nice geo

by Melid, Apr 23, 2025, 3:01 PM

Let ABCD be a cyclic quadrilateral, which is AB=7 and BC=6. Let E be a point on segment CD so that BE=9. Line BE and AD intersect at F. Suppose that A, D, and F lie in order. If AF=11 and DF:DE=7:6, find the length of segment CD.

geometry

A game optimization on a graph

by Assassino9931, Apr 8, 2025, 1:59 PM

Let $X_0, X_1, \dots, X_{n-1}$ be $n \geq 2$ given points in the plane, and let $r > 0$ be a real number. Alice and Bob play the following game. Firstly, Alice constructs a connected graph with vertices at the points $X_0, X_1, \dots, X_{n-1}$ , i.e., she connects some of the points with edges so that from any point you can reach any other point by moving along the edges.Then, Alice assigns to each vertex $X_i$ a non-negative real number $r_i$ , for $i = 0, 1, \dots, n-1$ , such that $\sum_{i=0}^{n-1} r_i = 1$ . Bob then selects a sequence of distinct vertices $X_{i_0} = X_0, X_{i_1}, \dots, X_{i_k}$ such that $X_{i_j}$ and $X_{i_{j+1}}$ are connected by an edge for every $j = 0, 1, \dots, k-1$ . (Note that the length $k \geq 0$ is not fixed and the first selected vertex always has to be $X_0$ .) Bob wins if
$\frac{1}{k+1} \sum_{j=0}^{k} r_{i_j} \geq r;$ otherwise, Alice wins. Depending on $n$ , determine the largest possible value of $r$ for which Bobby has a winning strategy.

product of all integers of form i^3+1 is a perfect square

by AlastorMoody, Apr 6, 2020, 12:09 PM

Determine all integers $1 \le m, 1 \le n \le 2009$ , for which
$\begin{align*} \prod_{i=1}^n \left( i^3 +1 \right) = m^2 \end{align*}$

number theory

Concurrency

by Dadgarnia, Mar 12, 2020, 10:54 AM

Let $ABC$ be an isosceles triangle ( $AB=AC$ ) with incenter $I$ . Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$ . $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$ , respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$ . Prove that $AD$ , $MN$ and $BC$ are concurrent.

Proposed by Alireza Dadgarnia

Nice inequality

by sqing, Apr 24, 2019, 1:01 PM

Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be real numbers . Prove that : There exist positive integer $k\in \{1,2,\cdots,n\}$ such that $\sum_{i=1}^{n}\{kx_i\}(1-\{kx_i\})<\frac{n-1}{6}.$ Where $\{x\}=x-\left \lfloor x \right \rfloor.$

inequalities

China

Tiling rectangle with smaller rectangles.

by MarkBcc168, Jul 10, 2018, 11:25 AM

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.

Proposed by Jeck Lim, Singapore

This post has been edited 2 times. Last edited by MarkBcc168, Jul 15, 2018, 12:57 PM

Problem 1

by SpectralS, Jul 10, 2012, 5:24 PM

Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A.$ This excircle is tangent to the side $BC$ at $M$ , and to the lines $AB$ and $AC$ at $K$ and $L$ , respectively. The lines $LM$ and $BJ$ meet at $F$ , and the lines $KM$ and $CJ$ meet at $G.$ Let $S$ be the point of intersection of the lines $AF$ and $BC$ , and let $T$ be the point of intersection of the lines $AG$ and $BC.$ Prove that $M$ is the midpoint of $ST.$

(The excircle of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$ , to the ray $AB$ beyond $B$ , and to the ray $AC$ beyond $C$ .)

Proposed by Evangelos Psychas, Greece

geometry

IMO Shortlist

Composite sum

by rohitsingh0812, Jun 3, 2006, 5:39 AM

Let $a$ , $b$ , $c$ , $d$ , $e$ , $f$ be positive integers and let $S = a+b+c+d+e+f$ .
Suppose that the number $S$ divides $abc+def$ and $ab+bc+ca-de-ef-df$ . Prove that $S$ is composite.

A magician has one hundred cards numbered 1 to 100

by Valentin Vornicu, Oct 24, 2005, 10:21 AM

A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn.

How many ways are there to put the cards in the three boxes so that the trick works?

IMO ShortList 1998, combinatorics theory problem 1

by orl, Oct 22, 2004, 3:22 PM

A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number $x$ in the array can be changed into either $\lceil x\rceil$ or $\lfloor x\rfloor$ so that the row-sums and column-sums remain unchanged. (Note that $\lceil x\rceil$ is the least integer greater than or equal to $x$ , while $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$ .)

