Difference between revisions of "2022 SSMO Team Round Problems"
(Created page with "==Problem 1== In triangle <math>ABC</math>, circumcircle <math>\omega</math> is drawn. Let <math>I</math> be the incenter of <math>\triangle{ABC}</math>. Let <math>H_A</math>...") |
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Line 42: | Line 42: | ||
==Problem 4== | ==Problem 4== | ||
+ | Let <math>a_1=2,b_0=3, a_n=\left(a_{n-1}\right)^2,</math> and <math>b_n=\left(b_{n-1}\right)^3.</math> If <math>c_n=a_n+b_n,</math> find the last two digits of <math>c_1+c_2+\dots+c_{2022}.</math> | ||
[[2022 SSMO Team Round Problems/Problem 4|Solution]] | [[2022 SSMO Team Round Problems/Problem 4|Solution]] | ||
==Problem 5== | ==Problem 5== | ||
+ | Consider the following rectangle <math>ABCD</math> where <math>BC=8.</math> If <cmath>CD=CT_2, 4T_2P_2=2DP_1=AC, [ADP_1C]=33, \text{ and }[ABP_2C]=34,</cmath> find the value of <math>[P_1CP_2A].</math> (Note that <math>[ABC]</math> is the area of <math>ABC</math>.) | ||
+ | <center> | ||
+ | <asy> | ||
+ | unitsize(0.25cm); | ||
+ | dot((0,0)); | ||
+ | dot((12,0)); | ||
+ | dot((12,16)); | ||
+ | dot((0,16)); | ||
+ | draw((0,0)--(12,0)--(12,16)--(0,16)--(0,0)--cycle); | ||
+ | dot((8,22)); | ||
+ | dot((17,4)); | ||
+ | dot((8,16)); | ||
+ | dot((12,4)); | ||
+ | label("$A$", (0,0), SW); | ||
+ | label("$B$", (12,0), SE); | ||
+ | label("$C$", (12,16), NE); | ||
+ | label("$D$", (0,16), NW); | ||
+ | label("$P_1$", (8,22), NE); | ||
+ | label("$P_2$", (17,4), SE); | ||
+ | label("$T_1$", (8,16), SE); | ||
+ | label("$T_2$", (12,4), SW); | ||
+ | draw((8,22)--(8,16),dashed+linewidth(.5)); | ||
+ | draw((12,4)--(17,4),dashed+linewidth(.5)); | ||
+ | </asy> | ||
+ | </center> | ||
[[2022 SSMO Team Round Problems/Problem 5|Solution]] | [[2022 SSMO Team Round Problems/Problem 5|Solution]] | ||
==Problem 6== | ==Problem 6== | ||
+ | Let <math>n</math> be a positive integer, and let <math>x</math> be some variable. Define <math>P_{x,n}</math> as the maximum fraction of elements in the set of the first <math>x</math> natural numbers that may be contained in a subset <math>S</math> such that if <math>k</math> is an element of <math>S</math>, then <math>nk</math> is not. For example, <math>P_{3, 3}=\frac{2}{3}</math>, since we take the set <math>\{1, 2\}</math>. As <math>x</math> approaches infinity, <math>P_{x,n}</math> approaches a value <math>P_n</math>. Given that <math>\displaystyle{\prod_{n=2}^{100}P_n}=\frac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers, find <math>a+b.</math> | ||
[[2022 SSMO Team Round Problems/Problem 6|Solution]] | [[2022 SSMO Team Round Problems/Problem 6|Solution]] | ||
==Problem 7== | ==Problem 7== | ||
+ | Let <math>\cos(2A)</math>, <math>\cos(2B)</math>, and <math>\cos(2C)</math> be the not necessarily distinct roots of a monic cubic <math>f</math>. Given that <math>f(1)= \frac{17}{30}</math>, the value of <math>\sin(A)\sin(B)\sin(C)</math> can be expressed as <math>\frac{\sqrt{m}}{n},</math> with <math>m</math> squarefree. Find <math>m+n</math>. | ||
[[2022 SSMO Team Round Problems/Problem 7|Solution]] | [[2022 SSMO Team Round Problems/Problem 7|Solution]] | ||
==Problem 8== | ==Problem 8== | ||
+ | A frog is at <math>0</math> on a number line and wants to go to <math>9</math>. On each turn, if the frog is at <math>n</math>, the frog hops to one of the numbers from <math>n</math> to <math>9</math>, inclusive, with equal probability (staying in place counts as a hop). It is then teleported to the largest multiple of <math>3</math> that is less than or equal to the frog's position. The expected number of hops it takes for the frog to reach <math>9</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
[[2022 SSMO Team Round Problems/Problem 8|Solution]] | [[2022 SSMO Team Round Problems/Problem 8|Solution]] | ||
==Problem 9== | ==Problem 9== | ||
+ | Given real numbers <math>a,b,x,y</math> such that | ||
+ | <align*> | ||
+ | a^2+b^2&=1,\\ | ||
+ | x^2+y^2&=1,\\ | ||
+ | abxy-\frac{1}{8}&=b^2y^2, | ||
+ | </align*> | ||
+ | find the sum of all distinct values of <math>(a+b+x+y)^2</math>. | ||
[[2022 SSMO Team Round Problems/Problem 9|Solution]] | [[2022 SSMO Team Round Problems/Problem 9|Solution]] | ||
==Problem 10== | ==Problem 10== | ||
+ | If <math>p, q, r</math> are the roots of the polynomial <math>x^3-2x^2-4</math>, find <cmath>(p^3+qr)(q^3+pr)(r^3+pq).</cmath> | ||
[[2022 SSMO Team Round Problems/Problem 10|Solution]] | [[2022 SSMO Team Round Problems/Problem 10|Solution]] | ||
