Aime type Geo

by ehuseyinyigit, May 5, 2025, 9:04 PM

In a scalene triangle $ABC$, let $M$ be the midpoint of side $BC$. Let the line perpendicular to $AC$ at point $C$ intersect $AM$ at $N$. If $(BMN)$ is tangent to $AB$ at $B$, find $AB/MA$.
geometry
L

4 wise men and 100 hats. 3 must guess their numbers

by NO_SQUARES, May 5, 2025, 5:44 PM

There are four wise men in a row, each sees only those following him in the row, i.e. the $1$st sees the other three, the $2$nd sees the $3$rd and $4$th, and the $3$rd sees only the $4$th. The devil has $100$ hats, numbered from $1$ to $100$, he puts one hat on each wise man, and hides the extra $96$ hats. After that, each wise man (in turn: first the first, then the second, etc.) loudly calls a number, trying to guess the number of his hat. The numbers mentioned should not be repeated. When all the wise men have spoken, they take off their hats and check which one of them has guessed. Can the sages to act in such a way that at least three of them knowingly guessed?
combinatorics
L

\sqrt{2-a}+\sqrt{2-b}+\sqrt{2-c}\geqslant 2+\sqrt{(2-a)(2-b)(2-c)}

by NO_SQUARES, May 5, 2025, 5:06 PM

Positive numbers $a$, $b$ and $c$ are such that $a^2+b^2+c^2+abc=4$. Prove that \[\sqrt{2-a}+\sqrt{2-b}+\sqrt{2-c}\geqslant 2+\sqrt{(2-a)(2-b)(2-c)}.\]
inequalities
algebra
L

BMO 2024 SL A4

by MuradSafarli, Apr 27, 2025, 12:43 PM

A4.
Let \(a \geq b \geq c \geq 0\) be real numbers such that \(ab + bc + ca = 3\).
Prove that:
\[ 3 + (2 - \sqrt{3}) \cdot \frac{(b-c)^2}{b+(\sqrt{3}-1)c} \leq a+b+c \]and determine all the cases when the equality occurs.
This post has been edited 1 time. Last edited by MuradSafarli, Apr 27, 2025, 12:43 PM
algebra
inequalities proposed
L

Inequality with a,b,c

by GeoMorocco, Apr 11, 2025, 10:05 PM

Let $ a,b,c $ be positive real numbers such that : $ ab+bc+ca=3 $ . Prove that : $$\frac{\sqrt{1+a^2}}{1+ab}+\frac{\sqrt{1+b^2}}{1+bc}+\frac{\sqrt{1+c^2}}{1+ca}\ge \sqrt{\frac{3(a+b+c)}{2}}$$
inequalities
L

1996 St. Petersburg City Mathematical Olympiad

by Sadece_Threv, Jul 29, 2024, 8:33 PM

Find all positive integers $n$ such that $3^{n-1}+5^{n-1}$ divides $3^{n}+5^{n}$
number theory
L

NT ineq: sum 1/a_i < (m+n)/m , {a_1,a_2,...,a_n} subset of {1,2,...,m}

by parmenides51, Apr 12, 2020, 9:53 PM

Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that
$$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$
divisible
number theory
inequalities
Subset
L

IMO 2010 Problem 5

by mavropnevma, Jul 8, 2010, 8:15 AM

Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed

Type 1) Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$;

Type 2) Choose a non-empty box $B_k$, $1\leq k \leq 4$, remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$.

Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins.

Proposed by Hans Zantema, Netherlands
IMO
IMO 2010
combinatorics
invariant
game
IMO Shortlist
L

Never 8

by chess64, May 14, 2006, 11:50 PM

Given the polynomial \[ f(x)=x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_n \] with integer coefficients $a_1,a_2,\ldots,a_n$, and given also that there exist four distinct integers $a$, $b$, $c$ and $d$ such that \[ f(a)=f(b)=f(c)=f(d)=5, \] show that there is no integer $k$ such that $f(k)=8$.
algebra
polynomial
L

old and easy imo inequality

by Valentin Vornicu, Oct 24, 2005, 10:12 AM

Let $ a, b, c$ be positive real numbers so that $ abc = 1$. Prove that
\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1. \]
inequalities
IMO
three variable inequality
imo 2000
IMO Shortlist
ineqstd
L

You can either Bash or Think..... Choose Wisely

avatar

pro_4_ever
Shouts
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  • Hey, what a nice blog!!!!! :D

    by Power_Set, Jun 24, 2018, 9:53 PM

  • Is your blog dead ? No new post in almost 2 months.

