Nice problem

by hanzo.ei, Mar 23, 2025, 2:58 PM

Given two positive integers $m, n$ satisfying $m > n$ and their sum is an even number, consider the quadratic polynomial:

$P(x) = x^2 - (m^2 - m + 1)x + (m^2 - n^2 - m)(n^2 + 1).$
Prove that all roots of $P(x)$ are positive integers but are not perfect squares.

Polynomial number theory

number theory

Number of sign change in cos ka

by Rohit-2006, Mar 23, 2025, 2:48 PM

Let $0\leq\alpha\leq\pi$ . Denote by $V_n(\alpha)$ the number of changes of signs in the
sequence
$1, cos \alpha, cos 2\alpha, . . . , cos n\alpha.$ Then prove that
$\lim_{n\rightarrow\infty}\frac{V_n(\alpha)}{n}=\frac{\alpha}{\pi}$ .

This post has been edited 3 times. Last edited by Rohit-2006, an hour ago

algebra

Very hard math problem

by slimshady360, Mar 23, 2025, 2:40 PM

In a chess tournament with n ≥ 5 players, each player played all other players. One gets a point for a
win, half a point for a draw, and zero points for a loss. At the end of the tournament, each player had
a different number of points. Prove that the second and third ranked players had together more points
than the winner of the tournament.

combinatorics

Obsolete NT

by GreekIdiot, Mar 23, 2025, 1:29 PM

Find all $n \in \mathbb{N}$ greater than $1$ , such that, if $gcd(a,b)=1$ , then $a \equiv b \: mod \: n \iff ab \equiv 1 \: mod \: n$

number theory

1/sqrt(5) ???

by navi_09220114, Mar 22, 2025, 1:10 PM

Two circles $\omega_1$ and $\omega_2$ are externally tangent at a point $A$ . Let $\ell$ be a line tangent to $\omega_1$ at $B\neq A$ and $\omega_2$ at $C\neq A$ . Let $BX$ and $CY$ be diameters in $\omega_1$ and $\omega_2$ respectively. Suppose points $P$ and $Q$ lies on $\omega_2$ such that $XP$ and $XQ$ are tangent to $\omega_2$ , and points $R$ and $S$ lies on $\omega_1$ such that $YR$ and $YS$ are tangent to $\omega_1$ .

a) Prove that the points $P$ , $Q$ , $R$ , $S$ lie on a circle $\Gamma$ .

b) Prove that the four segments $XP$ , $XQ$ , $YR$ , $YS$ determine a quadrilateral with an incircle $\gamma$ , and its radius is $\displaystyle\frac{1}{\sqrt{5}}$ times the radius of $\Gamma$ .

Proposed by Ivan Chan Kai Chin

geometry

Checkerboard

by Ecrin_eren, Mar 21, 2025, 5:20 AM

On an 8×8 checkerboard, what is the minimum number of squares that must be marked (including the marked ones) so that every square has exactly one marked neighbor? (We define neighbors as squares that share a common edge, and a square is not considered a neighbor of itself.)

This post has been edited 1 time. Last edited by Ecrin_eren, Mar 21, 2025, 5:21 AM
Reason: Checkerboard

combinatorics unsolved

combinatorics

a^2+b^2+c^2=2(ab+bc+ca)

by sqing, Jul 17, 2022, 6:35 AM

Let $a,b,c>0$ and $a^2+b^2+c^2=2(ab+bc+ca).$ Prove that
$a^3+b^3+c^3\geq \frac{33}{2}abc$ $a^2(b+c)+c^2(a+b)\geq \frac{5}{2}abc$ $a(b^2+c^2)+b(c^2+a^2)+c(a^2+b^2)\geq \frac{21}{2}abc$ $\left( \frac{a}{b} + \frac{b}{c} \right) \left( \frac{b}{a} + \frac{a}{c} \right) \geq\frac{25}{16}$ $\left( \frac{a}{b} + \frac{b}{c} \right) \left( \frac{b}{a} + \frac{a}{c} \right)\left( \frac{c}{a} + \frac{c}{b} \right)\geq \frac{25}{2}$

algebra

inequalities proposed

Inequality

2, 4, 5-Nim

by cjquines0, Jan 21, 2017, 3:47 PM

Two players, $A$ (first player) and $B$ , take alternate turns in playing a game using 2016 chips as follows: the player whose turn it is, must remove $s$ chips from the remaining pile of chips, where $s \in \{ 2,4,5 \}$ . No one can skip a turn. The player who at some point is unable to make a move (cannot remove chips from the pile) loses the game. Who among the two players can force a win on this game?

