Difference between revisions of "2023 SSMO Accuracy Round Problems"
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Let the roots of <math>P(x) = x^3 - 2023x^2 + 2023^{2023}</math> be <math>\alpha, \beta, \gamma</math>. | Let the roots of <math>P(x) = x^3 - 2023x^2 + 2023^{2023}</math> be <math>\alpha, \beta, \gamma</math>. | ||
Find | Find | ||
− | <cmath>\frac{alpha^2 + \beta^2}{\alpha + \beta} + \frac{\beta^2 + \gamma^2}{\beta + \gamma} + \frac{\gamma^2 + \alpha^2}{\gamma + \alpha}</cmath> | + | <cmath>\frac{\alpha^2 + \beta^2}{\alpha + \beta} + \frac{\beta^2 + \gamma^2}{\beta + \gamma} + \frac{\gamma^2 + \alpha^2}{\gamma + \alpha}</cmath> |
[[2023 SSMO Accuracy Round Problems/Problem 6|Solution]] | [[2023 SSMO Accuracy Round Problems/Problem 6|Solution]] |
Latest revision as of 03:41, 23 December 2023
Contents
Problem 1
Mr. Sammy proposes a Hamburger Proclamation, which has lines, divided into paragraphs of
lines each. It takes him
seconds to read each line. Additionally, he adds a
second pause between two lines in a paragraph, and a
second pause between paragraphs. If it takes him
minutes to read the whole Hamburger Proclamation, find
Problem 2
Suppose that the average of all -digit palindromes is denoted by
and the average of all
-digit numbers is denoted by
Find
Problem 3
Suppose that are real numbers such
Find the sum of all possible values of
.
Problem 4
In square point
is selected on diagonal
Let
be the intersection of the circumcircles of triangles
and
Given that
and
find the maximum possible area of triangle
(A circumcircle of some triangle
is the circle containing
,
, and
)
Problem 5
Define the between two numbers
and
to be
where
is the number of divisors of
. Find the sum of integers
which have a relationship of
with
.
Problem 6
Let the roots of be
.
Find
Problem 7
Concentric circles and
are drawn, with radii
and
respectively. Chords
and
of
are both tangent to
and intersect at
If
then the sum of all possible distinct values of
can be expressed as
for relatively prime positive integers
and
Find
Problem 8
There is a quadrilateral inscribed in a circle
with center
. In quadrilateral
, diagonal
is a diameter of the circle,
and
Let
be the base of the altitude from
onto side
. Let
be the base of the altitude from
onto
. Given that
and that the product of the lengths of the diagonals of
is
for some squarefree
find
Problem 9
Consider a grid called
. We take one of the four smaller
grids
located in
as
. We repeat the process of taking smaller grids until we eventually converge at the unit square
![[asy] size(7cm); filldraw((0,0)--(0,10)--(10,10)--(10,0)--cycle, opacity(0.2)+lightblue, blue); fill((0,0)--(0,9)--(9,9)--(9,0)--cycle, opacity(0.1)+lightblue); draw((0,9)--(9,9)--(9,0), blue); fill((1,0)--(1,8)--(9,8)--(9,0)--cycle, opacity(0.1)+lightblue); draw((1,0)--(1,8)--(9,8), blue); label("$A = P_{2023}$", (8.3, 9.52)); label("$P_{2022}$", (6.8, 8.52)); label("$\dots$", (4.78, 7.52)); [/asy]](http://latex.artofproblemsolving.com/d/6/8/d68b27269494c8ba543d444a4cc6fc9c95b4cd37.png)
Of the distinct tuples of shrinking grids
, let
be the number of these tuples such that their last element is the center square of the original grid
. Find the largest integer
such
Problem 10
Let be a triangle such
,
,
. Let the incircle of
touch
at
,
at
, and
at
. Let
be the line through the midpoints of
and
. Define
and
similarily. Let the area of the star created by the union of
and the triangle bound by
,
, and
be
for relatively prime
and
. Find
.