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
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
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.
+ 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
regions of a three-set Venn Diagram with
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


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
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
, Amanda will list the numbers
, then stop.
Suppose that the last two numbers in Amanda's list were
and
. If
, how many possible values are there for
?



Suppose that the last two numbers in Amanda's list were




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
be
, and
. What is the sum
?
p18. Two students are playing a game. They take a deck of five cards numbered
through
, 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
such that
is also prime.
p20. In how many ways can one color the
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.




p18. Two students are playing a game. They take a deck of five cards numbered


p19. Compute the sum of all primes


p20. In how many ways can one color the

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
are all real positive number that:
. Prove that:




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
be a circle and let
be a line such that
and
have no common points. Further, let
be a diameter of the circle
; assume that this diameter
is perpendicular to the line
, and the point
is nearer to the line
than the point
. Let
be an arbitrary point on the circle
, different from the points
and
. Let
be the point of intersection of the lines
and
. One of the two tangents from the point
to the circle
touches this circle
at a point
; hereby, we assume that the points
and
lie in the same halfplane with respect to the line
. Denote by
the point of intersection of the lines
and
. Let the line
intersect the circle
at a point
, different from
. Prove that the reflection of the point
in the line
lies on the line
.



































Solution
![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(15cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -7.976002749378333, xmax = 34.86531199295848, ymin = -10.481189671238441, ymax = 12.239223899756556; /* image dimensions */
/* draw figures */
draw(circle((4.02,0.64), 4.671316730858656));
draw((xmin, -1.4436090225563913*xmin + 27.16711976319434)--(xmax, -1.4436090225563913*xmax + 27.16711976319434)); /* line */
draw((0.18,-2.02)--(17.401986181172095,2.0454555016526608));
draw((17.401986181172095,2.0454555016526608)--(5.17527077355485,5.166206959449599));
draw((5.17527077355485,5.166206959449599)--(24.587486816833866,-8.327598047573366));
draw((0.18,-2.02)--(24.587486816833866,-8.327598047573366));
draw((0.18,-2.02)--(13.720717655728016,7.359767959436592));
draw((7.86,3.3)--(8.644181009288122,-0.021928994182223827));
draw((0.8275136207952526,4.050165790484705)--(24.587486816833866,-8.327598047573366));
draw((0.18,-2.02)--(5.17527077355485,5.166206959449599));
/* dots and labels */
dot((7.86,3.3),dotstyle);
label("$B$", (6.338195399361469,3.090290649029001), NE * labelscalefactor);
dot((0.18,-2.02),dotstyle);
label("$A$", (-0.5150645633606531,-2.5813727683961987), NE * labelscalefactor);
dot((13.720717655728016,7.359767959436592),dotstyle);
label("$P$", (13.866653387967544,7.681637225039878), NE * labelscalefactor);
dot((8.644181009288122,-0.021928994182223827),dotstyle);
label("$C$", (8.90394789772049,-1.5685757295702702), NE * labelscalefactor);
dot((17.401986181172095,2.0454555016526608),linewidth(3.pt) + dotstyle);
label("$D$", (17.917841543271265,1.9762139063204798), NE * labelscalefactor);
dot((5.17527077355485,5.166206959449599),linewidth(3.pt) + dotstyle);
label("$E$", (5.257878557947144,5.487243640917033), NE * labelscalefactor);
dot((24.587486816833866,-8.327598047573366),linewidth(3.pt) + dotstyle);
label("$F$", (25.074940617641165,-8.489355494880781), NE * labelscalefactor);
dot((6.090428041178006,-3.547424951721978),linewidth(3.pt) + dotstyle);
label("$G$", (5.865556781242702,-4.6069668460480555), NE * labelscalefactor);
dot((0.8275136207952526,4.050165790484705),linewidth(3.pt) + dotstyle);
label("$G'$", (0.2614131664058928,4.440686700796906), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/0/c/6/0c60f19dac196db75f17aa7938bb3bf472112cf8.png)
Let




![\[\angle{G'AB}=\angle{G'CB}=\angle{BAG}\]](http://latex.artofproblemsolving.com/1/7/9/179c53ddd54aecfb450ca523aa47cd766a93ae02.png)
![\[90-\angle{DCF}=\angle{BAG} \leftrightarrow 90-\angle{BAG}=\angle{PFA}=\angle{DCF} \leftrightarrow \triangle{DCF} \sim \triangle{DFA}\]](http://latex.artofproblemsolving.com/4/3/a/43af76ca0e56f19dc4afd15233ad15fa2fd2c5ef.png)
![\[\frac{DF}{DC}=\frac{AD}{DF} \Rightarrow AD \cdot DC = DF^2.\]](http://latex.artofproblemsolving.com/7/b/5/7b5aad8b4f78989c99e7d393085642ae73997a5c.png)


![\[\angle{DFE}=\angle{PFE}\stackrel{AEPF \text{cyclic}}{=}\angle{BAE}=\angle{DEB},\]](http://latex.artofproblemsolving.com/c/5/1/c51263bb41ac43d7f258345954acd490af1a6a8f.png)

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
are constructed on diameter
of a semicircle of radius
. The centers of the small semicircles divide
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]](//latex.artofproblemsolving.com/0/3/3/03396c4453b99be98addd190d3160624b32f3613.png)






![[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]](http://latex.artofproblemsolving.com/0/3/3/03396c4453b99be98addd190d3160624b32f3613.png)


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