Complex Number Geometry
by gauss202, May 14, 2025, 12:21 PM
Describe the locus of complex numbers, 
, such that 
.
, such that 
.
. Prove that
and 
Show that
Let 
Show that
Inequalities
by sqing, May 13, 2025, 9:04 AM
Let 
and 
Show that
Let and 
Show that
Let and 
Show that
and 
Show that
Let and 
Show that
Let and 
Show that
This post has been edited 2 times. Last edited by sqing, Tuesday at 9:13 AM
2024 Mock AIME 1 ** p15 (cheaters' trap) - 128 | n^{\sigma (n)} - \sigma(n^n)
by parmenides51, Jan 29, 2025, 11:38 PM
Let 
be the number of positive integers 
such that divides 
and 
divides
where 
denotes the number of positive integers that divide , including 
and . Find the remainder when is divided by 
.
be the number of positive integers 
such that divides 
and 
divides
where 
denotes the number of positive integers that divide , including 
and . Find the remainder when is divided by 
.Assam Mathematics Olympiad 2022 Category III Q14
by SomeonecoolLovesMaths, Sep 12, 2024, 11:40 AM
The following sum of three four digits numbers is divisible by 
, 
, where 
are decimal digits. Find the necessary conditions in .
, 
, where 
are decimal digits. Find the necessary conditions in .2019 SMT Team Round - Stanford Math Tournament
by parmenides51, Feb 6, 2022, 10:23 AM
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
Find function
by trito11, Nov 11, 2019, 3:46 AM
Find 
such that
i) f(x)>f(y)
x>y>0
ii) f(2x)
2f(x)x>0
iii)
x,y>0
such thati) f(x)>f(y)
x>y>0ii) f(2x)
2f(x)x>0iii)
x,y>0This post has been edited 1 time. Last edited by trito11, Nov 11, 2019, 3:46 AM
Stay insane,Coz it's your will, labour and pain,which takes you to the top of the mountain.
September 2017
February 2017
September 2016
July 2016
April 2016
March 2016
December 2015
November 2015
August 2015
April 2015
March 2015
February 2015
January 2015
October 2014
September 2014
August 2014
July 2014