time from a bag containing 6 red and 4 blue marbles
2.(omitted)

I thought it was independent, since the events happen at the same time, but the textbook answer said dependent.
Can someone help me understand(or prove the textbook wrong)?
13 replies
RYang2
Mar 14, 2018
FabulousSpider24
Apr 18, 2025
J
Annoying Probability Math Problem
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Reveal topic tags
Math Problem
probability
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G H BBookmark kLocked kLocked NReply
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RYang2
1936 posts
RYang2
#1 h  Mar 14, 2018, 5:07 PM • 2 Y
Y by Adventure10, Mango247
I was working in my math textbook(not the AoPS one) when I came across this math problem:

Determine if the events are dependent or independent.
1. Drawing a red and a blue marble at the same time from a bag containing 6 red and 4 blue marbles
2.(omitted)

I thought it was independent, since the events happen at the same time, but the textbook answer said dependent.
Can someone help me understand(or prove the textbook wrong)?
This post has been edited 1 time. Last edited by RYang2, Mar 14, 2018, 5:10 PM
Reason: Too general subject
Z K Y
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math_is_life_2004
1374 posts
math_is_life_2004
#2 h  Mar 14, 2018, 5:10 PM • 1 Y
Y by Adventure10
It’s dependent
Z K Y
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RYang2
1936 posts
RYang2
#3 h  Mar 14, 2018, 5:11 PM • 1 Y
Y by Adventure10
Explain?
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math_is_life_2004
1374 posts
math_is_life_2004
#4 h  Mar 14, 2018, 5:14 PM • 2 Y
Y by Adventure10, Mango247
Actually I think its independent what grade textbook you have?
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brainpopper
1052 posts
brainpopper
#5 h  Mar 14, 2018, 5:15 PM • 2 Y
Y by Adventure10, Mango247
Independent is only when they are replace or when the probability doesn't change, right? In #1, If we draw a blue marble first, the probability is 6/10 but the next red marble is 4/9
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RYang2
1936 posts
RYang2
#6 h  Mar 14, 2018, 5:19 PM • 2 Y
Y by Adventure10, Mango247
Pre-Algebra Textbook
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RYang2
1936 posts
RYang2
#7 h  Mar 14, 2018, 5:19 PM • 2 Y
Y by Adventure10, Mango247
And, the problem said at the same time, so there would be no "first" or "second"
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math_is_life_2004
1374 posts
math_is_life_2004
#8 h  Mar 14, 2018, 5:53 PM • 2 Y
Y by Adventure10, Mango247
RYang2 wrote:
Pre-Algebra Textbook

Oh I haven’t done that for 2 years. After you memorized the law of cosine you kinda forget everything. #2ndquadraticequation
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creeperhissboom
224 posts
creeperhissboom
#9 h  Mar 14, 2018, 9:17 PM • 2 Y
Y by Adventure10, Mango247
It is dependant.

It is the same as taking one out at a time.
It is only independant if there is replacement.
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A-Username
70 posts
A-Username
#10 h  Mar 14, 2018, 9:26 PM • 2 Y
Y by Adventure10, Mango247
RYang2 wrote:
I was working in my math textbook(not the AoPS one) when I came across this math problem:

Determine if the events are dependent or independent.
1. Drawing a red and a blue marble at the same time from a bag containing 6 red and 4 blue marbles
2.(omitted)

I thought it was independent, since the events happen at the same time, but the textbook answer said dependent.
Can someone help me understand(or prove the textbook wrong)?

@RYang2 Easy explanation for #1 is
Independent-Event 1 doesn't influence Event 2.
Dependent-Event 1 does influence Event 2.
Since this bag contains 6 red and 4 blues, there are 10 marbles in the bag. Now, we know that the color doesn't really matter because we know there is more than 1 for each color. Then we can determine which definition it fits better. First, we know it can't be dependent because if we take it at the same time, there is no sense of order and that the taking of the blue doesn't affect the number of reds there are and vice versa.
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EthanNg6
35 posts
EthanNg6
#11 h  Apr 16, 2025, 1:07 AM
Y by
It's probably dependent
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Jaxman8
122 posts
Jaxman8
#12 h  Apr 16, 2025, 1:31 AM
Y by
I think dependent, if you were to calculate the probabilities for blue first vs red first it’s the same
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giratina3
555 posts
giratina3
#13 h  Apr 16, 2025, 2:55 AM • 1 Y
Y by EthanNg6
The answer is dependent. When you pick 2 marbles, the first pick affects the second one. Although you pick the two at the same time, the change in marble just occurs instantly, so it is still dependent (most average school textbook problem)
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FabulousSpider24
200 posts
FabulousSpider24
#14 h  Apr 18, 2025, 2:00 AM • 1 Y
Y by EthanNg6
You can think about it like this: It is dependent because as soon as the first marble is taken, the number of marbles in the bag decreases and so the probability of picking a marble (either red or blue) will change immediately afterwards. .
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