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2022 SSMO Relay Round 2 Problems

Problem 1

Let $P$ be a randomly selected point on a circle, and let $A$ be a randomly selected point inside the same circle. A dilation centered at $P$ with a scale factor of $2$ sends $A$ to $A'.$ Given that the probability that $PA'$ is less than the length of the diameter of the circle can be expressed as $\frac{a\pi+b\sqrt{c}}{d\pi},$ where $a,b,c,d$ are integers such that $a$ and $d$ are positive, $c$ is squarefree, and $\gcd{(a, b, d)}=1$, find the value of $a+b+c+d.$

Solution

Problem 2

Let $T=$ TNYWR. Suppose that the monic quadratic $f(x)$ is tangent to the function $g(x)=|x+2|-T$ at two points, when graphed on the coordinate plane. Then $|f(1)|$ can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $10m+n$.

Solution

Problem 3

Let $T =$ TNYWR. Let $a+b = \lfloor \sqrt{T} \rfloor$. If $a^5 + b^5 = 15$, then $ab$ has two possible values. The absolute difference of these values is $\frac{x\sqrt{y}}{z}$, where $x,y$ and $z$ are positive integers, $x$ and $z$ are relatively prime, and $y$ is not divisible by the square of any prime. What is $x+y+z?$

Solution

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