[/quote]
,
![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
and 
Prove that
be a triangle with incentre 
and circumcircle 
.
be the midpoints of the arcs 
of 
respectively. Let 
be the point of 
.
and the midpoint 
of 
lie on a line.
from the set of positive integers to the set of positive integers such that, for all positive integers 
and 
,![\[ a, f(b) \text{ and } f(b + f(a) - 1).\]](http://latex.artofproblemsolving.com/4/a/b/4ab97da27e05d0f9e7762f74d0d0bd2dc8b9842f.png)
be integers such that, for each 
,
with![\[
\sum_{j \in S} a_j \equiv i \pmod{2048}.
\]](http://latex.artofproblemsolving.com/8/b/e/8be39aaf83795f611b1c3fa0d5971905bd6ebdfe.png)
Show that for each 
,
such that 
is divisible by 
but not by 
.
is the summation notation, for instance, 
,
,
.
,
.
and radius 
,
and 
at 
and 
respectively. Construct a circle with center 
,
and 
at 
.
be the intersection of 
and 
,
be the intersection of 
and 
,
)
)
and the circle with center 
be a natural number. At a mathematical olympiad training camp the same 
be natural numbers such that![\[
\frac{a^n - 1}{(a - 1)^n + 1}
\]](http://latex.artofproblemsolving.com/6/2/7/627f1d5f55e7594f2e6dcf5d541e796b7a5bfae2.png)
is a natural number.
is odd.
be a prime divisor of ![\[
a^{(q - 1)/2} \equiv 1 \pmod{q}.
\]](http://latex.artofproblemsolving.com/b/2/1/b2135d8c5526e577722718e6d02bcc52559b0555.png)
3. Prove that if a is prime and 
,![\[
2^{(a - 1)/2} \equiv 1 \pmod{a}.
\]](http://latex.artofproblemsolving.com/2/d/c/2dc6cc1fe27cc72e72d63b13cf07939e19051245.png)
such that 
and 
then 
and 
then find the value of 
where 
be a regular heptagon with its center 
.
is the orthocenter of triangle 
,
.
and 
be the intersection point of 
and 
.
.
concur at a single point.
at 
,
and 
at 
,
is perpendicular to 
if and only if 
be an inscribed quadrilateral. Circles of diameters 
and 
intersect at points 
and 
,
and 
intersect at points 
and 
.
and 
intersect in two points 
and 
.
and 
are concurrent.
Y by PikaPika999
p1. How many functions 
take on exactly 
distinct values?
p2. Let
be one of the numbers 
. Suppose that for all positive integers 
, the number 
never has remainder upon division by 
. List all possible values of .
p3. A card is an ordered 4-tuple
where each 
is chosen from 
. A line is an (unordered) set of three (distinct) cards 
,
,
such that for each , the numbers 
are either all the same or all different. How many different lines are there?
p4. We say that the pair of positive integers
, where 
, is a 
-tangent pair if we have
. Compute the second largest integer that appears in a 
-tangent pair.
p5. Regular hexagon
has side length 
. For 
, choose 
to be a point on the segment 
uniformly at random, assuming the convention that 
for all integers 
. What is the expected value of the area of hexagon 
?
p6. Evaluate
.
p7. A plane in-dimensional space passes through the point 
, with 
, 
, and 
all positive. The plane also intersects all three coordinate axes with intercepts greater than zero (i.e. there exist positive numbers 
, 
, 
such that 
, 
, and 
all lie on this plane). Find, in terms of , , , the minimum possible volume of the tetrahedron formed by the origin and these three intercepts.
p8. The left end of a rubber band e meters long is attached to a wall and a slightly sadistic child holds on to the right end. A point-sized ant is located at the left end of the rubber band at time
, when it begins walking to the right along the rubber band as the child begins stretching it. The increasingly tired ant walks at a rate of 
centimeters per second, while the child uniformly stretches the rubber band at a rate of one meter per second. The rubber band is infinitely stretchable and the ant and child are immortal. Compute the time in seconds, if it exists, at which the ant reaches the right end of the rubber band. If the ant never reaches the right end, answer 
.
p9. We say that two lattice points are neighboring if the distance between them is. We say that a point lies at distance d from a line segment if 
is the minimum distance between the point and any point on the line segment. Finally, we say that a lattice point 
is nearby a line segment if the distance between and the line segment is no greater than the distance between the line segment and any neighbor of . Find the number of lattice points that are nearby the line segment connecting the origin and the point 
.
p10. A permutation of the first n positive integers is valid if, for all
, comes after 
in the permutation. What is the probability that a random permutation of the first 
integers is valid?
p11. Given that
and 
, find the range of all possible values of
.
p12. A triangle has sides of length
, 
, and 
. Compute the area of the smallest regular polygon that has three vertices coinciding with the vertices of the given triangle.
p13. How many positive integers are there such that for any natural numbers 
, we have 
implies 
?
p14. Find constants
and 
such that the following limit is finite and nonzero: 
.
