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| − | ==Problem 1==
| + | '''2021 WSMO Accuracy Round''' problems and solutions. '''This test was held on ZOOM on 12/21/2021 at 6:40 PM EST.''' The first link contains the full set of test problems. The rest contain each individual problem and its solution. |
| − | 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.
| + | *[[2021 WSMO Accuracy Round Problems]] |
| − | | + | *[[2021 WSMO Accuracy Round Answer Key]] |
| − | [[2021 WSMO Accuracy Round/Problem 1|Solution]] | + | **[[2021 WSMO Accuracy Round/Problem 1|Problem 1]] |
| − | | + | **[[2021 WSMO Accuracy Round/Problem 2|Problem 2]] |
| − | ==Problem 2==
| + | **[[2021 WSMO Accuracy Round/Problem 3|Problem 3]] |
| − | 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.
| + | **[[2021 WSMO Accuracy Round/Problem 4|Problem 4]] |
| − | | + | **[[2021 WSMO Accuracy Round/Problem 5|Problem 5]] |
| − | [[2021 WSMO Accuracy Round/Problem 2|Solution]] | + | **[[2021 WSMO Accuracy Round/Problem 6|Problem 6]] |
| − | | + | **[[2021 WSMO Accuracy Round/Problem 7|Problem 7]] |
| − | ==Problem 3==
| + | **[[2021 WSMO Accuracy Round/Problem 8|Problem 8]] |
| − | 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>.
| + | **[[2021 WSMO Accuracy Round/Problem 9|Problem 9]] |
| − | | + | **[[2021 WSMO Accuracy Round/Problem 10|Problem 10]] |
| − | [[2021 WSMO Accuracy Round/Problem 3|Solution]] | |
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| − | ==Problem 4==
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| − | 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]] | |
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| − | ==Problem 5==
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| − | 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]] | |
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| − | ==Problem 6==
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| − | 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]] | |
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| − | ==Problem 7==
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| − | 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]] | |
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| − | ==Problem 8==
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| − | 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]] | |
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| − | ==Problem 9==
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| − | Let <math>x=1+\frac{5}{2+\frac{3}{2+\frac{3}{2+\ldots}}}.</math>
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| − | 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]] | |
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| − | ==Problem 10==
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| − | 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]] | |
