Trig Multiplication

by szhang7, Mar 16, 2025, 3:50 PM

Find the exact value of $(2-\sin^2(\frac{\pi}{7}))(2-\sin^2(\frac{3\pi}{7}))(2-\sin^2(\frac{3\pi}{7}))$.

interesting problem

by sausagebun, Mar 16, 2025, 3:21 PM

Six points, labeled A, B, C, D, E, and F, are positioned consecutively on a straight line. Let G be a point not located on this line. The following distances are given: AC = 26, BD = 22, CE = 31, DF = 33, AF = 73, CG = 40, and DG = 30. Determine the area of triangle BGE.
I brute forced this with trig, was wondering if theres a more elegant way of doing this

Algebra-1

by JetFire008, Mar 16, 2025, 3:19 PM

Find real numbers $p$,$q$ if $1+i$ is a root of $x^3+px^2+qx+6=0$ and solve the equation.

Cooked for AMC 10?

by Dream9, Mar 16, 2025, 2:00 PM

So I'm like a 8th grader so almost 9th over the summer and I suck at AMC 10. I got like a 75 for my first time but I can do like almost all the problems from AMC 8 with enough time which I find really weird because most other ppl who can do that get higher AMC 10 scores. I do like the first 11 problems a day from past years to try to at least get down the first 10 questions and move on from there. Does anyone have any good suggestions on how I can boost my AMC 10 scores?
+ something annoying that often happens is like I don't even know where to start when I see a problem.

Find the value

by yt12, Mar 16, 2025, 3:33 AM

PIE Help

by Rice_Farmer, Mar 15, 2025, 11:47 PM

How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and reflection.

Non ai way btw pls
This post has been edited 2 times. Last edited by Rice_Farmer, Yesterday at 3:35 AM

[PMO27 Areas] I.19 Geo sequence mod 100

by aops-g5-gethsemanea2, Jan 25, 2025, 1:20 PM

Let $a_1,a_2,a_3,\dots$ be an infinite geometric sequence where all the terms are positive integers. Amanda lists down the two-digit numbers formed by the last two digits of each term in the sequence in order, adding a leading zero if necessary. When she is about to write a two-digit number that she has already written before, she stops. For example, if the sequence was $15,45,135,405,1215,\dots$, Amanda will list the numbers $14,45,35,05$, then stop.

Suppose that the last two numbers in Amanda's list were $76$ and $52$. If $a_2\le 1000$, how many possible values are there for $a_2$?

2021 SMT Guts Round 5 p17-20 - Stanford Math Tournament

by parmenides51, Feb 11, 2022, 3:13 PM

p17. Let the roots of the polynomial $f(x) = 3x^3 + 2x^2 + x + 8 = 0$ be $p, q$, and $r$. What is the sum $\frac{1}{p} +\frac{1}{q} +\frac{1}{r}$ ?


p18. Two students are playing a game. They take a deck of five cards numbered $1$ through $5$, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play?


p19. Compute the sum of all primes $p$ such that $2^p + p^2$ is also prime.


p20. In how many ways can one color the $8$ vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?


PS. You should use hide for answers. Collected here.
This post has been edited 1 time. Last edited by parmenides51, Aug 11, 2023, 9:36 AM

inequalities

by LuvThoConBoiRoi, Aug 1, 2019, 10:27 AM

With $a,b,c$ are all real positive number that: $abc=a+b+c+2$. Prove that:
$\frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a} \leq \frac{3}{4}$
This post has been edited 3 times. Last edited by LuvThoConBoiRoi, Aug 1, 2019, 11:04 AM

Whoever wrote this... doesn't know what concise means :\

by shiningsunnyday, Apr 11, 2017, 3:46 PM

2004 ISL G2 wrote:
Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

Solution

Can anyone spot the projective solution (since I found this in a projective chapter in Lemmas).

Four Semicircles

by worthawholebean, Jan 5, 2009, 2:31 AM

Three semicircles of radius $ 1$ are constructed on diameter $ AB$ of a semicircle of radius $ 2$. The centers of the small semicircles divide $ \overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?
[asy]import graph;
unitsize(14mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dashed=linetype("4 4");
dotfactor=3;

pair A=(-2,0), B=(2,0);

fill(Arc((0,0),2,0,180)--cycle,mediumgray);
fill(Arc((-1,0),1,0,180)--cycle,white);
fill(Arc((0,0),1,0,180)--cycle,white);
fill(Arc((1,0),1,0,180)--cycle,white);
draw(Arc((-1,0),1,60,180));
draw(Arc((0,0),1,0,60),dashed);
draw(Arc((0,0),1,60,120));
draw(Arc((0,0),1,120,180),dashed);
draw(Arc((1,0),1,0,120));
draw(Arc((0,0),2,0,180)--cycle);

dot((0,0));
dot((-1,0));
dot((1,0));

draw((-2,-0.1)--(-2,-0.3),gray);
draw((-1,-0.1)--(-1,-0.3),gray);
draw((1,-0.1)--(1,-0.3),gray);
draw((2,-0.1)--(2,-0.3),gray);

label("$A$",A,W);
label("$B$",B,E);
label("1",(-1.5,-0.1),S);
label("2",(0,-0.1),S);
label("1",(1.5,-0.1),S);[/asy]$ \textbf{(A)}\ \pi-\sqrt3 \qquad
\textbf{(B)}\ \pi-\sqrt2 \qquad
\textbf{(C)}\ \frac{\pi+\sqrt2}{2} \qquad
\textbf{(D)}\ \frac{\pi+\sqrt3}{2}$
$ \textbf{(E)}\ \frac{7}{6}\pi-\frac{\sqrt3}{2}$

To share with readers my favorite problem I came across today :) (Shout for contrib.)

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  • 2021 post

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  • Let $ ABC$ be an equilateral triangle of side length $ 1$. Let $ D$ be the point such that $ C$ is the midpoint of $ BD$, and let $ I$ be the incenter of triangle $ ACD$. Let $ E$ be the point on line $ AB$ such that $ DE$ and $ BI$ are perpendicular. $ \

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  • oh my gosh it's been so longggggg.... contrib? what does that mean?

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  • 2019 post

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  • hi contrib please

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  • hihihihihi contrib plzzzzz

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  • contrib /charmander

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  • for contrib

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