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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>...")
 
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 20:48, 31 May 2023

Problem 1

In triangle $ABC$, circumcircle $\omega$ is drawn. Let $I$ be the incenter of $\triangle{ABC}$. Let $H_A$ be the intersection of the $A$-altitude and $\omega.$ Given that $AB=13,AC=15,$ and $BC=14,$ the area of triangle $AIH_A$ can be expressed as $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$

Solution

Problem 2

Consider $8$ 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 $6$. For example, $bwwbbbww$ is a good ordering because we may consider the lineup $bwwbb$, which has a length equal to $5.$ If the probability that the marbles form a good ordering can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Solution

Problem 3

Let $ABCD$ be an isosceles trapezoid such that $AB\parallel CD.$ Let $E$ be a point on $CD$ such that $AB=CE.$ Let the midpoint of $DE$ be $M$ such that $BD$ intersects $AM$ at $G$ and $AE$ at $F.$ If $DC=36, AB=24,$ and $AD=10,$ then $[AGF]$ can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

[asy] unitsize(2mm); fill((6,8/5)--(10,8/3)--(6,8)--cycle,lightgray); draw((0,0)--(36,0)--(30,8)--(6,8)--cycle); draw((6,8)--(12,0)--(0,0)); draw((0,0)--(30,8)); draw((6,8)--(6,0)); label("A", (6,8), NW); dot((6,8)); label("B", (30,8), NE); dot((30,8)); label("C", (36,0), SE); dot((36,0)); label("D", (0,0), SW); dot((0,0)); label("E",(12,0),S); dot((12,0)); label("M",(6,0),S); dot((6,0)); dot((6,8/5)); dot((10,8/3)); [/asy]

Solution

Problem 4

Let $a_1=2,b_0=3, a_n=\left(a_{n-1}\right)^2,$ and $b_n=\left(b_{n-1}\right)^3.$ If $c_n=a_n+b_n,$ find the last two digits of $c_1+c_2+\dots+c_{2022}.$

Solution

Problem 5

Consider the following rectangle $ABCD$ where $BC=8.$ If \[CD=CT_2, 4T_2P_2=2DP_1=AC, [ADP_1C]=33, \text{ and }[ABP_2C]=34,\] find the value of $[P_1CP_2A].$ (Note that $[ABC]$ is the area of $ABC$.)

[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]

Solution

Problem 6

Let $n$ be a positive integer, and let $x$ be some variable. Define $P_{x,n}$ as the maximum fraction of elements in the set of the first $x$ natural numbers that may be contained in a subset $S$ such that if $k$ is an element of $S$, then $nk$ is not. For example, $P_{3, 3}=\frac{2}{3}$, since we take the set $\{1, 2\}$. As $x$ approaches infinity, $P_{x,n}$ approaches a value $P_n$. Given that $\displaystyle{\prod_{n=2}^{100}P_n}=\frac{a}{b},$ where $a$ and $b$ are relatively prime positive integers, find $a+b.$

Solution

Problem 7

Let $\cos(2A)$, $\cos(2B)$, and $\cos(2C)$ be the not necessarily distinct roots of a monic cubic $f$. Given that $f(1)= \frac{17}{30}$, the value of $\sin(A)\sin(B)\sin(C)$ can be expressed as $\frac{\sqrt{m}}{n},$ with $m$ squarefree. Find $m+n$.

Solution

Problem 8

A frog is at $0$ on a number line and wants to go to $9$. On each turn, if the frog is at $n$, the frog hops to one of the numbers from $n$ to $9$, inclusive, with equal probability (staying in place counts as a hop). It is then teleported to the largest multiple of $3$ that is less than or equal to the frog's position. The expected number of hops it takes for the frog to reach $9$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 9

Given real numbers $a,b,x,y$ 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 $(a+b+x+y)^2$.

Solution

Problem 10

If $p, q, r$ are the roots of the polynomial $x^3-2x^2-4$, find \[(p^3+qr)(q^3+pr)(r^3+pq).\]

Solution

Problem 11

Find the number of solutions to $a^3+36ab+64=27b^3,$ where $a$ is an integer, $b$ is a real number, and $|b|\le 50$.

Solution

Problem 12

Regular pentagon $ABCDE$ is inscribed in circle $\omega_1$ with radius $5\sqrt{5}$. Circle $\omega_2$ is the reflection of $\omega_1$ across $\overline{AB}$. Let $I$ be the intersection of $\overline{AD}$ and $\overline{BE}$, let $P$ be an intersection of $\overline{DO}$ and $\omega_2$, and let line $\ell$ be the tangent to $\omega_2$ at $P$. The sum of the possible distances from point $I$ to line $\ell$ can be expressed as $m\sqrt{n}$, where $n$ is a squarefree positive integer. Find $m+n$.

Solution

Problem 13

In regular hexagon $ABCDEF$ with side length 1, an electron starts at point $A$. 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 $CD$. The electron bounces off of a total of $5$ 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 $\sqrt{m}$, find $m$.

Solution

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?

Solution

Problem 15

Consider two externally tangent circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$. Suppose that $\omega_1$ and $\omega_2$ have radii of $1$ and $3$ respectively. There exist points $A, B$ on $\omega_1$ and points $C, D$ on $\omega_2$ such that $AC$ and $BD$ are the external tangents of $\omega_1$ and $\omega_2$. The circumcircle of $\triangle BO_2D$ intersects $AC$ at two points $X$ and $Y$ such that $AX < AY$. If $CX$ can be expressed as $\frac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Solution

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