Difference between revisions of "2021 WSMO Accuracy Round"
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==Problem 1== | ==Problem 1== | ||
Find the sum of all the positive integers <math>n</math> such that <math>n</math> is <math>\frac{2n^2-5n+5}{n-5}</math> an integer. | Find the sum of all the positive integers <math>n</math> such that <math>n</math> is <math>\frac{2n^2-5n+5}{n-5}</math> an integer. | ||
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| + | [[2021 WSMO Accuracy Round/Problem 1|Solution]] | ||
==Problem 2== | ==Problem 2== | ||
A fair 20-sided die has faces labeled with the numbers <math>1,3,6,\dots,210</math>. Find the expected value of a single roll of this die. | A fair 20-sided die has faces labeled with the numbers <math>1,3,6,\dots,210</math>. Find the expected value of a single roll of this die. | ||
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| + | [[2021 WSMO Accuracy Round/Problem 2|Solution]] | ||
==Problem 3== | ==Problem 3== | ||
If <math>f</math> is a monic polynomial of minimal degree with rational coefficients satisfying <math>f\left(3+\sqrt{5}\right)=0</math> and <math>f\left(4-\sqrt{7}\right)=0,</math> find the value of <math>|f(1)|</math>. | If <math>f</math> is a monic polynomial of minimal degree with rational coefficients satisfying <math>f\left(3+\sqrt{5}\right)=0</math> and <math>f\left(4-\sqrt{7}\right)=0,</math> find the value of <math>|f(1)|</math>. | ||
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| + | [[2021 WSMO Accuracy Round/Problem 3|Solution]] | ||
==Problem 4== | ==Problem 4== | ||
A 12-hour clock has a minute hand that is the same length as the second hand, and an hour hand half the length of the minute hand. In a day, the tip of the minute hand travels a distance of <math>m,</math> the tip of the second hand travels a distance of <math>s,</math> and the tip of the hour hand travels a distance of <math>h.</math> The value of <math>\frac{m^2}{hs}</math> can be expressed as <math>\frac{a}{b}</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers. Find <math>a+b</math>. | A 12-hour clock has a minute hand that is the same length as the second hand, and an hour hand half the length of the minute hand. In a day, the tip of the minute hand travels a distance of <math>m,</math> the tip of the second hand travels a distance of <math>s,</math> and the tip of the hour hand travels a distance of <math>h.</math> The value of <math>\frac{m^2}{hs}</math> can be expressed as <math>\frac{a}{b}</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers. Find <math>a+b</math>. | ||
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| + | [[2021 WSMO Accuracy Round/Problem 4|Solution]] | ||
==Problem 5== | ==Problem 5== | ||
Suppose regular octagon <math>ABCDEFGH</math> has side length <math>5.</math> If the distance from the center of the octagon to one of the sides 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> | Suppose regular octagon <math>ABCDEFGH</math> has side length <math>5.</math> If the distance from the center of the octagon to one of the sides 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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| + | [[2021 WSMO Accuracy Round/Problem 5|Solution]] | ||
==Problem 6== | ==Problem 6== | ||
Roy is baking a circular three tier cake. All of the tiers are centered around the same point. Each tier's radius is <math>\frac{3}{4}</math> of the radius of the tier below it, but the height of each tier stays constant. Roy wants to ice the cake, but only on the curved surfaces of the cake and the top of the smallest tier. The diameter of the lowest tier is <math>128</math> centimeters and its height is <math>10</math> centimeters. If the surface area that is iced can be expressed as <math>m\pi,</math> find <math>m.</math> | Roy is baking a circular three tier cake. All of the tiers are centered around the same point. Each tier's radius is <math>\frac{3}{4}</math> of the radius of the tier below it, but the height of each tier stays constant. Roy wants to ice the cake, but only on the curved surfaces of the cake and the top of the smallest tier. The diameter of the lowest tier is <math>128</math> centimeters and its height is <math>10</math> centimeters. If the surface area that is iced can be expressed as <math>m\pi,</math> find <math>m.</math> | ||
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| + | [[2021 WSMO Accuracy Round/Problem 6|Solution]] | ||
==Problem 7== | ==Problem 7== | ||
Find the value of <math>\sum_{n=1}^{100}\left(\sum_{i=1}^{n}r_i\right),</math> where <math>r_i</math> is the remainder when <math>2^i+3^i</math> is divided by 10. | Find the value of <math>\sum_{n=1}^{100}\left(\sum_{i=1}^{n}r_i\right),</math> where <math>r_i</math> is the remainder when <math>2^i+3^i</math> is divided by 10. | ||
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| + | [[2021 WSMO Accuracy Round/Problem 7|Solution]] | ||
