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Difference between revisions of "2021 WSMO Team Round Problems"

(Created page with "==Problem 1== How many ways are there to pick a three-letter word using only letters from "Winter Solstice" such that the word is a capital letter followed by two lowercase le...")
 
 
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==Problem 1==
 
==Problem 1==
 
How many ways are there to pick a three-letter word using only letters from "Winter Solstice" such that the word is a capital letter followed by two lowercase letters? (A word does not have to be an English word, and the word can only use a letter in "Winter Solstice" as many times as it appears)
 
How many ways are there to pick a three-letter word using only letters from "Winter Solstice" such that the word is a capital letter followed by two lowercase letters? (A word does not have to be an English word, and the word can only use a letter in "Winter Solstice" as many times as it appears)
 +
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''Proposed by sanaops9''
  
 
[[2021 WSMO Team Round/Problem 1|Solution]]
 
[[2021 WSMO Team Round/Problem 1|Solution]]
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==Problem 2==
 
==Problem 2==
 
Bobby has some pencils. When he tries to split them into 5 equal groups, he has 2 left over. When he tries to split them into groups of 8, he has 6 left over. What is the second smallest number of pencils that Bobby could have?
 
Bobby has some pencils. When he tries to split them into 5 equal groups, he has 2 left over. When he tries to split them into groups of 8, he has 6 left over. What is the second smallest number of pencils that Bobby could have?
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''Proposed by pinkpig''
  
 
[[2021 WSMO Team Round/Problem 2|Solution]]
 
[[2021 WSMO Team Round/Problem 2|Solution]]
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==Problem 3==
 
==Problem 3==
 
Farmer Sam has <math>n</math> dollars. He knows that this is exactly enough to buy either 50 pounds of grass and 32 ounces of hay or 96 ounces of grass and 24 pounds of hay. However, he must save 4 dollars for tax. After some quick calculations, he finds that he has exactly enough to buy 18 pounds of grass and 16 pounds of hay (and still have money left over for tax!). Find <math>n.</math>
 
Farmer Sam has <math>n</math> dollars. He knows that this is exactly enough to buy either 50 pounds of grass and 32 ounces of hay or 96 ounces of grass and 24 pounds of hay. However, he must save 4 dollars for tax. After some quick calculations, he finds that he has exactly enough to buy 18 pounds of grass and 16 pounds of hay (and still have money left over for tax!). Find <math>n.</math>
 +
 +
''Proposed by pinkpig''
  
 
[[2021 WSMO Team Round/Problem 3|Solution]]
 
[[2021 WSMO Team Round/Problem 3|Solution]]
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==Problem 4==
 
==Problem 4==
 
Consider a triangle <math>A_1B_1C_1</math> satisfying <math>A_1B_1=3,B_1C_1=3\sqrt{3},A_1C_1=6</math>. For all successive triangles <math>A_nB_nC_n</math>, we have <math>A_nB_nC_n\sim B_{n-1}A_{n-1}C_{n-1}</math> and <math>A_n=B_{n-1},C_n=C_{n-1}</math>, where <math>A_nB_nC_n</math> is outside of <math>A_{n-1}B_{n-1}C_{n-1}</math>. Find the value of <cmath>\left(\sum_{i=1}^{\infty}[A_iB_iC_i]\right)^2,</cmath> where <math>[A_iB_iC_i]</math> is the area of <math>A_iB_iC_i</math>.
 
