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Difference between revisions of "2021 WSMO Speed Round"

(Created page with "==Problem 1== Let <math>f^1(x)=(x-1)^2</math>, and let <math>f^n(x)=f^1(f^{n-1}(x))</math>. Find the value of <math>|f^7(2)|</math>. ==Problem 2== A square with side length o...")
 
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==Problem 1==
 
==Problem 1==
 
Let <math>f^1(x)=(x-1)^2</math>, and let <math>f^n(x)=f^1(f^{n-1}(x))</math>. Find the value of <math>|f^7(2)|</math>.
 
Let <math>f^1(x)=(x-1)^2</math>, and let <math>f^n(x)=f^1(f^{n-1}(x))</math>. Find the value of <math>|f^7(2)|</math>.
 +
 +
[[2021 WSMO Speed Round/Problem 1|Solution]]
  
 
==Problem 2==
 
==Problem 2==
 
A square with side length of <math>4</math> units is rotated around one of its sides by <math>90^{\circ}</math>. If the volume the square sweeps out can be expressed as <math>m\pi</math>, find <math>m</math>.
 
A square with side length of <math>4</math> units is rotated around one of its sides by <math>90^{\circ}</math>. If the volume the square sweeps out can be expressed as <math>m\pi</math>, find <math>m</math>.
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[[2021 WSMO Speed Round/Problem 2|Solution]]
  
 
==Problem 3==
 
==Problem 3==
 
Let <math>a@b=\frac{a^2-b^2}{a+b}</math>. Find the value of <math>1@(2@(\dots(2020@2021)\dots)</math>.
 
Let <math>a@b=\frac{a^2-b^2}{a+b}</math>. Find the value of <math>1@(2@(\dots(2020@2021)\dots)</math>.
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[[2021 WSMO Speed Round/Problem 3|Solution]]
  
 
==Problem 4==
 
==Problem 4==
 
A square <math>ABCD</math> with side length <math>10</math> is placed inside of a right isosceles triangle <math>XYZ</math> with <math>\angle XYZ=90^{\circ}</math> such that <math>A</math> and <math>B</math> are on <math>XZ</math>, <math>C</math> is on <math>YZ</math>, and <math>D</math> is on <math>XY</math>. Find the area of <math>XYZ</math>.
 
A square <math>ABCD</math> with side length <math>10</math> is placed inside of a right isosceles triangle <math>XYZ</math> with <math>\angle XYZ=90^{\circ}</math> such that <math>A</math> and <math>B</math> are on <math>XZ</math>, <math>C</math> is on <math>YZ</math>, and <math>D</math> is on <math>XY</math>. Find the area of <math>XYZ</math>.
 +
 +
[[2021 WSMO Speed Round/Problem 4|Solution]]
  
 
==Problem 5==
 
==Problem 5==
 
The number of ways to arrange the characters in "delicious greenbeans" into two separate strings of letters can be expressed as <math>a\cdot b!,</math> where <math>b</math> is maximized and both <math>a</math> and <math>b</math> are positive integers. Find <math>a+b.</math> (A string of letters is defined as a group of consecutive letters with no spaces between them.)
 
The number of ways to arrange the characters in "delicious greenbeans" into two separate strings of letters can be expressed as <math>a\cdot b!,</math> where <math>b</math> is maximized and both <math>a</math> and <math>b</math> are positive integers. Find <math>a+b.</math> (A string of letters is defined as a group of consecutive letters with no spaces between them.)
 +
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[[2021 WSMO Speed Round/Problem 5|Solution]]
  
 
==Problem 6==
 
==Problem 6==
−
A bag weighs 1 pound and can hold 16 pounds of food at maximum. Danny buys 100 packages of tomatoes and 300 packages of potatoes. Tomatoes come in packages that are <math>12</math> ounces each and potatoes come in packages that are <math>24</math> ounces each. If all of Danny's food must go in bags, how many pounds does Danny's total luggage weigh, including the bags? (Danny will use only as many bags as he needs)
+
A bag weighs 1 pound and can hold 16 pounds of food at maximum. Danny buys 100 packages of tomatoes and 300 packages of potatoes. Tomatoes come in packages that are <math>12</math> ounces each and potatoes come in packages that are <math>24</math> ounces each. If all of Danny's food must go in bags, how many pounds does Danny's total luggage weigh, including the bags? (Danny will use only as many bags as he needs).
 +
 
 +
[[2021 WSMO Speed Round/Problem 6|Solution]]
  
 
==Problem 7==
 
==Problem 7==
 
Consider triangle <math>ABC</math> with side lengths <math>AB=13,AC=14,BC=15</math> and incircle <math>\omega</math>. A second circle <math>\omega_2</math> is drawn which is tangent to <math>AB,AC</math> and externally tangent to <math>\omega</math>. The radius of <math>\omega_2</math> can be expressed as <math>\frac{a-b\sqrt{c}}{d}</math>, where <math>\gcd{(a,b,d)}=1</math> and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c+d</math>.
 