==Problem 8== | ==Problem 8== | ||
20 unit spheres are stacked in a triangular pyramid formation, such that the first layer has 1 sphere, the second layer has 3 spheres, the third layer has 6 spheres, and the fourth layer has 10 spheres. The radius of the smallest sphere that fully contains all of these spheres is <math>\frac{a\sqrt{b}+c}{d},</math> where <math>\gcd{(a,c,d)}=1</math> and <math>b</math> is not divisible by the square of any prime. Find <math>a+b+c+d.</math> | 20 unit spheres are stacked in a triangular pyramid formation, such that the first layer has 1 sphere, the second layer has 3 spheres, the third layer has 6 spheres, and the fourth layer has 10 spheres. The radius of the smallest sphere that fully contains all of these spheres is <math>\frac{a\sqrt{b}+c}{d},</math> where <math>\gcd{(a,c,d)}=1</math> and <math>b</math> is not divisible by the square of any prime. Find <math>a+b+c+d.</math> | ||
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| + | [[2021 WSMO Accuracy Round/Problem 8|Solution]] | ||
==Problem 9== | ==Problem 9== | ||
Let <math>x=1+\frac{5}{2+\frac{3}{2+\frac{3}{2+\ldots}}}.</math> | Let <math>x=1+\frac{5}{2+\frac{3}{2+\frac{3}{2+\ldots}}}.</math> | ||
If <math>\sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}</math> can be written as <math>\frac{a+\sqrt{b}}{c},</math> where <math>b</math> is not divisible by the square of any prime, find <math>a+b+c.</math> | If <math>\sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}</math> can be written as <math>\frac{a+\sqrt{b}}{c},</math> where <math>b</math> is not divisible by the square of any prime, find <math>a+b+c.</math> | ||
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| + | [[2021 WSMO Accuracy Round/Problem 9|Solution]] | ||
==Problem 10== | ==Problem 10== | ||
The largest value of <math>x</math> that satisfies the equation <math>5x^2-7\lfloor x\rfloor\{x\}=\frac{26\lfloor x\rfloor^2}{5}</math> can be expressed as <math>\frac{a+b\sqrt{c}}{d},</math> where <math>c</math> is not divisible by the square of any prime and <math>\gcd(a,b,d)=1.</math> Find <math>a+b+c+d.</math> (<math>\{x\}</math> denotes the fractional part of <math>x</math>, or <math>x-\lfloor x\rfloor</math>.) | The largest value of <math>x</math> that satisfies the equation <math>5x^2-7\lfloor x\rfloor\{x\}=\frac{26\lfloor x\rfloor^2}{5}</math> can be expressed as <math>\frac{a+b\sqrt{c}}{d},</math> where <math>c</math> is not divisible by the square of any prime and <math>\gcd(a,b,d)=1.</math> Find <math>a+b+c+d.</math> (<math>\{x\}</math> denotes the fractional part of <math>x</math>, or <math>x-\lfloor x\rfloor</math>.) | ||
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| + | [[2021 WSMO Accuracy Round/Problem 10|Solution]] | ||
Revision as of 16:09, 22 December 2021
Contents
Problem 1
Find the sum of all the positive integers 
such that is 
an integer.
Problem 2
A fair 20-sided die has faces labeled with the numbers 
. Find the expected value of a single roll of this die.
Problem 3
If 
is a monic polynomial of minimal degree with rational coefficients satisfying 
and 
find the value of 
.
Problem 4
A 12-hour clock has a minute hand that is the same length as the second hand, and an hour hand half the length of the minute hand. In a day, the tip of the minute hand travels a distance of 
the tip of the second hand travels a distance of 
and the tip of the hour hand travels a distance of 
The value of 
can be expressed as 
, where 
and 
are relatively prime positive integers. Find 
.
Problem 5
Suppose regular octagon 
has side length 
If the distance from the center of the octagon to one of the sides can be expressed as 
where 
and 
is not divisible by the square of any prime, find 
Problem 6
Roy is baking a circular three tier cake. All of the tiers are centered around the same point. Each tier's radius is 
of the radius of the tier below it, but the height of each tier stays constant. Roy wants to ice the cake, but only on the curved surfaces of the cake and the top of the smallest tier. The diameter of the lowest tier is 
centimeters and its height is 
centimeters. If the surface area that is iced can be expressed as 
find 
Problem 7
Find the value of 
where 
is the remainder when 
is divided by 10.
Problem 8
20 unit spheres are stacked in a triangular pyramid formation, such that the first layer has 1 sphere, the second layer has 3 spheres, the third layer has 6 spheres, and the fourth layer has 10 spheres. The radius of the smallest sphere that fully contains all of these spheres is 
where 
and is not divisible by the square of any prime. Find
Problem 9
Let 
If 
can be written as 
where is not divisible by the square of any prime, find 
Problem 10
The largest value of 
that satisfies the equation 
can be expressed as 
where is not divisible by the square of any prime and 
Find (
denotes the fractional part of , or 
.)