    My blog has died due to malnutrition of problems. :P

    by integrated_JRC, May 2, 2018, 5:52 AM

  • @ below, I will do it soon

    by pro_4_ever, Mar 1, 2018, 10:37 AM

  • Hi, I think we should 'hide' our solutions, as it is difficult to go up and down the blog :D

    by Drunken_Master, Mar 1, 2018, 10:36 AM

  • Yeah, I created a test blog, it seems to be an AoPS problem not CSS, can't do anything about it. I'll post it in site support forum later.

    by Vrangr, Feb 20, 2018, 5:48 AM

  • The problem is not in the CSS, I believe, as the CSS only looks after the appearance of the blog, and not the $\text{\LaTeX}$ rendering.

    by WizardMath, Feb 19, 2018, 11:34 PM

  • I'll fix the CSS after boards of need be (I'll probably be starting my own blog too then)

    by Vrangr, Feb 19, 2018, 4:41 PM

  • @ below,
    Pls ask WizardMath...
    The CSS is taken from his amazing Blog, "An Olympiad Journey"...
    I know nothing abt programming.
    (BTW I got permission for taking the CSS. No Copyright Issues!)

    by pro_4_ever, Feb 19, 2018, 4:16 PM

  • Huge flaw in website design: comment gets erased if while typing $\LaTeX$ has an error and needs to be fixed.

    by Vrangr, Feb 19, 2018, 12:26 PM

  • Upon WizardMath's Request, let us all post synthetic solutions. Well, let's declare that this blog is intended to give the reader some good problems with simple synthetic solutions. All the best. Not trying to be rude,but I might delete upcoming Bashes :)

    by pro_4_ever, Feb 13, 2018, 4:50 PM

  • Here's an idea. Ban all bashes on this blog from now on. Also try posting your own previous geo solutions as storage (I hope this is allowed). This would make the quality of the blog better. Bashes are basically write-once, read-never things. Why bother?

    by WizardMath, Feb 13, 2018, 4:33 PM

  • Somebody type Egmo 2013 Problem 1.
    No bash,purely troll problem.:)

    by QWERTYphysics, Feb 13, 2018, 4:11 PM

  • I hate having to see a pure geometry blog being swamped to death with bashes.

    Obviously a bash is a super terrible idea. Don't do one.
    Unless you are bad at synthetic.

    by WizardMath, Feb 13, 2018, 2:58 PM

  • @ayan Done:)

    by pro_4_ever, Feb 13, 2018, 1:38 PM

  • 9th shout :P

    by MEGAKNIGHT, Feb 13, 2018, 11:23 AM

  • @below Pls don't post a bash if you are thinking about posting a new entry!

    by pro_4_ever, Feb 13, 2018, 11:11 AM

  • Wait, I thought bash is frowned upon in this blog.

    by WizardMath, Feb 12, 2018, 6:57 PM

  • Hmmm...
    Combi Geometry is accepted...

    by pro_4_ever, Feb 10, 2018, 7:32 AM

  • Can we post Combi here?

    by ayan.nmath, Feb 10, 2018, 7:31 AM

  • Parag dey

    by Paragdey12, Feb 6, 2018, 1:17 PM

  • More posts please :icecream:

    by AnArtist, Feb 6, 2018, 2:38 AM

  • Second Shout!

    by ccx09, Feb 3, 2018, 9:20 PM

  • 1st shout! :D

    by AnArtist, Jan 31, 2018, 9:20 AM

23 shouts
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About Owner
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