weird permutation problem

by Sedro, Apr 20, 2025, 2:09 AM

Let $\sigma$ be a permutation of $1,2,3,4,5,6,7$ such that there are exactly $7$ ordered pairs of integers $(a,b)$ satisfying $1\le a < b \le 7$ and $\sigma(a) < \sigma(b)$ . How many possible $\sigma$ exist?

Some problems

by hashbrown2009, Apr 19, 2025, 11:01 PM

1. Real numbers a,b,c are satisfy a+1/b = b+1/c = c+1/a =x. If a,b,c are distinct, what is the value of x?
2. If x^2+y^2=1, then what is the value of : root(x^2-2x+1) + root(xy-2x+y-2) ?
3. Find the value of the sequence 2^2 + (3^2+1) + (4^2+2) + … + (97^2+95) + (98^2+96).
4. If x^2+x-1=0, then evaluate (1-x^2-x^3-x^4-…-x^2022-x^2023)/x^2022 .
5. If triangle XYZ has 3 sides that are all whole numbers, and the perimeter of XYZ is 24, what is the probability XYZ is a right triangle?

Note: If someone can latex-ify this it would help.

This post has been edited 1 time. Last edited by hashbrown2009, Yesterday at 11:02 PM

probability

Twin Primes Digital Root

by FerMath10, Apr 19, 2025, 8:51 PM

Hi,

I noticed something interesting while playing around with twin primes (pairs of primes that differ by 2). Here is what I noticed:
Conjecture: The product of twin primes—excluding the pair (3, 5)—always has a digital root of 8.

Just to clarify, the digital root of a number is the single-digit value you get by repeatedly summing its digits until only one digit remains. For example, the digital root of 77 is 7 + 7 = 14, and then 1 + 4 = 5.

I tested this on several examples, and it seems to hold, but I’m not sure if it’s a well-known result or something that breaks down for larger primes.

Is this an obvious consequence of some known number theory property? Would love to hear your thoughts!

Good Problem??

by pedronis, Apr 19, 2025, 6:05 PM

Let $m \geq 2$ be a positive integer and $p \geq 5$ a prime number. Consider the sequence $a_1, a_2, \ldots$ defined by
$a_n = (p - 2)^n + p^n - m, \quad \text{for } n \geq 1.$ Let $q$ be the smallest prime such that some term of the sequence is divisible by $q$ . Determine all pairs $(m, p)$ for which $q \geq m$ .

Calculus BC help

by needcalculusasap45, Apr 19, 2025, 1:51 PM

So basically, I have the AP Calculus BC exam in less than a month, and I have only covered until Unit 6 or 7 of the cirriculum. I am self studying this course (no teacher) and have not had much time to study bc of 6 other APs. I need to finish 8, 9, and 10 in less than 2 weeks. What can I do ? I would appreciate any help or resources anyone could provide. Could I just learn everything from barrons and princeton? Also, I have not taken AP Calculus AB before.

calculus

ez problem....

by Cobedangiu, Apr 18, 2025, 11:07 AM

Let $x,y \in Z$ and $xy \cancel \vdots7$
Find $n \in Z^+$ .
$x^2+y^2+xy=7^n$

This post has been edited 1 time. Last edited by Cobedangiu, Friday at 11:08 AM
Reason: .

Geometry

by AlexCenteno2007, Apr 17, 2025, 6:28 PM

Let A, B, C, and D be four distinct points on a straight line, in that order. The circles with diameters AC and BD intersect at X and Y. The straight line XY intersects BC at Z. Let P be a point on XY distinct from Z. The straight line CP intersects the circle with diameter AC at C and M, and the straight line BP intersects the circle with diameter BD at B and N. Show that AM, DN, and XY are aligned.

power of a point

geometry

Geometry

by AlexCenteno2007, Apr 17, 2025, 3:54 AM

Let C be a point on a semicircle of diameter AB and let D be the mid-length of arc AC. Let E be the projection of point D on BC and F the intersection of AE with the semicircle. Prove that BF bisects segment DE.

This post has been edited 1 time. Last edited by AlexCenteno2007, Apr 17, 2025, 5:42 PM
Reason: Error

geometry

power of a point

Inequalities

by sqing, Apr 16, 2025, 4:52 AM

Let $a,b$ be reals such that $a^2+ab+b^2 =3$ . Prove that
$\frac{4}{ 3}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{11}{4 }$ $\frac{13}{ 4}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{2}{3 }$ $\frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq -\frac{17}{6 }$ $\frac{19}{ 6}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2}$ Let $a,b$ be reals such that $a^2-ab+b^2 =1$ . Prove that
$\frac{3}{ 2}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }+ab \geq \frac{4}{15 }$ $\frac{14}{ 15}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }-ab \geq -\frac{1}{2 }$ $\frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq \frac{13}{42 }$ $\frac{41}{ 42}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2 }$

inequalities

Inscribed Semi-Circle!!!

by ehz2701, Sep 11, 2022, 7:29 AM

A right triangle $ABC$ with legs $AB = a$ and $BC = b$ is drawn with a semicircle inscribed into the triangle. What is the smallest possible radius of the semi-circle and the largest possible radius?

geometry

semicircle

Combinatorics #2

by utkarshgupta, Feb 18, 2017, 10:46 AM

Problem (ISL 2004 C5) :
$A$ and $B$ play a game, given an integer $N$ , $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$ , but his number cannot be bigger than $N$ . The player who writes $N$ wins. For which values of $N$ does $B$ win?

