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2021 WSMO Accuracy Round Problems

Revision as of 11:36, 6 June 2022 by Pinkpig (talk | contribs)

Problem 1

Find the sum of all the positive integers $n$ such that $n$ is $\frac{2n^2-5n+5}{n-5}$ an integer.

Proposed by pinkpig

Solution

Problem 2

A fair 20-sided die has faces labeled with the numbers $1,3,6,\dots,210$. Find the expected value of a single roll of this die.

[i]Proposed by pinkpig[i]

Solution

Problem 3

If $f$ is a monic polynomial of minimal degree with rational coefficients satisfying $f\left(3+\sqrt{5}\right)=0$ and $f\left(4-\sqrt{7}\right)=0,$ find the value of $|f(1)|$.

[i]Proposed by pinkpig[i]

Solution

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 $m,$ the tip of the second hand travels a distance of $s,$ and the tip of the hour hand travels a distance of $h.$ The value of $\frac{m^2}{hs}$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$.

[i]Proposed by pinkpig[i]

Solution

Problem 5

Suppose regular octagon $ABCDEFGH$ has side length $5.$ If the distance from the center of the octagon to one of the sides 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 $a+b+c+d.$

[i]Proposed by mahaler[i]

Solution

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 $\frac{3}{4}$ 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 $128$ centimeters and its height is $10$ centimeters. If the surface area that is iced can be expressed as $m\pi,$ find $m.$

[i]Proposed by sanaops9[i]

Solution

Problem 7

Find the value of $\sum_{n=1}^{100}\left(\sum_{i=1}^{n}r_i\right),$ where $r_i$ is the remainder when $2^i+3^i$ is divided by 10.

[i]Proposed by pinkpig[i]

Solution

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 $\frac{a\sqrt{b}+c}{d},$ where $\gcd{(a,c,d)}=1$ and $b$ is not divisible by the square of any prime. Find $a+b+c+d.$

[i]Proposed by pinkpig[i]

Solution

Problem 9

Let $x=1+\frac{5}{2+\frac{3}{2+\frac{3}{2+\ldots}}}.$ If $\sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}$ can be written as $\frac{a+\sqrt{b}}{c},$ where $b$ is not divisible by the square of any prime, find $a+b+c.$

[i]Proposed by mahaler[i]

Solution

Problem 10

The largest value of $x$ that satisfies the equation $5x^2-7\lfloor x\rfloor\{x\}=\frac{26\lfloor x\rfloor^2}{5}$ can be expressed as $\frac{a+b\sqrt{c}}{d},$ where $c$ is not divisible by the square of any prime and $\gcd(a,b,d)=1.$ Find $a+b+c+d.$ ($\{x\}$ denotes the fractional part of $x$, or $x-\lfloor x\rfloor$.)

[i]Proposed by pinkpig[i]

Solution

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