Consider a triangle <math>A_1B_1C_1</math> satisfying <math>A_1B_1=3,B_1C_1=3\sqrt{3},A_1C_1=6</math>. For all successive triangles <math>A_nB_nC_n</math>, we have <math>A_nB_nC_n\sim B_{n-1}A_{n-1}C_{n-1}</math> and <math>A_n=B_{n-1},C_n=C_{n-1}</math>, where <math>A_nB_nC_n</math> is outside of <math>A_{n-1}B_{n-1}C_{n-1}</math>. Find the value of <cmath>\left(\sum_{i=1}^{\infty}[A_iB_iC_i]\right)^2,</cmath> where <math>[A_iB_iC_i]</math> is the area of <math>A_iB_iC_i</math>.
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''Proposed by pinkpig''
  
 
[[2021 WSMO Team Round/Problem 4|Solution]]
 
[[2021 WSMO Team Round/Problem 4|Solution]]
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==Problem 5==
 
==Problem 5==
 
Two runners are running at different speeds. The first runner runs at a consistent 12 miles per hour. The second runner runs at <math>t+4</math> miles per hour, where <math>t</math> is the number of hours that have passed. After <math>n</math> hours, the runners have run the same distance, where <math>n</math> is positive. Find <math>n</math>.
 
Two runners are running at different speeds. The first runner runs at a consistent 12 miles per hour. The second runner runs at <math>t+4</math> miles per hour, where <math>t</math> is the number of hours that have passed. After <math>n</math> hours, the runners have run the same distance, where <math>n</math> is positive. Find <math>n</math>.
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''Proposed by pinkpig''
  
 
[[2021 WSMO Team Round/Problem 5|Solution]]
 
[[2021 WSMO Team Round/Problem 5|Solution]]
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</asy>
 
</asy>
 
</center>
 
</center>
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''Proposed by mahaler''
  
 
[[2021 WSMO Team Round/Problem 6|Solution]]
 
[[2021 WSMO Team Round/Problem 6|Solution]]
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==Problem 7==
 
==Problem 7==
 
A frog makes one hop every minute on the first quadrant of the coordinate plane (this means that the frog's <math>x</math> and <math>y</math> coordinates are positive). The frog can hop up one unit, right one unit, left one unit, down one unit, or it can stay in place, and will always randomly choose a valid hop from these 5 directions (a valid hop is a hop that does not place the frog outside the first quadrant). Given that the frog starts at <math>(1,1)</math>, the expected number of minutes until the frog reaches the line <math>x+y=5</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>.
 
A frog makes one hop every minute on the first quadrant of the coordinate plane (this means that the frog's <math>x</math> and <math>y</math> coordinates are positive). The frog can hop up one unit, right one unit, left one unit, down one unit, or it can stay in place, and will always randomly choose a valid hop from these 5 directions (a valid hop is a hop that does not place the frog outside the first quadrant). Given that the frog starts at <math>(1,1)</math>, the expected number of minutes until the frog reaches the line <math>x+y=5</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>.
 +
 +
''Proposed by asdf334''
  
 
[[2021 WSMO Team Round/Problem 7|Solution]]
 
[[2021 WSMO Team Round/Problem 7|Solution]]
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==Problem 8==
 
==Problem 8==
 
Isaac, Gottfried, Carl, Euclid, Albert, Srinivasa, René, Adihaya, and Euler sit around a round table (not necessarily in that order). Then, Hypatia takes a seat. There are <math>a\cdot b!</math> possible seatings where Euler doesn't sit next to Hypatia and Isaac doesn't sit next to Gottfried, where <math>b</math> is maximized. Find <math>a+b</math>. (Rotations are not distinct, but reflections are).
 
Isaac, Gottfried, Carl, Euclid, Albert, Srinivasa, René, Adihaya, and Euler sit around a round table (not necessarily in that order). Then, Hypatia takes a seat. There are <math>a\cdot b!</math> possible seatings where Euler doesn't sit next to Hypatia and Isaac doesn't sit next to Gottfried, where <math>b</math> is maximized. Find <math>a+b</math>. (Rotations are not distinct, but reflections are).
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''Proposed by mahaler''
  
 
[[2021 WSMO Team Round/Problem 8|Solution]]
 
[[2021 WSMO Team Round/Problem 8|Solution]]
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==Problem 9==
 
==Problem 9==
 
In triangle <math>ABC,</math> points <math>D</math> and <math>E</math> trisect side <math>BC</math> such that <math>D</math> is closer to <math>C</math> than <math>E.</math> If <math>\angle{CAD}=\angle{EAD},ED=3,</math> and <math>[AEB]=6,</math> then find <math>[ABC],</math> where <math>[ABC]</math> is the area of <math>ABC</math>.
 
