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2023 WSMO Accuracy Round Problems

Problem 1

Let $x = \sqrt{69+\sqrt{69+\sqrt{69\dots}}}.$ Find the value of $(2x-1)^2.$

Solution

Problem 2

When Bob is in precalculus, he gets bored and writes all the permutations in "precal". Since he is not very smart, it takes him 5 seconds to write each permutation. When Bob advances to calculus, he gets bored and writes all the permutations in "calculus". He is smart and can now write each permutation in 2 seconds. Find the positive difference in minutes between the time it takes for him to write the permutations of "precal" and "calculus".

Solution

Problem 3

$f(x)=x^3-8x^2+10x-4$ has complex roots $a,b,c$. Denote $P(n) = a^n+b^n+c^n.$ Find $P(-1)P(0)P(1).$

Solution

Problem 4

Bob and his 3 friends are standing in a line of 10 people. Given that Bob is not on either end of the line, then the probability the person in front and behind Bob are both his friends is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$

Solution

Problem 5

Bob flips $x+1$ coins and Bobby flips $x$ coins, where $x$ is a random integer chosen between the range of $[27,100].$ The expected probability that Bob gets more heads than Bobby is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n$.

Solution

Problem 6

In quadrilateral $ABCD,$ there exists a point $O$ such that $AO = BO = CO = DO$ and $\angle(AOB)+\angle(COD) = 120^{\circ}.$ Let $K,L,M,N$ be the foot of the perpendiculars from $A$ to $BD,$ $B$ to $AC,$ $C$ to $BD,$ and $D$ to $AC.$ If $[ABCD] = 20,$ find $\left([KLMN]\right)^2.$

Solution

Problem 7

How many ordered triplets of integers $(a, b, c)$ satisfy $a^2 + 2ab + b^2 = c^2 - 6c + 9$ and $-2 \le a, b, c \le 7$?

Solution

Problem 8

Let $f(x)=x^3-3x^2+4x+5$ have complex roots $a,b,c$. Then, the value of $\frac{1}{a^2+b^2}+\frac{1}{a^2+c^2}+\frac{1}{b^2+c^2}$ is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n$

Solution

Problem 9

Given circles $\omega_1, \omega_2$ with radius $1,4$ respectively, they are externally tangent to each other. The diameters of $\omega_1$ , $\omega_2$ are $AB,CD$ respectively, satisfying $AB\parallel CD$ and $BD$ is an external tangent of the circles. The third circle $\omega_3$ passes through $A,C$ and is tangent to $BD$. If the minimum possible value of the radius of $\omega_3$ is $\frac{a+b\sqrt{c}}{d}$, where $\gcd(a,b,d) = 1,$ $a$ is positive, and $c$ is squarefree, find $a+b+c+d.$

Solution

Problem 10

In tetrahedron $T$ of side length $12,$ let $S_1$ be the sphere inscribed in $T$ and let $S_2$ be the sphere circumscribed around $T.$ Let $R$ be a rectangular prism such that all points on $S_1$ lie strictly inside or are touching $R$ and all points on $R$ lie strictly inside or are touching $S_2.$ The minimum possible volume of $R$ is $m\sqrt{n}.$ Find $m+n.$

Solution

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