bank accounts

by cloventeen, Mar 23, 2025, 2:43 AM

edgar has three bank accounts, each with an integer amount of dollars in it. He is only allowed to transfer money from one account to another if, by doing so, the latter ends up with double the money it had previously. Prove that edgar can always transfer all of his money into two accounts. Will he always be able to transfer all of his money into a single account?

number theory

permutations of sets

by cloventeen, Mar 23, 2025, 2:36 AM

Find the number of permutations of the set $A = (1, 2, \dots, n)$ with the set $B = (1, 1, 2, 3, \dots, n)$ such that each element in the permutations has at most one immediate neighbor greater than itself.

combinatorics

Prove that P1(x), P2(x) ,... Pn(x) = k has no root

by truongphatt2668, Mar 23, 2025, 2:26 AM

Let $n \in \mathbb{N}^*$ and $P_1(x),P_2(x), \ldots P_n(x) \in \mathbb{Z}[x]$ such that $\mathrm{deg} P_i = 2, \forall i = \overline{1,n}$ . Prove that exists many $k \in \mathbb{N}$ such that every equation: $P_i(x) = k, \forall i = \overline{1,n}$ has no real roots

polynomial

algebra

number theory

Number theory

by XAN4, Mar 23, 2025, 1:52 AM

Find the smallest $n$ such that there exists $x,y\in\mathbb Z^+$ satisfying $x^3-y^2=n$ .
Hint

number theory

number theory proposed

A cyclic inequality

by JK1603JK, Mar 23, 2025, 1:35 AM

Let a,b,c be real numbers. Prove that a^6+b^6+c^6\ge 2(a+b+c)(ab+bc+ca)(a-b)(b-c)(c-a).

inequalities

Cyclic ine

by m4thbl3nd3r, Mar 22, 2025, 3:17 PM

Let $a,b,c>0$ such that $a+b+c=3$ . Prove that $a^3b+b^3c+c^3a+9abc\le 12$

inequalities

Inspired by A Romanian competition question

by sqing, Mar 18, 2025, 12:44 PM

Let $a,b,c$ be reals such that $a^2+b^2 +ab+bc+ca=1.$ Prove that
$(a+ b) c- a b \leq1$ Let $a,b,c$ be reals such that $a^2+b^2+c^2+ab+bc+ca =1.$ Prove that
$29(a+ b) c - 10a b \leq 10$ Let $a,b,c$ be reals such that $a^2+b^2+c^2+bc+ca=1.$ Prove that
$149(a+ b) c- 100a b \leq50$

This post has been edited 1 time. Last edited by sqing, Mar 18, 2025, 1:57 PM

inequalities proposed

algebra

inequalities

Sharygin 2025 CR P14

by Gengar_in_Galar, Mar 10, 2025, 12:09 PM

A point $D$ lies inside a triangle $ABC$ on the bisector of angle $B$ . Let $\omega_{1}$ and $\omega_{2}$ be the circles touching $AD$ and $CD$ at $D$ and passing through $B$ ; $P$ and $Q$ be the common points of $\omega_{1}$ and $\omega_{2}$ with the circumcircle of $ABC$ distinct from $B$ . Prove that the circumcircles of the triangles $PQD$ and $ACD$ are tangent.
Proposed by: L Shatunov

This post has been edited 1 time. Last edited by Gengar_in_Galar, Mar 11, 2025, 9:54 AM

geometry

Sharygin 2025 CR P8

by Gengar_in_Galar, Mar 10, 2025, 12:04 PM

The diagonals of a cyclic quadrilateral $ABCD$ meet at point $P$ . Points $K$ and $L$ lie on $AC$ , $BD$ respectively in such a way that $CK=AP$ and $DL=BP$ . Prove that the line joining the common points of circles $ALC$ and $BKD$ passes through the mass-center of $ABCD$ .
Proposed by:V.Konyshev

This post has been edited 1 time. Last edited by Gengar_in_Galar, Mar 11, 2025, 9:52 AM

geometry

IMO 2024 Prediction

by GreenTea2593, Jul 10, 2024, 1:56 PM

Hello Aops! Since IMO 2024 is less than a week away,
What are your predictions for the category of each problem at IMO 2024?