In triangle <math>ABC,</math> points <math>D</math> and <math>E</math> trisect side <math>BC</math> such that <math>D</math> is closer to <math>C</math> than <math>E.</math> If <math>\angle{CAD}=\angle{EAD},ED=3,</math> and <math>[AEB]=6,</math> then find <math>[ABC],</math> where <math>[ABC]</math> is the area of <math>ABC</math>.
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''Proposed by pinkpig''
  
 
[[2021 WSMO Team Round/Problem 9|Solution]]
 
[[2021 WSMO Team Round/Problem 9|Solution]]
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==Problem 10==
 
==Problem 10==
 
The minimum possible value of <cmath>\sqrt{m^2+n^2}+\sqrt{3m^2+3n^2-6m+12n+15}</cmath> can be expressed as <math>a.</math> Find <math>a^2.</math>
 
The minimum possible value of <cmath>\sqrt{m^2+n^2}+\sqrt{3m^2+3n^2-6m+12n+15}</cmath> can be expressed as <math>a.</math> Find <math>a^2.</math>
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''Proposed by pinkpig''
  
 
[[2021 WSMO Team Round/Problem 10|Solution]]
 
[[2021 WSMO Team Round/Problem 10|Solution]]
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==Problem 11==
 
==Problem 11==
 
Find the remainder when <cmath>\sum_{x+y+z\leq10}\frac{(x+y+z)!}{x!y!z!}</cmath> is divided by <math>100</math>. (<math>x,y,z\geq 0</math>)
 
Find the remainder when <cmath>\sum_{x+y+z\leq10}\frac{(x+y+z)!}{x!y!z!}</cmath> is divided by <math>100</math>. (<math>x,y,z\geq 0</math>)
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''Proposed by pinkpig''
  
 
[[2021 WSMO Team Round/Problem 11|Solution]]
 
[[2021 WSMO Team Round/Problem 11|Solution]]
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==Problem 12==
 
==Problem 12==
 
Choose three integers <math>x,y,z</math> randomly and independently from the nonnegative integers. The probability that the sum of the factors of <math>2^x3^y5^z</math> is divisible by <math>6</math> is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 
Choose three integers <math>x,y,z</math> randomly and independently from the nonnegative integers. The probability that the sum of the factors of <math>2^x3^y5^z</math> is divisible by <math>6</math> is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 +
 +
''Proposed by pinkpig''
  
 
[[2021 WSMO Team Round/Problem 12|Solution]]
 
[[2021 WSMO Team Round/Problem 12|Solution]]
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==Problem 13==
 
==Problem 13==
 
Square <math>BCDE</math> is drawn outside of equilateral triangle <math>ABC.</math> Regular hexagon <math>DEFGHI</math> is drawn outside of square <math>BCDE.</math> If the area of triangle <math>AED</math> is 3, then the area of triangle <math>AGH</math> can be expressed as <math>a\sqrt{b}-c,</math> where <math>b</math> is not divisible by the square of any prime. Find <math>a+b+c.</math>
 
Square <math>BCDE</math> is drawn outside of equilateral triangle <math>ABC.</math> Regular hexagon <math>DEFGHI</math> is drawn outside of square <math>BCDE.</math> If the area of triangle <math>AED</math> is 3, then the area of triangle <math>AGH</math> can be expressed as <math>a\sqrt{b}-c,</math> where <math>b</math> is not divisible by the square of any prime. Find <math>a+b+c.</math>
 +
 +
''Proposed by pinkpig''
  
 
[[2021 WSMO Team Round/Problem 13|Solution]]
 
[[2021 WSMO Team Round/Problem 13|Solution]]
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==Problem 14==
 
==Problem 14==
 
Suppose that <math>x</math> is a complex number such that <math>x+\frac{1}{x}=\frac{\sqrt{6}+\sqrt{2}}{2}</math> and the imaginary part of <math>x</math> is nonnegative. Find the sum of the five smallest nonnegative integers <math>n</math> such that <math>x^{n}+\frac{1}{x^n}</math> is an integer.
 