Consider triangle <math>ABC</math> with side lengths <math>AB=13,AC=14,BC=15</math> and incircle <math>\omega</math>. A second circle <math>\omega_2</math> is drawn which is tangent to <math>AB,AC</math> and externally tangent to <math>\omega</math>. The radius of <math>\omega_2</math> can be expressed as <math>\frac{a-b\sqrt{c}}{d}</math>, where <math>\gcd{(a,b,d)}=1</math> and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c+d</math>.
 +
 +
[[2021 WSMO Speed Round/Problem 7|Solution]]
  
 
==Problem 8==
 
==Problem 8==
 
Let <math>n</math> be the number of ways to seat <math>12</math> distinguishable people around a regular hexagon such that rotations do not matter (but reflections do), and two people are seated on each side (the order in which they are seated matters). Find the number of divisors of <math>n</math>.
 
Let <math>n</math> be the number of ways to seat <math>12</math> distinguishable people around a regular hexagon such that rotations do not matter (but reflections do), and two people are seated on each side (the order in which they are seated matters). Find the number of divisors of <math>n</math>.
 +
 +
[[2021 WSMO Speed Round/Problem 8|Solution]]
  
 
==Problem 9==
 
==Problem 9==
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</asy>
 
</asy>
 
</center>
 
</center>
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[[2021 WSMO Speed Round/Problem 9|Solution]]
  
 
==Problem 10==
 
==Problem 10==
 
Find the remainder when <math>\underbrace{2021^{2022^{\ldots^{2022^{2021}}}}}_{2021\text{ } 2021\text{'}s}\cdot\underbrace{2022^{2021^{\ldots^{2021^{2022}}}}}_{2022\text{ }2022\text{'}s}</math> is divided by 11.
 
Find the remainder when <math>\underbrace{2021^{2022^{\ldots^{2022^{2021}}}}}_{2021\text{ } 2021\text{'}s}\cdot\underbrace{2022^{2021^{\ldots^{2021^{2022}}}}}_{2022\text{ }2022\text{'}s}</math> is divided by 11.
 +
 +
[[2021 WSMO Speed Round/Problem 10|Solution]]

Revision as of 16:07, 22 December 2021

Problem 1

Let $f^1(x)=(x-1)^2$, and let $f^n(x)=f^1(f^{n-1}(x))$. Find the value of $|f^7(2)|$.

Solution

Problem 2

A square with side length of $4$ units is rotated around one of its sides by $90^{\circ}$. If the volume the square sweeps out can be expressed as $m\pi$, find $m$.

Solution

Problem 3

Let $a@b=\frac{a^2-b^2}{a+b}$. Find the value of $1@(2@(\dots(2020@2021)\dots)$.

Solution

Problem 4

A square $ABCD$ with side length $10$ is placed inside of a right isosceles triangle $XYZ$ with $\angle XYZ=90^{\circ}$ such that $A$ and $B$ are on $XZ$, $C$ is on $YZ$, and $D$ is on $XY$. Find the area of $XYZ$.

Solution

Problem 5

The number of ways to arrange the characters in "delicious greenbeans" into two separate strings of letters can be expressed as $a\cdot b!,$ where $b$ is maximized and both $a$ and $b$ are positive integers. Find $a+b.$ (A string of letters is defined as a group of consecutive letters with no spaces between them.)

Solution

Problem 6

A bag weighs 1 pound and can hold 16 pounds of food at maximum. Danny buys 100 packages of tomatoes and 300 packages of potatoes. Tomatoes come in packages that are $12$ ounces each and potatoes come in packages that are $24$ ounces each. If all of Danny's food must go in bags, how many pounds does Danny's total luggage weigh, including the bags? (Danny will use only as many bags as he needs).

Solution

Problem 7

Consider triangle $ABC$ with side lengths $AB=13,AC=14,BC=15$ and incircle $\omega$. A second circle $\omega_2$ is drawn which is tangent to $AB,AC$ and externally tangent to $\omega$. The radius of $\omega_2$ can be expressed as $\frac{a-b\sqrt{c}}{d}$, where $\gcd{(a,b,d)}=1$ and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.

Solution

Problem 8

Let $n$ be the number of ways to seat $12$ distinguishable people around a regular hexagon such that rotations do not matter (but reflections do), and two people are seated on each side (the order in which they are seated matters). Find the number of divisors of $n$.

Solution

Problem 9

Bobby is going to throw 20 darts at the dartboard shown below. It is formed by 4 concentric circles, with radii of $1,3,5,$ and $7$, with the largest circle being inscribed in a square. Each point on the dartboard has an equally likely chance of being hit by a dart, and Bobby is guaranteed to hit the dartboard. Each region is labeled with its point value (the number of points Bobby will get if he hits that region). The expected number of points Bobby will get after throwing the 20 darts can be expressed as $\frac{m}{n}\pi,$ where $\gcd{(m,n)}=1$. Find $m+n.$\newline

[asy] size(5cm); filldraw(shift(-7,-7)*((0,0)--(14,0)--(14,14)--(0,14)--cycle),gray); filldraw(circle((0,0),7),white); filldraw(circle((0,0),5),white); filldraw(circle((0,0),3),white); filldraw(circle((0,0),1),white); label("$70$",(0,0)); label("$-28$",(0,2)); label("$49$",(0,4)); label("$-21$",(0,6)); label("$0$",(-6,6)); label("$0$",(6,6)); label("$0$",(-6,-6)); label("$0$",(6,-6)); [/asy]

Solution

Problem 10

Find the remainder when $\underbrace{2021^{2022^{\ldots^{2022^{2021}}}}}_{2021\text{ } 2021\text{'}s}\cdot\underbrace{2022^{2021^{\ldots^{2021^{2022}}}}}_{2022\text{ }2022\text{'}s}$ is divided by 11.

Solution

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