no. of divisors of the form

by S_14159, Apr 10, 2025, 9:25 AM

If $P=2^5 \cdot 3^6 \cdot 5^4 \cdot 7^3$ then number of positive integral divisors of $``P"$

(A) of form $(2 n+3), n \in \mathbb{N}$ , is $=138$
(B) of form $(4 n+1), n \in \mathbb{W}$ , is $=70$
(C) of form $(6 n+3), n \in \mathbb{W}$ , is $=120$
(D) of form $(4 n+3), n \in \mathbb{W}$ , is $=56$

(more than one option may be correct)

number theory

Divisors

max n with n times n square are black

by NicoN9, Apr 10, 2025, 9:19 AM

Find the maximum positive integer $n$ such that for $45\times 45$ grid, no matter how you paint $2022$ unit squares black, there exists $n\times n$ square with all unit square painted black.

grid

combinatorics

AIME

BDE tangent to EF

by NicoN9, Apr 10, 2025, 9:10 AM

Let $ABC$ be a triangle with $AB=5$ , $BC=7$ , $CA=6$ . Let $D, E$ , and $F$ be points lying on sides $BC, CA, AB$ , respectively. Given that $A, B, D, E$ , and $B, C, E, F$ are cyclic respectively, and the circumcircle of $BDE$ are tangent to line $EF$ , find the length of segment $AE$ .

This post has been edited 2 times. Last edited by NicoN9, an hour ago
Reason: Source

geometry

circumcircle

AIME

Inspired by old results

by sqing, Apr 10, 2025, 8:28 AM

Let $a ,b,c \geq 0$ and $a+b+c=1$ . Prove that
$\frac{1}{2}\leq \frac{ \left(1-a^{2}\right)^2+2\left(1-b^{2}\right) \left(1-c^{2}\right) }{\left(1+a\right)\left(1+b\right)\left(1+c\right)}\leq 1$ $1 \leq \frac{\left(1-a^{2}\right)^{2}+2\left(1-b^{2}\right) +\left(1-c^{2}\right)^{2}}{\left(1+a\right)\left(1+b\right)\left(1+c\right)}\leq \frac{3}{2}$

This post has been edited 1 time. Last edited by sqing, 2 hours ago

inequalities proposed

algebra

Prove that x1=x2=....=x2025

by Rohit-2006, Apr 9, 2025, 5:22 AM

The real numbers $x_1,x_2,\cdots,x_{2025}$ satisfy,
$x_1+x_2=2\bar{x_1}, x_2+x_3=2\bar{x_2},\cdots, x_{2025}+x_1=2\bar{x_{2025}}$ Where { $\bar{x_1},\cdots,\bar{x_{2025}}$ } is a permutation of $x_1,x_2,\cdots,x_{2025}$ . Prove that $x_1=x_2=\cdots=x_{2025}$

This post has been edited 1 time. Last edited by Rohit-2006, Yesterday at 5:23 AM

algebra

Orthocenter config once again

by Assassino9931, Apr 8, 2025, 1:53 PM

Let $ABC$ be an acute triangle with $AB < AC$ , midpoint $M$ of side $BC$ , altitude $AD$ ( $D \in BC$ ), and orthocenter $H$ . A circle passes through points $B$ and $D$ , is tangent to line $AB$ , and intersects the circumcircle of triangle $ABC$ at a second point $Q$ . The circumcircle of triangle $QDH$ intersects line $BC$ at a second point $P$ . Prove that the lines $MH$ and $AP$ are perpendicular.

geometry

interesting ineq

by nikiiiita, Jan 29, 2025, 10:27 AM

Given $a,b,c$ are positive real numbers satisfied $a^3+b^3+c^3=3$ . Prove that:
$\sqrt{2ab+5c^{2}+2a}+\sqrt{2bc+5a^{2}+2b}+\sqrt{2ac+5b^{2}+2c}\le3\sqrt{3\left(a+b+c\right)}$

This post has been edited 1 time. Last edited by nikiiiita, Jan 29, 2025, 10:29 AM

inequalities

Maximum number of m-tastic numbers

by Tsukuyomi, Jul 10, 2018, 1:11 PM

Call a rational number short if it has finitely many digits in its decimal expansion. For a positive integer $m$ , we say that a positive integer $t$ is $m-$ tastic if there exists a number $c\in \{1,2,3,\ldots ,2017\}$ such that $\dfrac{10^t-1}{c\cdot m}$ is short, and such that $\dfrac{10^k-1}{c\cdot m}$ is not short for any $1\le k<t$ . Let $S(m)$ be the set of $m-$ tastic numbers. Consider $S(m)$ for $m=1,2,\ldots{}.$ What is the maximum number of elements in $S(m)$ ?

This post has been edited 2 times. Last edited by v_Enhance, Jan 20, 2022, 1:01 AM
Reason: fix "the the"

IMO Shortlist

number theory

decimal representation

Digits

IMO

IMO 2017

Matrices :D :D :D

by utkarshgupta, Jul 4, 2016, 3:56 AM

Long time people

This sure is nice and I wouldn't have been able to solve it without first solving the $3 \times 3$ version.

Problem (Romanian Mathematical Olympiad Grade XI) :
Let $A=(a_{ij})_{1\leq i,j\leq n}$ be a real $n\times n$ matrix, such that $a_{ij} + a_{ji} = 0$ , for all $i,j$ . Prove that for all non-negative real numbers $x,y$ we have $\det(A+xI_n)\cdot \det(A+yI_n) \geq \det (A+\sqrt{xy}I_n)^2.$
Solution :
Let $f(x)=det(A+xI_{n})$
Obviously $f$ is a polynomial.
So we have to show that $f(x)f(y) \ge f(\sqrt{xy})^2$

But the above result will follow directly once we establish that all the coefficients of $f$ are positive.
$f(0)=0$ obviously.

I will call the kind of matrices asked in the question as ski matrix.
To show this I will use induction on the order of the matrix.
Let the result be true for all such $n-1 \times n-1$ ski matrices.

Now consider any $n \times n$ such ski matrix and call it $f_n(x)$ .
Then,
$f_n(x) = \int^{x}_{0} f_{n}^{'}(x)$ But obviously since $f_{n}^{'}(x)$ is a sum of determinants of ski matrices of the order $n-1 \times n-1$ , all it's coefficients by the induction hypothesis are positive and hence so are it's integrals (except the constants which we have already established as zero).

Hence $f(x)$ has all it's coefficients positive and hence by Cauchy Schwartz inequality, we are done.

linear algebra

Matrices

romanian mathematical olympiad

Angle EBA is equal to Angle DCB

by WakeUp, Nov 6, 2011, 1:25 PM

Let $ABCD$ be a convex quadrilateral such that $\angle ADB=\angle BDC$ . Suppose that a point $E$ on the side $AD$ satisfies the equality
$AE\cdot ED + BE^2=CD\cdot AE.$
Show that $\angle EBA=\angle DCB$ .

geometry proposed

geometry

if xy+xz+yz+2xyz+1 prove that...

by behdad.math.math, Sep 25, 2008, 8:17 PM

if xy+xz+yz+2xyz+1 prove that x+y+z>=3/2

inequalities

inequalities unsolved

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