Trig Multiplication
by szhang7, Mar 16, 2025, 3:50 PM
Find the exact value of 
.
.Algebra-1
by JetFire008, Mar 16, 2025, 3:19 PM
Find real numbers 
,
if 
is a root of 
and solve the equation.
,
if 
is a root of 
and solve the equation.
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
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
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 
?
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 
?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.
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.
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:
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 
.
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 
.![[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)
We show 
is the reflection of 
We'll proceed with straight angle chasing. It suffices to show ![\[\angle{G'AB}=\angle{G'CB}=\angle{BAG}\]](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)
But by power of a point, 
so it suffices to show 
which follows from a final angle chase: ![\[\angle{DFE}=\angle{PFE}\stackrel{AEPF \text{cyclic}}{=}\angle{BAE}=\angle{DEB},\]](http://latex.artofproblemsolving.com/c/5/1/c51263bb41ac43d7f258345954acd490af1a6a8f.png)
showing 
is isosceles and we're done.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)
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]](http://latex.artofproblemsolving.com/0/3/3/03396c4453b99be98addd190d3160624b32f3613.png)
To share with readers my favorite problem I came across today :) (Shout for contrib.)
May 2017
April 2017
March 2017
February 2017
January 2017
December 2016
November 2016
October 2016
September 2016
August 2016
July 2016
June 2016