2023 WSMO Team Round Problems
Contents
Problem 1
Bob has a number of pencils. When Bob splits them into groups of 10, he has 3 left over. When he splits them into groups of 12, he has 5 left over. Find the smallest number of pencils Bob can have.
Problem 2
Integers and
satisfy
Find the maximum possible value of
Problem 3
In the figure below, there are total circles. The area between circles alternate between shaded and non-shaded. If the area of the shaded region is
find the remainder when
is divided by
Problem 4
Honko the hamster is in his cage. He wants to find the smallest distance needed to travel to reach four tennis balls. His current position is . The tennis balls are located at
and
. The length of the shortest path can be expressed as
where
is minimal. Find
Problem 5
A monkey is throwing darts at the dart board pictured below. The dart is equally likely to land anywhere on the board. Point values for the three regions are labeled and the radii the three circles are respectively. If the expected value of points the monkey gets from 5 dart throws is
for relatively prime positive integers
and
find
Problem 6
A quartic real polynomial satisfying
has 3 distinct roots. If the sum of the three roots is
find their product.
Problem 7
In triangle with
, and
a rectangle
is inscribed such that the area of
is maximized. If the minimum possible value of
is
for relatively prime positive integers
and
find
Problem 8
Let Find the modulo 1000 on the minimum integer
such
is a positive integer.
Problem 9
A circle has three chords of equal length, which intersect forming a triangle with side lengths 2, 2, and
. If the radius of the circle is
then
for positive integers
and squarefree
Find
Problem 10
Let square be a square with side length
. Define ellipse
as the ellipse that is able to be inscribed inside
such that 2 of its vertices on its minor axis and 1 of its vertices on its major axis form an equilateral triangle. The largest possible area of
is
for squarefree
Find
Problem 11
When is expressed in base
the digits are
where
are decimal digits. Let
denote the minimum positive integer
such that
for
and
for
Find
Problem 12
Consider parabola pointing upwards with vertex at the origin of the Cartesian plane. Denote the focus of it as
and the directrix of it as
. Point
with
-coordinate
is selected in
. The perpendicular bisector of
meets
at
. Given the
-coordinate of
is
, then
for relatively prime positive integers
and
Find
Problem 13
Gerry the Squirrel is at a corner of a grid. 1 acorn
is placed. There are four coins placed randomly on the
grid. Gerry will go the shortest path to
and if there are multiple shortest paths, Gerry will pick one randomly. Given that Gerry does not start on top of a coin, the acorn is not on a coin, find the expected value of coins Gerry passes during his trip to
is
for relatively prime positive integers
and
Find
Problem 14
Consider quadrilateral with circumcircle centered at
. Given
, the circumcircle of
is tangent to the circumcircle of
. The circumcircle of
passes through the center of the circumcircle of
. Given the radius of the circumcircle of
is
. Denote
as the area and perimeter of
respectively, compute
. The answer can be expressed in the simplest form of
. Find
Problem 15
On a number line labeled and old man starts at
and tries to reach
Initially, he knows to walk right. However, he has dementia. On each move, there is a
chance he forgets which direction he is supposed to go, resulting in him walking the opposite direction. If the probability the old man reaches
without dying is
for relatively prime positive integers
and
Find
(Note: if the old man tries to walk left when he is at 0, he falls off a cliff and dies.)