and 
Prove that 
be real numbers such that 
.
Prove that 
and ![$e\in [-2,2]$](http://latex.artofproblemsolving.com/b/b/7/bb78c9dedb18af5e6a904f5492917921e41a7ebc.png)
such that 
Find the value of 
:
Y by
p1. Given 
, find the value of x that minimizes 
.
p2. There are real numbers
and 
such that the only 
-intercept of 
equals its 
-intercept. Compute 
.
p3. Consider the set of
digit numbers 
(with 
) such that 
, 
, and 
. What’s the size of this set?
p4. Let
be the midpoint of 
in 
. A line perpendicular to D intersects 
at 
. If the area of is four times that of the area of 
, what is 
in degrees?
p5. Define the sequence
with 
and 
for 
. Find 
.
p6. Find the maximum possible value of
.
p7. Let
. Let 
(x) be the 
th derivative of 
. What is the largest integer 
such that 
divides 
?
p8. Let
be the set of vectors 
where 
are all real numbers. Let 
denote 
. Let 
be the set in 
given by 
If a point 
is uniformly at random from , what is ![$E[||z||^2]$](//latex.artofproblemsolving.com/e/1/a/e1aa966627c05d1f317981d82094e07d1b1ce0e1.png)
?
p9. Let
be the unique integer between 
and 
, inclusive, that is equivalent modulo to 
. Let be the set of primes between 
and 
, inclusive. Find 
.
p10. In the Cartesian plane, consider a box with vertices
,
,
,
. We pick an integer between 
and 
, inclusive, uniformly at random. We shoot a puck from in the direction of 
and the puck bounces perfectly around the box (angle in equals angle out, no friction) until it hits one of the four vertices of the box. What is the expected number of times it will hit an edge or vertex of the box, including both when it starts at and when it ends at some vertex of the box?
p11. Sarah is buying school supplies and she has
. She can only buy full packs of each of the following items. A pack of pens is 
, a pack of pencils is 
, and any type of notebook or stapler is 
. Sarah buys at least pack of pencils. She will either buy stapler or no stapler. She will buy at most college-ruled notebooks and at most 
graph paper notebooks. How many ways can she buy school supplies?
p12. Let
be the center of the circumcircle of right triangle 
with 
. Let 
be the midpoint of minor arc 
and let 
be a point on line such that 
. Let 
be the intersection of line 
and the Circle and let 
be the intersection of line 
and 
. If 
and 
, compute the radius of the Circle .
p13. Reduce the following expression to a simplified rational
p14. Compute the following integral
.
p15. Define
to be the maximum possible least-common-multiple of any sequence of positive integers which sum to . Find the sum of all possible odd
PS. You should use hide for answers. Collected here.
, find the value of x that minimizes 
.p2. There are real numbers
and 
such that the only 
-intercept of 
equals its 
-intercept. Compute 
.p3. Consider the set of
digit numbers 
(with 
) such that 
, 
, and 
. What’s the size of this set?p4. Let
be the midpoint of 
in 
. A line perpendicular to D intersects 
at 
. If the area of is four times that of the area of 
, what is 
in degrees?p5. Define the sequence
with 
and 
for 
. Find 
.p6. Find the maximum possible value of
.p7. Let
. Let 
(x) be the 
th derivative of 
. What is the largest integer 
such that 
divides 
?p8. Let
be the set of vectors 
where 
are all real numbers. Let 
denote 
. Let 
be the set in 
given by 
If a point 
is uniformly at random from , what is ![$E[||z||^2]$](http://latex.artofproblemsolving.com/e/1/a/e1aa966627c05d1f317981d82094e07d1b1ce0e1.png)
?p9. Let
be the unique integer between 
and 
, inclusive, that is equivalent modulo to 
. Let be the set of primes between 
and 
, inclusive. Find 
.p10. In the Cartesian plane, consider a box with vertices
,
,
,
. We pick an integer between 
and 
, inclusive, uniformly at random. We shoot a puck from in the direction of 
and the puck bounces perfectly around the box (angle in equals angle out, no friction) until it hits one of the four vertices of the box. What is the expected number of times it will hit an edge or vertex of the box, including both when it starts at and when it ends at some vertex of the box?p11. Sarah is buying school supplies and she has
. She can only buy full packs of each of the following items. A pack of pens is 
, a pack of pencils is 
, and any type of notebook or stapler is 
. Sarah buys at least pack of pencils. She will either buy stapler or no stapler. She will buy at most college-ruled notebooks and at most 
graph paper notebooks. How many ways can she buy school supplies?p12. Let
be the center of the circumcircle of right triangle 
with 
. Let 
be the midpoint of minor arc 
and let 
be a point on line such that 
. Let 
be the intersection of line 
and the Circle and let 
be the intersection of line 
and 
. If 
and 
, compute the radius of the Circle .p13. Reduce the following expression to a simplified rational
p14. Compute the following integral
.p15. Define
to be the maximum possible least-common-multiple of any sequence of positive integers which sum to . Find the sum of all possible odd PS. You should use hide for answers. Collected here.
This post has been edited 3 times. Last edited by parmenides51, Feb 6, 2022, 10:58 AM
and the 
intercepts must be 
then 
also and 
We have 
The size of the desired set is 
.
onto 
.
.
.![\[\lim_{k\rightarrow\infty}\frac{c_k}{8^k}=2.\overline{5}_8\]](http://latex.artofproblemsolving.com/e/3/8/e388942755bac05bf4383d5f635ca84024f6192f.png)
so the answer is just ![\[2+\frac{5}{7}=\boxed{\frac{19}{7}}.\]](http://latex.artofproblemsolving.com/e/6/5/e658ade982e72084a3c5669a5890849a5599923e.png)
represents Pens, Pencils, College-Ruled Notebooks, Graph Paper Notebook, and Staplers, respectively. We have
We can do casework for combinations of 
.
cases. Good Luck.
and 
.
:
which is a minimum.
we get![\[\int_0^\infty \sum_{k=1}^\infty \frac{(-1)^{k-1}e^{-kt}}{k}\,dt.\]](http://latex.artofproblemsolving.com/2/0/8/20828fa6e4c3293806df3d40a103138c32a2d440.png)
Everything converges nicely, so we can swap the sum and integral to get![\[\sum_{k=1}^\infty \int_0^\infty \frac{(-1)^{k-1}e^{-kt}}{k}\,dt=\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k^2}=\boxed{\frac{\pi^2}{12}}.\]](http://latex.artofproblemsolving.com/0/c/7/0c73c987fa152b3beb6569dd6eae8f3d2c268fe4.png)
.
.
.
.
.
and we are given that 
.
.
So the X-intercept is 
Y-intercept is 
which is given to only have one root. Thus, the discriminate condition is as follows. 
Another statement we are given is the Y-intercept and the X-intercept are the same. So using the fact that the discriminate is 0 we get, 
.
.
So 
So 
.
is extraneous, so 
gives the answer of 48.
as 
,
as 
.
,
.
which is 90.