Attachments:

This post has been edited 1 time. Last edited by orl, Oct 23, 2004, 12:58 PM

Submit

I still exist as well.
by G.G.Otto, Aug 11, 2023, 2:44 AM
hello I'm still here lol
by player01, Aug 6, 2022, 6:24 PM
[REVIVAL] I will start posting more calculator relating posts very soon. Even though school has been busy, I have been programming my calculators a decent amount, so I have a lot to share...
by sonone, Feb 18, 2022, 10:29 PM
wow its been like 2.5 years since geo class
by pieMax2713, Feb 4, 2022, 8:38 PM
@violin21, I've been very busy with school lately and haven't been able to add another lesson. I will when i get a free moment
by sonone, Aug 19, 2021, 12:45 AM
ORZ CODER
by samrocksnature, Aug 9, 2021, 9:57 PM
Could you make more Asymptote lessons on your "How to do Asymptote" blog?
by violin21, Aug 9, 2021, 7:26 PM
You can take it, just C&P the CSS into your CSS area
by sonone, Apr 17, 2021, 10:08 PM
how can we take the CSS if we have permission to not take it?
by GoogleNebula, Apr 17, 2021, 5:22 PM
That is awesome!
by sonone, Apr 15, 2021, 10:09 PM
I modified your dodecahedron and got:
[asy]
import three;
import solids;
size(300);
currentprojection=orthographic(0,1.3,1.2);
light(0,5,10);

real phi=(sqrt(6)+1)/3;
real g=(phi-1)/2;
real s=1/2;
real a=sqrt(1-phi*phi/4-g*g)+phi/2;

triple[] d;
d[0]=(phi
by Andrew2019, Mar 26, 2021, 12:15 AM
Not too many, just changing the color here and there. I really like your CSS!
by sonone, Feb 2, 2021, 10:35 AM
Nice!

I see you're making changes to the CSS.
by G.G.Otto, Feb 1, 2021, 9:26 PM
I'm learning Java now!
by sonone, Feb 1, 2021, 5:56 PM
And I took part of it from CaptainFlint and then added a ton of modifications.
by G.G.Otto, Dec 1, 2020, 8:56 AM
fixed $\text{[math]}$
by sonone, Apr 25, 2020, 8:43 PM
Change in progress...
by sonone, Apr 25, 2020, 8:42 PM
When it says "At least 11 users clicked that button" it's incorrect since one person could have just posted 11 times. It should say "At most 11 users click that button".
by G.G.Otto, Apr 25, 2020, 8:26 PM
see this PM
by sonone, Apr 25, 2020, 7:03 PM
I can't get the cursor to work.
by G.G.Otto, Apr 25, 2020, 6:38 PM
If the cursor is confusing, where you are actually pointing in a front of the ship
by sonone, Apr 25, 2020, 11:04 AM
It's ok $\text{[math]}$
by sonone, Apr 23, 2020, 5:30 PM
Sorry I haven't been posting. I've got a Research Paper (for Roman History) due in a week.
by G.G.Otto, Apr 22, 2020, 11:15 PM
You should download it, as it's really useful when learning python. You can find out more here
by G.G.Otto, Apr 19, 2020, 10:04 PM
No. I don't have IDLE.
by sonone, Apr 18, 2020, 7:49 PM
Do you have IDLE installed on your computer? If so, I can give you a couple really cool codes that made that don't work in the AoPS widgets.
by G.G.Otto, Apr 14, 2020, 6:47 AM
I don't know how to change it....
by G.G.Otto, Apr 5, 2020, 5:58 PM
Do you know how the change the into thing above my avatar a different color? It's too hard to see.
by sonone, Apr 5, 2020, 2:08 PM
WELCOME TO MY BLOG!
by sonone, Mar 31, 2020, 4:56 PM

98 shouts

sonone's place

by Melid, Apr 23, 2025, 3:01 PM

by Assassino9931, Apr 8, 2025, 1:59 PM

by AlastorMoody, Apr 6, 2020, 12:09 PM

by Dadgarnia, Mar 12, 2020, 10:54 AM

by sqing, Apr 24, 2019, 1:01 PM

by MarkBcc168, Jul 10, 2018, 11:25 AM

by SpectralS, Jul 10, 2012, 5:24 PM

by rohitsingh0812, Jun 3, 2006, 5:39 AM

by Valentin Vornicu, Oct 24, 2005, 10:21 AM

by orl, Oct 22, 2004, 3:22 PM