Proposed by A. Slinko & S. Marshall, New Zealand

Idea of Solution

combinatorics

ISL 2004

2004

Combinatorics #1

by utkarshgupta, Feb 17, 2017, 1:57 PM

JEE preps are adversely impacting my thinking ability.
So I will try and do something that I didn't even do when I was actually preparing for the olympiads

Actually think !
It's trivial I know I know...
But it was fun !

Problem (ISL 2015 C1) :
In Lineland there are $n\geq1$ towns, arranged along a road running from left to right. Each town has a left bulldozer (put to the left of the town and facing left) and a right bulldozer (put to the right of the town and facing right). The sizes of the $2n$ bulldozers are distinct. Every time when a left and right bulldozer confront each other, the larger bulldozer pushes the smaller one off the road. On the other hand, bulldozers are quite unprotected at their rears; so, if a bulldozer reaches the rear-end of another one, the first one pushes the second one off the road, regardless of their sizes.

Let $A$ and $B$ be two towns, with $B$ to the right of $A$ . We say that town $A$ can sweep town $B$ away if the right bulldozer of $A$ can move over to $B$ pushing off all bulldozers it meets. Similarly town $B$ can sweep town $A$ away if the left bulldozer of $B$ can move over to $A$ pushing off all bulldozers of all towns on its way.

Prove that there is exactly one town that cannot be swept away by any other one.

Solution :
Let the statement be true for $k \le n$ .

Let the towns be labelled $T_i$ from left to right and their left and right bulldozer $l_i,r_i$ respectively.

Now we have to prove the statement for $n+1$ towns..
Consider the rightmost town $T_{n+1}$ and let some $r_j$ collide with $l_{n+1}$

Then there are two cases :
$l_{n+1}$ derails all such $r_j$ . Then obviously $T_{n+1}$ is the new winner town !

If some $r_j$ derails $l_{n+1}$ . Then obviously since there is no other bulldozer between this point and $T_{n+1}$ ,
$r_j$ sweeps $T_{n+1}$
Since there are no bulldozers between $T_j$ and $T_{n+1}$ , The first $j$ towns live unaffected by the remaining towns. And hence by inducton we are done.

ISL

2015

combinatorics

Submit

Here goes first post of 2025! Great blog.
by math_holmes15, Jan 14, 2025, 8:53 AM
First post of 2024
by Yiyj1, Feb 8, 2024, 5:40 AM
First post of 2023
by HoRI_DA_GRe8, Jul 22, 2023, 7:45 AM
Nice blog ! Your isogonality lemma is really powerful !
by 554183, Oct 14, 2021, 8:55 AM
Post plss....
by samrocksnature, Apr 11, 2021, 10:12 PM
alas,this is ded
by Hamroldt, Mar 18, 2021, 4:13 PM
Thanks for the nice blog.
by Feridimo, Mar 6, 2020, 4:17 PM
I think this might be silly but ... when should we expect to have another post ?? I am very keen to see it
by gamerrk1004, Nov 4, 2019, 4:54 PM
Let's all echo what's written in the blog description - Stay Insane / 'Cause it's your labor, will and pain/ That takes you to the top of soda fountain
by Kayak, Oct 2, 2017, 7:18 PM
hey utkarsh jee is over now ... continue your elementary blog pleaseeeeeee!
by kk108, Jun 17, 2017, 11:19 AM
Congrats on becoming a contest moderator!
by Ankoganit, Mar 9, 2017, 5:22 AM
INTERSTING BLOG
by kk108, Feb 19, 2017, 2:04 PM
I have no plans for this blog right now....
No time here people !
But lets see....
I may try some combinatorics
by utkarshgupta, Feb 15, 2017, 12:47 PM
Thanks for the nice blog!
by Orkhan-Ashraf_2002, Feb 13, 2017, 6:34 PM
Revive it!!!
Best blog out there, for sure!
by rmtf1111, Jan 12, 2017, 6:02 PM
Oh you certainly will..
Bu I don't think I will
by achilles04, Feb 11, 2015, 6:24 PM
Thanks
Hope you are there too ...

And hope I get selected
by utkarshgupta, Feb 11, 2015, 11:14 AM
Nice blog ..
keep it up and you might one day become a mathematician lIke me
( just joking )
Btw don't forget us after going to imotc.
by achilles04, Feb 10, 2015, 4:49 PM
nice blog!!!!!
by pradhit, Feb 7, 2015, 11:51 AM
No ideas about the cut off but should be less than 60 according to me...
So you never know
by utkarshgupta, Feb 6, 2015, 4:06 PM
Hi utkarsh,
what do u expect the cutoff to be this year??

by kgdpsdurg, Feb 6, 2015, 12:41 PM
Hi utkarsh
How many did you clear in INMO 2015?
by moldevort, Feb 1, 2015, 3:59 PM
@ Anonymous Bunny
If we both (that should have included me) are lucky enough !
by utkarshgupta, Sep 17, 2014, 2:44 AM
Nice blog. Great job!

If I am lucky enough, hopefully we shall meet at IMOTC.
by AnonymousBunny, Sep 16, 2014, 8:08 PM
Thanx all !!

If anyone wish to contribute anything just pm !

I will add u to the contributors' list !
by utkarshgupta, Sep 16, 2014, 6:35 AM
doing good progress!! btw nice blog.
by rachitgoel, Sep 15, 2014, 4:21 PM
Nice blog.
by Kezer, Aug 5, 2014, 4:04 PM
Hi Utkarsh! Wonderful blog. Keep it up.
by YESMAths, Jul 20, 2014, 1:36 PM
I need some shouts and characters

by utkarshgupta, Jul 20, 2014, 4:22 AM