Give your answer in the form
.
p15. Sean thinks packing is hard, so he decides to do math instead. He has a rectangular sheet that he wants to fold so that it fits in a given rectangular box. He is curious to know what the optimal size of a rectangular sheet is so that it’s expected to fit well in any given box. Let a and b be positive reals with
, and let 
and be independently and uniformly distributed random variables in the interval 
. For the ordered 
-tuple 
, let 
denote the ratio between the area of a sheet with dimension a×b and the area of the horizontal cross-section of the box with dimension 
after the sheet has been folded in halves along each dimension until it occupies the largest possible area that will still fit in the box (because Sean is picky, the sheet must be placed with sides parallel to the box’s sides). Compute the smallest value of b/a that maximizes the expectation 
.
PS. You had better use hide for answers.
take on exactly 
distinct values?p2. Let
be one of the numbers 
. Suppose that for all positive integers 
, the number 
never has remainder upon division by 
. List all possible values of .p3. A card is an ordered 4-tuple
where each 
is chosen from 
. A line is an (unordered) set of three (distinct) cards 
,
,
such that for each , the numbers 
are either all the same or all different. How many different lines are there?p4. We say that the pair of positive integers
, where 
, is a 
-tangent pair if we have
. Compute the second largest integer that appears in a 
-tangent pair.p5. Regular hexagon
has side length 
. For 
, choose 
to be a point on the segment 
uniformly at random, assuming the convention that 
for all integers 
. What is the expected value of the area of hexagon 
?p6. Evaluate
.p7. A plane in
-dimensional space passes through the point 
, with 
, 
, and 
all positive. The plane also intersects all three coordinate axes with intercepts greater than zero (i.e. there exist positive numbers 
, 
, 
such that 
, 
, and 
all lie on this plane). Find, in terms of , , , the minimum possible volume of the tetrahedron formed by the origin and these three intercepts.p8. The left end of a rubber band e meters long is attached to a wall and a slightly sadistic child holds on to the right end. A point-sized ant is located at the left end of the rubber band at time
, when it begins walking to the right along the rubber band as the child begins stretching it. The increasingly tired ant walks at a rate of 
centimeters per second, while the child uniformly stretches the rubber band at a rate of one meter per second. The rubber band is infinitely stretchable and the ant and child are immortal. Compute the time in seconds, if it exists, at which the ant reaches the right end of the rubber band. If the ant never reaches the right end, answer 
.p9. We say that two lattice points are neighboring if the distance between them is
. We say that a point lies at distance d from a line segment if 
is the minimum distance between the point and any point on the line segment. Finally, we say that a lattice point 
is nearby a line segment if the distance between and the line segment is no greater than the distance between the line segment and any neighbor of . Find the number of lattice points that are nearby the line segment connecting the origin and the point 
.p10. A permutation of the first n positive integers is valid if, for all
, comes after 
in the permutation. What is the probability that a random permutation of the first 
integers is valid?p11. Given that
and 
, find the range of all possible values of
.p12. A triangle has sides of length
, 
, and 
. Compute the area of the smallest regular polygon that has three vertices coinciding with the vertices of the given triangle.p13. How many positive integers
are there such that for any natural numbers 
, we have 
implies 
?p14. Find constants
and 
such that the following limit is finite and nonzero: 
.Give your answer in the form
.p15. Sean thinks packing is hard, so he decides to do math instead. He has a rectangular sheet that he wants to fold so that it fits in a given rectangular box. He is curious to know what the optimal size of a rectangular sheet is so that it’s expected to fit well in any given box. Let a and b be positive reals with
, and let 
and be independently and uniformly distributed random variables in the interval 
. For the ordered 
-tuple 
, let 
denote the ratio between the area of a sheet with dimension a×b and the area of the horizontal cross-section of the box with dimension 
after the sheet has been folded in halves along each dimension until it occupies the largest possible area that will still fit in the box (because Sean is picky, the sheet must be placed with sides parallel to the box’s sides). Compute the smallest value of b/a that maximizes the expectation 
.PS. You had better use hide for answers.
This post has been edited 6 times. Last edited by parmenides51, Jan 24, 2022, 11:28 PM
and 
and 
be the number of elements of elements in the domain that map to 
is a permutation of 
Then, there are 
is 
ways to choose what those inputs are and then 
ways to choose the inputs for the remaining two outputs, for a total of 
Then, there are 
is 
ways to choose what that input is, and then 
ways to choose the inputs for the remaining two outputs, for a total of 
are 
and the cycle repeats.
cannot be obtained as remainders.
.
We can telescope this sum to get 
.
Note that its clearly divisible by 
triangles, each of which has an area proportional to the product of 
and 
,
and 
and area 