If you want to write your prediction, please write it in the form ABC DEF
Where A,B,C,D,E,F are problems 1,2,3,4,5,6 respectively. Each letter should be A,C,G or N.

Rules:
1. Problems 1,2,4,5 are distinct categories.
2. Each day consists of 3 distinct categories.

Edit : the answer is ANC GCA

This post has been edited 4 times. Last edited by GreenTea2593, Aug 12, 2024, 2:15 PM

IMO 2024

IMO

Submit

!!!!!!!!
by stroller, Mar 10, 2020, 8:15 PM
cooooooool
nice blog
by Navansh, Jun 4, 2019, 7:47 AM
Victor is one of the GOATs.
by awesomemathlete, Oct 2, 2018, 11:48 PM
This blog is truly amazing.
by sunfishho, Feb 20, 2018, 6:32 AM
what a gem.
by vjdjmathaddict, May 10, 2017, 3:26 PM
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by ahaanomegas, Dec 15, 2013, 7:21 PM
yo dawg i heard you imo
by Mewto55555, Aug 1, 2013, 10:25 PM
Good job on 5 problems on USAMO. I know that is a pretty bad score for someone as awesome as you on a normal USAMO, but no one got all 6 this year so it's GREAT!

VICTOR WANG 42 ON IMO 2013 LET'S GO.

See you at MOP this year (probably ).
by yugrey, May 3, 2013, 10:32 PM
you can just write "Solution
" instead of "Click to reveal hidden text
"

by math154, Feb 6, 2013, 6:33 PM
Hello. May I ask a stupid question: how to turn "Hidden Text" into hidden "Solution" tag?
by ysymyth, Feb 1, 2013, 12:59 PM
This is a good blog.I like it.
by Lingqiao, Jul 17, 2012, 4:21 PM
hi.i like the blog.
by Dranzer, Feb 9, 2012, 12:25 PM
sorry, i've decided this blog is just for the random stuff i do. don't take it personally... you can always post things on your own blog
by math154, Nov 23, 2011, 10:26 PM
what i got uncontribbed for posting a nice geo problem?
by yugrey, Nov 23, 2011, 1:32 AM
Can I be a contributor?
by Binomial-theorem, Oct 5, 2011, 12:39 AM
by gauss1181, Mar 8, 2009, 5:09 PM
i hate you
by alsyy, Mar 8, 2009, 4:37 PM
Fine. I don't care.
by math154, Mar 8, 2009, 1:34 AM
Math154: It was West Somethign Middle School from Columbia that got 2nd.
by Mewto55555, Mar 8, 2009, 1:20 AM
Oh dang, I already feel like deleting this blog. Maybe I should not do that for a while, though.
by math154, Mar 6, 2009, 2:50 AM
math154 mew and tinytim tied for first at chapter...
by AIME15, Mar 6, 2009, 1:06 AM
Thanks! Good luck to you too.
by math154, Mar 5, 2009, 1:00 AM
good luck on saturday
by gauss1181, Mar 5, 2009, 12:51 AM
Yeah gauss, our chapter sends the top 50 individuals and the top 10 teams.
by math154, Mar 4, 2009, 10:49 PM
Me wants be contrib.
by AIME15, Mar 4, 2009, 5:07 PM
I'll contrib in the future when I feel like it. I'm not gonna forget you, though, so don't worry.
by math154, Mar 2, 2009, 11:55 PM
third! contrib?
by nikeballa96, Mar 2, 2009, 10:40 PM
2nd Shout!
by gauss1181, Feb 28, 2009, 5:25 PM
1st Shout!
by mathemagician1729, Feb 28, 2009, 4:02 AM