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Difference between revisions of "2021 WSMO Speed Round Problems"

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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>.
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''Proposed by pinkpig''
  
 
[[2021 WSMO Speed Round Problems/Problem 1|Solution]]
 
[[2021 WSMO Speed Round Problems/Problem 1|Solution]]
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==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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''Proposed by mahaler''
  
 
[[2021 WSMO Speed Round Problems/Problem 2|Solution]]
 
[[2021 WSMO Speed Round Problems/Problem 2|Solution]]
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==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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''Proposed by asimov''
  
 
[[2021 WSMO Speed Round Problems/Problem 3|Solution]]
 
[[2021 WSMO Speed Round Problems/Problem 3|Solution]]
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==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>.
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''Proposed by asdf334''
  
 
[[2021 WSMO Speed Round Problems/Problem 4|Solution]]
 
[[2021 WSMO Speed Round Problems/Problem 4|Solution]]
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==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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''Proposed by pinkpig''
  
 
[[2021 WSMO Speed Round Problems/Problem 5|Solution]]
 
[[2021 WSMO Speed Round Problems/Problem 5|Solution]]
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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? (Note that Danny will use only as many bags as he needs and that packages have to stay together).
 
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? (Note that Danny will use only as many bags as he needs and that packages have to stay together).
  
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''Proposed by pinkpig''
  
 
[[2021 WSMO Speed Round Problems/Problem 6|Solution]]
 
[[2021 WSMO Speed Round Problems/Problem 6|Solution]]
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==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>.
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''Proposed by pinkpig''
  
 
[[2021 WSMO Speed Round Problems/Problem 7|Solution]]
 
[[2021 WSMO Speed Round Problems/Problem 7|Solution]]
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==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>.
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''Proposed by captainnobody''
  
 
[[2021 WSMO Speed Round Problems/Problem 8|Solution]]
 
[[2021 WSMO Speed Round Problems/Problem 8|Solution]]
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</asy>
 
</asy>
 
</center>
 
</center>
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''Proposed by pinkpig''
  
 
[[2021 WSMO Speed Round Problems/Problem 9|Solution]]
 
[[2021 WSMO Speed Round Problems/Problem 9|Solution]]
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==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.
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''Proposed by pinkpig''
  
 
[[2021 WSMO Speed Round Problems/Problem 10|Solution]]
 
[[2021 WSMO Speed Round Problems/Problem 10|Solution]]

Latest revision as of 11:38, 6 June 2022

Problem 1

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

Proposed by pinkpig

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 $m$.

Proposed by mahaler

Solution

Problem 3

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

Proposed by asimov

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 $XYZ$.

Proposed by asdf334

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.)

Proposed by pinkpig

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? (Note that Danny will use only as many bags as he needs and that packages have to stay together).

Proposed by pinkpig

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 $a+b+c+d$.

Proposed by pinkpig

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 $n$.

Proposed by captainnobody

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]

Proposed by pinkpig

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.

Proposed by pinkpig

Solution

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