Difference between revisions of "2021 WSMO Team Round"
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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) | ||
+ | [[2021 WSMO Team Round/Problem 1|Solution]] | ||
==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? | ||
+ | [[2021 WSMO Team Round/Problem 2|Solution]] | ||
==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> | ||
+ | [[2021 WSMO Team Round/Problem 3|Solution]] | ||
==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>. | ||
+ | [[2021 WSMO Team Round/Problem 4|Solution]] | ||
==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>. | ||
+ | [[2021 WSMO Team Round/Problem 5|Solution]] | ||
==Problem 6== | ==Problem 6== | ||
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</asy> | </asy> | ||
</center> | </center> | ||
+ | [[2021 WSMO Team Round/Problem 6|Solution]] | ||
==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>. | ||
+ | [[2021 WSMO Team Round/Problem 7|Solution]] | ||
==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) | ||
+ | [[2021 WSMO Team Round/Problem 8|Solution]] | ||
==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>. | ||
+ | [[2021 WSMO Team Round/Problem 9|Solution]] | ||
==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> | ||
+ | [[2021 WSMO Team Round/Problem 10|Solution]] | ||
==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>) | ||
+ | [[2021 WSMO Team Round/Problem 11|Solution]] | ||
==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>. | ||
+ | [[2021 WSMO Team Round/Problem 12|Solution]] | ||
==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> | ||
+ | [[2021 WSMO Team Round/Problem 13|Solution]] | ||
==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. | ||
+ | [[2021 WSMO Team Round/Problem 14|Solution]] | ||
==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> | ||
+ | [[2021 WSMO Team Round/Problem 15|Solution]] |
Revision as of 17:04, 22 December 2021
Contents
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) 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? Solution
Problem 3
Farmer Sam has 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
Solution
Problem 4
Consider a triangle satisfying
. For all successive triangles
, we have
and
, where
is outside of
. Find the value of
where
is the area of
.
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 miles per hour, where
is the number of hours that have passed. After
hours, the runners have run the same distance, where
is positive. Find
.
Solution
Problem 6
Suppose that regular dodecagon has side length
The area of the shaded region can be expressed as
where
is not divisible by the square of any prime. Find
.
Problem 7
A frog makes one hop every minute on the first quadrant of the coordinate plane (this means that the frog's and
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
, the expected number of minutes until the frog reaches the line
can be expressed as
, where
and
are relatively prime positive integers. Find
.
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 possible seatings where Euler doesn't sit next to Hypatia and Isaac doesn't sit next to Gottfried, where
is maximized. Find
. (Rotations are not distinct, but reflections are)
Solution
Problem 9
In triangle points
and
trisect side
such that
is closer to
than
If
and
then find
where
is the area of
.
Solution
Problem 10
The minimum possible value of can be expressed as
Find
Solution
Problem 11
Find the remainder when is divided by
. (
)
Solution
Problem 12
Choose three integers randomly and independently from the nonnegative integers. The probability that the sum of the factors of
is divisible by
is
, where
and
are relatively prime positive integers. Find
.
Solution
Problem 13
Square is drawn outside of equilateral triangle
Regular hexagon
is drawn outside of square
If the area of triangle
is 3, then the area of triangle
can be expressed as
where
is not divisible by the square of any prime. Find
Solution
Problem 14
Suppose that is a complex number such that
and the imaginary part of
is nonnegative. Find the sum of the five smallest nonnegative integers
such that
is an integer.
Solution
Problem 15
Let and
(vertices labelled clockwise) be squares that intersect exactly once and with areas
and
respectively. There exists a constant
such that
where
is maximized. Find
Solution