==Problem 11== | ==Problem 11== | ||
+ | Find the number of solutions to <math>a^3+36ab+64=27b^3,</math> where <math>a</math> is an integer, <math>b</math> is a real number, and <math>|b|\le 50</math>. | ||
[[2022 SSMO Team Round Problems/Problem 11|Solution]] | [[2022 SSMO Team Round Problems/Problem 11|Solution]] | ||
==Problem 12== | ==Problem 12== | ||
+ | Regular pentagon <math>ABCDE</math> is inscribed in circle <math>\omega_1</math> with radius <math>5\sqrt{5}</math>. Circle <math>\omega_2</math> is the reflection of <math>\omega_1</math> across <math>\overline{AB}</math>. Let <math>I</math> be the intersection of <math>\overline{AD}</math> and <math>\overline{BE}</math>, let <math>P</math> be an intersection of <math>\overline{DO}</math> and <math>\omega_2</math>, and let line <math>\ell</math> be the tangent to <math>\omega_2</math> at <math>P</math>. The sum of the possible distances from point <math>I</math> to line <math>\ell</math> can be expressed as <math>m\sqrt{n}</math>, where <math>n</math> is a squarefree positive integer. Find <math>m+n</math>. | ||
[[2022 SSMO Team Round Problems/Problem 12|Solution]] | [[2022 SSMO Team Round Problems/Problem 12|Solution]] | ||
==Problem 13== | ==Problem 13== | ||
+ | In regular hexagon <math>ABCDEF</math> with side length 1, an electron starts at point <math>A</math>. When the electron hits an edge, it reflects off of it, with the angle of reflection equal to the angle of incidence. The electron first travels in a straight line to a point on edge <math>CD</math>. The electron bounces off of a total of <math>5</math> edges before hitting a vertex. The electron stops and its total distance traveled is measured. If the shortest possible distance the electron could have traveled can be expressed as <math>\sqrt{m}</math>, find <math>m</math>. | ||
[[2022 SSMO Team Round Problems/Problem 13|Solution]] | [[2022 SSMO Team Round Problems/Problem 13|Solution]] | ||
==Problem 14== | ==Problem 14== | ||
+ | On a hot summer day, three little piggies decide to play with water balloons. The three piggies travel to a 200-floor parking garage each armed with exactly one water balloon. | ||
+ | |||
+ | The game works as follows: | ||
+ | 1. If a piggie drops a water balloon from any floor of the building, it will either break, or it will survive the fall. | ||
+ | 2. If the water balloon breaks, then any greater fall would have broken it as well. | ||
+ | 3. If the water balloon survives, then it would have survived any lesser fall. | ||
+ | 4. Every water balloon is identical and interchangeable. | ||
+ | |||
+ | The goal for the piggies is to find the lowest floor that will break a water balloon. Assuming they play optimally, what is the minimum number of tries in which they are guaranteed to find the lowest balloon-breaking floor? | ||
[[2022 SSMO Team Round Problems/Problem 14|Solution]] | [[2022 SSMO Team Round Problems/Problem 14|Solution]] | ||
==Problem 15== | ==Problem 15== | ||
+ | Consider two externally tangent circles <math>\omega_1</math> and <math>\omega_2</math> with centers <math>O_1</math> and <math>O_2</math>. Suppose that <math>\omega_1</math> and <math>\omega_2</math> have radii of <math>1</math> and <math>3</math> respectively. There exist points <math>A, B</math> on <math>\omega_1</math> and points <math>C, D</math> on <math>\omega_2</math> such that <math>AC</math> and <math>BD</math> are the external tangents of <math>\omega_1</math> and <math>\omega_2</math>. The circumcircle of <math>\triangle BO_2D</math> intersects <math>AC</math> at two points <math>X</math> and <math>Y</math> such that <math>AX < AY</math>. If <math>CX</math> can be expressed as <math>\frac{\sqrt{m}}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>. | ||
[[2022 SSMO Team Round Problems/Problem 15|Solution]] | [[2022 SSMO Team Round Problems/Problem 15|Solution]] |
Revision as of 21:48, 31 May 2023
Contents
Problem 1
In triangle , circumcircle
is drawn. Let
be the incenter of
. Let
be the intersection of the
-altitude and
Given that
and
the area of triangle
can be expressed as
for relatively prime positive integers
and
Find
Problem 2
Consider marbles in a line, where the color of each marble is either black or white and is randomly chosen. Define the period of a lineup of 8 marbles to be the length of the smallest lineup of marbles such that if we consider the infinite repeating sequence of marbles formed by repeating that lineup, the original lineup of 8 marbles can be found within that sequence.