Suppose that <math>x</math> is a complex number such that <math>x+\frac{1}{x}=\frac{\sqrt{6}+\sqrt{2}}{2}</math> and the imaginary part of <math>x</math> is nonnegative. Find the sum of the five smallest nonnegative integers <math>n</math> such that <math>x^{n}+\frac{1}{x^n}</math> is an integer.
 +
 +
''Proposed by pinkpig''
  
 
[[2021 WSMO Team Round/Problem 14|Solution]]
 
[[2021 WSMO Team Round/Problem 14|Solution]]
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==Problem 15==
 
==Problem 15==
 
Let <math>ABCD</math> and <math>DEFG</math> (vertices labelled clockwise) be squares that intersect exactly once and with areas <math>1011^2</math> and <math>69^2</math> respectively. There exists a constant <math>M</math> such that <math>CE+AG>M</math> where <math>M</math> is maximized. Find <math>M.</math>
 
Let <math>ABCD</math> and <math>DEFG</math> (vertices labelled clockwise) be squares that intersect exactly once and with areas <math>1011^2</math> and <math>69^2</math> respectively. There exists a constant <math>M</math> such that <math>CE+AG>M</math> where <math>M</math> is maximized. Find <math>M.</math>
 +
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''Proposed by MathLuis''
  
 
[[2021 WSMO Team Round/Problem 15|Solution]]
 
[[2021 WSMO Team Round/Problem 15|Solution]]

Latest revision as of 12:40, 6 June 2022

Problem 1

How many ways are there to pick a three-letter word using only letters from "Winter Solstice" such that the word is a capital letter followed by two lowercase letters? (A word does not have to be an English word, and the word can only use a letter in "Winter Solstice" as many times as it appears)

Proposed by sanaops9

Solution

Problem 2

Bobby has some pencils. When he tries to split them into 5 equal groups, he has 2 left over. When he tries to split them into groups of 8, he has 6 left over. What is the second smallest number of pencils that Bobby could have?

Proposed by pinkpig

Solution

Problem 3

Farmer Sam has $n$ dollars. He knows that this is exactly enough to buy either 50 pounds of grass and 32 ounces of hay or 96 ounces of grass and 24 pounds of hay. However, he must save 4 dollars for tax. After some quick calculations, he finds that he has exactly enough to buy 18 pounds of grass and 16 pounds of hay (and still have money left over for tax!). Find $n.$

Proposed by pinkpig

Solution

Problem 4

Consider a triangle $A_1B_1C_1$ satisfying $A_1B_1=3,B_1C_1=3\sqrt{3},A_1C_1=6$. For all successive triangles $A_nB_nC_n$, we have $A_nB_nC_n\sim B_{n-1}A_{n-1}C_{n-1}$ and $A_n=B_{n-1},C_n=C_{n-1}$, where $A_nB_nC_n$ is outside of $A_{n-1}B_{n-1}C_{n-1}$. Find the value of \[\left(\sum_{i=1}^{\infty}[A_iB_iC_i]\right)^2,\] where $[A_iB_iC_i]$ is the area of $A_iB_iC_i$.

Proposed by pinkpig

Solution

Problem 5

Two runners are running at different speeds. The first runner runs at a consistent 12 miles per hour. The second runner runs at $t+4$ miles per hour, where $t$ is the number of hours that have passed. After $n$ hours, the runners have run the same distance, where $n$ is positive. Find $n$.

