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Let <math>N</math> be the number of complex numbers <math>z</math> with the properties that <math>|z|=1</math> and <math>z^{6!}-z^{5!}</math> is a real number. Find the remainder when <math>N</math> is divided by <math>1000</math>. | Let <math>N</math> be the number of complex numbers <math>z</math> with the properties that <math>|z|=1</math> and <math>z^{6!}-z^{5!}</math> is a real number. Find the remainder when <math>N</math> is divided by <math>1000</math>. | ||
| + | |||
| + | ==Problem 9== | ||
| + | |||
| + | Find the number of four-element subsets of <math>\{1,2,3,4,\dots, 20\}</math> with the property that two distinct elements of a subset have a sum of <math>16</math>, and two distinct elements of a subset have a sum of <math>24</math>. For example, <math>\{3,5,13,19\}</math> and <math>\{6,10,20,18\}</math> are two such subsets. | ||
Revision as of 19:28, 15 June 2024
Problem 5
Suppose that 
, 
, and 
are complex numbers such that 
, 
, and 
, where 
. Then there are real numbers 
and 
such that 
. Find 
.
Problem 6
A real number is chosen randomly and uniformly from the interval ![$[-20, 18]$](http://latex.artofproblemsolving.com/5/e/d/5ed3fd299f7f005431bef3d44f28cf5f7494b169.png)
. The probability that the roots of the polynomial
![\[x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2\]](http://latex.artofproblemsolving.com/3/8/a/38a3a5e9eaaa5a957b5f806c1179ce9a38a46d20.png)
are all real can be written in the form 
, where 
and 
are relatively prime positive integers. Find 
.
Problem 9
Octagon 
with side lengths 
and 
is formed by removing 6-8-10 triangles from the corners of a 
rectangle with side 
on a short side of the rectangle, as shown. Let 
be the midpoint of , and partition the octagon into 7 triangles by drawing segments 
, 
, 
, 
, 
, and 
. Find the area of the convex polygon whose vertices are the centroids of these 7 triangles.
![[asy] unitsize(6); pair P = (0, 0), Q = (0, 23), R = (27, 23), SS = (27, 0); pair A = (0, 6), B = (8, 0), C = (19, 0), D = (27, 6), EE = (27, 17), F = (19, 23), G = (8, 23), J = (0, 23/2), H = (0, 17); draw(P--Q--R--SS--cycle); draw(J--B); draw(J--C); draw(J--D); draw(J--EE); draw(J--F); draw(J--G); draw(A--B); draw(H--G); real dark = 0.6; filldraw(A--B--P--cycle, gray(dark)); filldraw(H--G--Q--cycle, gray(dark)); filldraw(F--EE--R--cycle, gray(dark)); filldraw(D--C--SS--cycle, gray(dark)); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(J); dot(H); defaultpen(fontsize(10pt)); real r = 1.3; label("$A$", A, W*r); label("$B$", B, S*r); label("$C$", C, S*r); label("$D$", D, E*r); label("$E$", EE, E*r); label("$F$", F, N*r); label("$G$", G, N*r); label("$H$", H, W*r); label("$J$", J, W*r); [/asy]](http://latex.artofproblemsolving.com/e/f/a/efa76a4cabaef0470bc7116a12bbab1ee037586e.png)
Problem 13
Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is where and are relatively prime positive integers. Find 
.
Problem 6
Let 
be the number of complex numbers with the properties that 
and 
is a real number. Find the remainder when is divided by 
.
Problem 9
Find the number of four-element subsets of 
with the property that two distinct elements of a subset have a sum of 
, and two distinct elements of a subset have a sum of 
. For example, 
and 
are two such subsets.