A good ordering of these marbles is defined to be an ordering such that the period of the ordering is at most . For example,
is a good ordering because we may consider the lineup
, which has a length equal to
If the probability that the marbles form a good ordering can be expressed as
where
and
are relatively prime positive integers, find
Problem 3
Let be an isosceles trapezoid such that
Let
be a point on
such that
Let the midpoint of
be
such that
intersects
at
and
at
If
and
then
can be expressed as
where
and
are relatively prime positive integers. Find
Problem 4
Let and
If
find the last two digits of
Problem 5
Consider the following rectangle where
If
find the value of
(Note that
is the area of
.)
Problem 6
Let be a positive integer, and let
be some variable. Define
as the maximum fraction of elements in the set of the first
natural numbers that may be contained in a subset
such that if
is an element of
, then
is not. For example,
, since we take the set
. As
approaches infinity,
approaches a value
. Given that
where
and
are relatively prime positive integers, find
Problem 7
Let ,
, and
be the not necessarily distinct roots of a monic cubic
. Given that
, the value of
can be expressed as
with
squarefree. Find
.
Problem 8
A frog is at on a number line and wants to go to
. On each turn, if the frog is at
, the frog hops to one of the numbers from
to
, inclusive, with equal probability (staying in place counts as a hop). It is then teleported to the largest multiple of
that is less than or equal to the frog's position. The expected number of hops it takes for the frog to reach
can be expressed as
, where
and
are relatively prime positive integers. Find
.
Problem 9
Given real numbers such that
<align*>
a^2+b^2&=1,\\
x^2+y^2&=1,\\
abxy-\frac{1}{8}&=b^2y^2,
</align*>
find the sum of all distinct values of
.
Problem 10
If are the roots of the polynomial
, find
Problem 11
Find the number of solutions to where
is an integer,
is a real number, and
.
Problem 12
Regular pentagon is inscribed in circle
with radius
. Circle
is the reflection of
across
. Let
be the intersection of
and
, let
be an intersection of
and
, and let line
be the tangent to
at
. The sum of the possible distances from point
to line
can be expressed as
, where
is a squarefree positive integer. Find
.
Problem 13
In regular hexagon with side length 1, an electron starts at point
. When the electron hits an edge, it reflects off of it, with the angle of reflection equal to the angle of incidence. The electron first travels in a straight line to a point on edge
. The electron bounces off of a total of
edges before hitting a vertex. The electron stops and its total distance traveled is measured. If the shortest possible distance the electron could have traveled can be expressed as
, find
.
Problem 14
On a hot summer day, three little piggies decide to play with water balloons. The three piggies travel to a 200-floor parking garage each armed with exactly one water balloon.
The game works as follows: 1. If a piggie drops a water balloon from any floor of the building, it will either break, or it will survive the fall. 2. If the water balloon breaks, then any greater fall would have broken it as well. 3. If the water balloon survives, then it would have survived any lesser fall. 4. Every water balloon is identical and interchangeable.
The goal for the piggies is to find the lowest floor that will break a water balloon. Assuming they play optimally, what is the minimum number of tries in which they are guaranteed to find the lowest balloon-breaking floor?
Problem 15
Consider two externally tangent circles and
with centers
and
. Suppose that
and
have radii of
and
respectively. There exist points
on
and points
on
such that
and
are the external tangents of
and
. The circumcircle of
intersects
at two points
and
such that
. If
can be expressed as
, where
and
are relatively prime positive integers, find
.