Proposed by pinkpig

Solution

Problem 6

Suppose that regular dodecagon $ABCDEFGHIJKL$ has side length $5.$ The area of the shaded region can be expressed as $a+b\sqrt{c},$ where $c$ is not divisible by the square of any prime. Find $a+b+c$.

[asy] size(150); filldraw(polygon(12),grey); filldraw(rotate(75)*(dir(60)--dir(150)--dir(240)--dir(330)--cycle),white); for(int i=30; i<=360; i+=30){ dot(rotate(75)*dir(i)); } label("$C$",dir(45),NE); label("$B$",dir(75),N); label("$A$",dir(105),N); label("$L$",dir(135),NW); label("$K$",dir(165),W); label("$J$",dir(195),W); label("$I$",dir(225),SW); label("$H$",dir(255),S); label("$G$",dir(285),S); label("$F$",dir(315),SE); label("$E$",dir(345),E); label("$D$",dir(375),E); [/asy]

Proposed by mahaler

Solution

Problem 7

A frog makes one hop every minute on the first quadrant of the coordinate plane (this means that the frog's $x$ and $y$ coordinates are positive). The frog can hop up one unit, right one unit, left one unit, down one unit, or it can stay in place, and will always randomly choose a valid hop from these 5 directions (a valid hop is a hop that does not place the frog outside the first quadrant). Given that the frog starts at $(1,1)$, the expected number of minutes until the frog reaches the line $x+y=5$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Proposed by asdf334

Solution

Problem 8

Isaac, Gottfried, Carl, Euclid, Albert, Srinivasa, René, Adihaya, and Euler sit around a round table (not necessarily in that order). Then, Hypatia takes a seat. There are $a\cdot b!$ possible seatings where Euler doesn't sit next to Hypatia and Isaac doesn't sit next to Gottfried, where $b$ is maximized. Find $a+b$. (Rotations are not distinct, but reflections are).

Proposed by mahaler

Solution

Problem 9

In triangle $ABC,$ points $D$ and $E$ trisect side $BC$ such that $D$ is closer to $C$ than $E.$ If $\angle{CAD}=\angle{EAD},ED=3,$ and $[AEB]=6,$ then find $[ABC],$ where $[ABC]$ is the area of $ABC$.

Proposed by pinkpig

Solution

Problem 10

The minimum possible value of \[\sqrt{m^2+n^2}+\sqrt{3m^2+3n^2-6m+12n+15}\] can be expressed as $a.$ Find $a^2.$

Proposed by pinkpig

Solution

Problem 11

Find the remainder when \[\sum_{x+y+z\leq10}\frac{(x+y+z)!}{x!y!z!}\] is divided by $100$. ($x,y,z\geq 0$)

Proposed by pinkpig

Solution

Problem 12

Choose three integers $x,y,z$ randomly and independently from the nonnegative integers. The probability that the sum of the factors of $2^x3^y5^z$ is divisible by $6$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Proposed by pinkpig

Solution

Problem 13

Square $BCDE$ is drawn outside of equilateral triangle $ABC.$ Regular hexagon $DEFGHI$ is drawn outside of square $BCDE.$ If the area of triangle $AED$ is 3, then the area of triangle $AGH$ can be expressed as $a\sqrt{b}-c,$ where $b$ is not divisible by the square of any prime. Find $a+b+c.$

Proposed by pinkpig

Solution

Problem 14

Suppose that $x$ is a complex number such that $x+\frac{1}{x}=\frac{\sqrt{6}+\sqrt{2}}{2}$ and the imaginary part of $x$ is nonnegative. Find the sum of the five smallest nonnegative integers $n$ such that $x^{n}+\frac{1}{x^n}$ is an integer.

Proposed by pinkpig

Solution

Problem 15

Let $ABCD$ and $DEFG$ (vertices labelled clockwise) be squares that intersect exactly once and with areas $1011^2$ and $69^2$ respectively. There exists a constant $M$ such that $CE+AG>M$ where $M$ is maximized. Find $M.$

Proposed by MathLuis

Solution

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