combinatorics

game

nim

Collinearity with orthocenter

by liberator, Jan 4, 2016, 9:38 PM

Let $ABC$ be an acute triangle with orthocenter $H$ , and let $W$ be a point on the side $BC$ , lying strictly between $B$ and $C$ . The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$ , respectively. Denote by $\omega_1$ is the circumcircle of $BWN$ , and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$ . Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$ , and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$ . Prove that $X,Y$ and $H$ are collinear.

Proposed by Warut Suksompong and Potcharapol Suteparuk, Thailand

geometry

IMO

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

Submit

The blog is locked right?
by First, Apr 14, 2018, 6:00 PM
Great, amazing, inspiring blog. Good luck in life, and just know I aspire to succeed as you will in the future.
by mgrimalo, Apr 7, 2018, 6:19 PM
Yesyesyes
by shiningsunnyday, Mar 29, 2018, 5:30 PM
:O a new background picture
by MathAwesome123, Mar 29, 2018, 3:39 PM
did you get into MIT?
by 15Pandabears, Mar 15, 2018, 10:42 PM
wait what new site?
by yegkatie, Feb 11, 2018, 1:49 AM
Yea, doing a bit of cleaning before migrating to new site
by shiningsunnyday, Jan 21, 2018, 2:43 PM
Were there posts made in December 2017 for this blog and then deleted?

I ask because I was purging my thunderbird inbox and I found emails indicating new blog posts of yours.

email do not lie
by jonlin1000, Jan 21, 2018, 12:12 AM
@below sorry not accepting contribs
by shiningsunnyday, Dec 11, 2017, 11:15 AM
contrib plez?
also wow this blog is very popular
by DavidUsa, Dec 10, 2017, 7:53 PM
@First: lol same

first shout of december
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XD this blog is hilarious
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@wu2481632: stop encouraging SSD to procrastinate(blog entries are fun but procrastination isn't).
by First, Aug 7, 2017, 5:02 PM
3.5 weeks without a post
by Flash12, Aug 4, 2017, 8:10 AM
First august shout!!
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"Also, I figured that whenever"

Darn, whenever what?

Belated (oops sorry) well-wishes on the 2016 AMC10A!
by DeathLlama9, Feb 3, 2016, 3:23 AM
what is a 8char?
lol 8char
by whiteawesomesun, Jan 30, 2016, 9:43 AM
lol 8char
by chocopuff, Jan 30, 2016, 3:38 AM
yadayadayadayadayadayada amc 10 is near... hope you get into JMO...
by whiteawesomesun, Jan 30, 2016, 2:45 AM
so much hype and stress!
by chocopuff, Jan 29, 2016, 2:14 PM
Lol lets reconcile man
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DUDUUDDUDUUDE STAPHHHHH. war? war? TF boys
by whiteawesomesun, Jan 19, 2016, 8:50 AM
dude dude stop stop war war
by Konigsberg, Jan 18, 2016, 3:44 AM
I'm excited!
by paperplates, Jan 15, 2016, 6:55 AM
nice! \8char
by adihaya, Jan 14, 2016, 6:47 AM
yay, looks like this is going to be a cool blog!
by jh235, Jan 7, 2016, 8:15 PM
Post something!
by whiteawesomesun, Jan 4, 2016, 3:54 PM
at least I claim second shot
by Kline, Dec 29, 2015, 1:45 AM
first shout >
by oceanair, Dec 28, 2015, 6:00 AM

416 shouts

SSD's Musings and Dreams

by hanzo.ei, Mar 23, 2025, 2:58 PM

by Rohit-2006, Mar 23, 2025, 2:48 PM

by slimshady360, Mar 23, 2025, 2:40 PM

by GreekIdiot, Mar 23, 2025, 1:29 PM

by navi_09220114, Mar 22, 2025, 1:10 PM

by Ecrin_eren, Mar 21, 2025, 5:20 AM

by sqing, Jul 17, 2022, 6:35 AM

by cjquines0, Jan 21, 2017, 3:47 PM

by liberator, Jan 4, 2016, 9:38 PM

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