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Expected number of flips
Bread10 8
N
by mathprodigy2011
An unfair coin has a 
probability of coming up heads and 
probability of coming up tails. The expected number of flips necessary to first see the sequence 
in that consecutive order can be written as 
for relatively prime positive integers 
, 
. Find the number of factors of .
probability of coming up heads and 
probability of coming up tails. The expected number of flips necessary to first see the sequence 
in that consecutive order can be written as 
for relatively prime positive integers 
, 
. Find the number of factors of .
8 replies
Help with Competitive Geometry?
REACHAW 3
N
by REACHAW
Hi everyone,
I'm struggling a lot with geometry. I've found algebra, number theory, and even calculus to be relatively intuitive. However, when I took geometry, I found it very challenging. I stumbled my way through the class and can do the basic 'textbook' geometry problems, but still struggle a lot with geometry in competitive math. I find myself consistently skipping the geometry problems during contests (even the easier/first ones).
It's difficult for me to see the solution path. I can do the simpler textbook tasks (eg. find congruent triangles) but not more complex ones (eg. draw these two lines to form similar triangles).
Do you have any advice, resources, or techniques I should try?
I'm struggling a lot with geometry. I've found algebra, number theory, and even calculus to be relatively intuitive. However, when I took geometry, I found it very challenging. I stumbled my way through the class and can do the basic 'textbook' geometry problems, but still struggle a lot with geometry in competitive math. I find myself consistently skipping the geometry problems during contests (even the easier/first ones).
It's difficult for me to see the solution path. I can do the simpler textbook tasks (eg. find congruent triangles) but not more complex ones (eg. draw these two lines to form similar triangles).
Do you have any advice, resources, or techniques I should try?
3 replies
fractional part
Ecrin_eren 3
N
by rchokler
{x^2}+{x}=0.64
How many positive real values of x satisfy this equation?
How many positive real values of x satisfy this equation?
3 replies
Angle oriented geometry
Problems_eater 0
Let 
be four distinct points in the plane.
Which of the following statements, expressed using oriented angles, are always true?
1.If lines
and 
are distinct and parallel, then
the oriented angle
is equal to the oriented angle DCB.
2.If
lies on the segment 
, then
the oriented angle
plus the oriented angle 
equals 
.
3.If the oriented angle
plus the oriented angle 
equals 0°, then
lines
and are parallel.
4.If the oriented angle plus the oriented angle equals , then
lines and are parallel.
be four distinct points in the plane.Which of the following statements, expressed using oriented angles, are always true?
1.If lines
and 
are distinct and parallel, thenthe oriented angle
is equal to the oriented angle DCB.2.If
lies on the segment 
, thenthe oriented angle
plus the oriented angle 
equals 
.3.If the oriented angle
plus the oriented angle 
equals 0°, thenlines
and are parallel.4.If the oriented angle
plus the oriented angle equals , thenlines
and are parallel.
0 replies
how many quadrilaterals ?
Ecrin_eren 6
N
by mathprodigy2011
"All the diagonals of an 11-gon are drawn. How many quadrilaterals can be formed using these diagonals as sides? (The vertices of the quadrilaterals are selected from the vertices of the 11-gon.)"
6 replies
Plane geometry problem with inequalities
ReticulatedPython 3
N
by vanstraelen
Let 
and be points on a plane such that 
Let 
be a point on that plane such that 
Prove that ![$$AP \in \left[\frac{5-\sqrt{5}}{10}, \frac{-1+\sqrt{5}}{2}\right] \cup \left[\frac{5+\sqrt{5}}{10}, \frac{1+\sqrt{5}}{2}\right].$$](//latex.artofproblemsolving.com/7/8/e/78e428cbc99ce4e9448f5c4cec86f50567123812.png)
Source: Own
and be points on a plane such that 
Let 
be a point on that plane such that 
Prove that ![$$AP \in \left[\frac{5-\sqrt{5}}{10}, \frac{-1+\sqrt{5}}{2}\right] \cup \left[\frac{5+\sqrt{5}}{10}, \frac{1+\sqrt{5}}{2}\right].$$](http://latex.artofproblemsolving.com/7/8/e/78e428cbc99ce4e9448f5c4cec86f50567123812.png)
Source: Own
3 replies
be reals such that 
. Prove that
idk12345678 Math Contest
idk12345678 21
N
by idk12345678
Welcome to the 1st idk12345678 Math Contest.
You have 4 hours. You do not have to prove your answers.
Post \signup username to sign up. Post your answers in a hide tag and I will tell you your score.*
The contest is attached to the post
*I mightve done them wrong feel free to ask about an answer
You have 4 hours. You do not have to prove your answers.
Post \signup username to sign up. Post your answers in a hide tag and I will tell you your score.*
The contest is attached to the post
specifies the greatest common divisor of 
and 
.*I mightve done them wrong feel free to ask about an answer
21 replies
purple comet math competition question
AVY2024 4
N
by K1mchi_
Given that (1 + tan 1)(1 + tan 2). . .(1 + tan 45) = 2n, find n
4 replies
and 
Prove that
Let 
Prove that
