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2022 SSMO Accuracy Round Problems

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Problem 1

Consider a bijective function (meaning each element in the domain maps to a distinct element in the range) $f:S\rightarrow S,$ where $S=\{1, 2, 3, 4, 5\}$. What is the average of $f(1) + f(2) + f(3)$, over all $f$?

Solution

Problem 2

Consider a cone with radius $5$ and height $12$, and a point $P$ in the same plane as the base of the cone, but a distance of $10$ from the center of the base of the cone. We rotate the cone $360^{\circ}$ about $P$ such that the plane that the base of the cone lies on stays the same. The volume of the region that the cone sweeps out can be expressed as $m\pi$. Find $m$.

Solution

Problem 3

Let $A=(6,3,2), B=(2,-9,-6),$ and $O=(0,0,0).$ Suppose that $D$ is a point in space such that $OD$ bisects $\angle{AOB}$ and $O,D,A,B$ are coplanar. In addition, $\angle{DAO}=90^{\circ}.$ If $DO$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers and $b$ is squarefree, find $a+b+c.$

Solution

Problem 4

A monic polynomial $f$ has real roots $r,s,t.$ A monic polynomial $g$ has roots $r^3,s^3,t^3.$ Given that the minimum possible value of $\frac{g(1)}{f(1)}$ is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n,$ find $m+n.$

Solution

Problem 5

Find the number of ordered pairs $(a, b)$, where $1\leq a, b\leq 9$, for which the largest integer $n$ that satisfies \[(a-b)^k\equiv a^k-b^k\pmod n\] for all $k\geq 1$ is $ab-b^2$.

Solution

Problem 6

Consider an unfair $6$-sided die labeled from $1$ to $6$, such that the probability of rolling a number $m$ is directly proportional to $7-m$. However, if we roll any number $n$, then the probability of rolling a number less than $n$ becomes $0$, and the probability of rolling any number $m$ from $n$ to $6$ inclusive remains directly proportional to $7-m$. The expected number of rolls until a $6$ is rolled can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 7

After a robber drives in a car for $t$ (not necessarily integral) minutes, the car goes at $120-t$ miles per hour. Whenever the car's speed drops below $60$ miles per hour, the robber switches into a new car with no time loss. A police car can drive at a constant speed of 117 miles per hour. Given that the robber starts 1 hour before the police car, how many minutes will pass between when the police car starts and when the police car catches up to the robber?

Solution

Problem 8

Let $ABCD$ be a trapezoid with $AB \parallel CD$. Suppose that $AD=1$, $DC=4$, $CB=2$, and $AB<CD$. Let $X$ be the midpoint of $AB$. If $E$ is the intersection of $AC$ and $BD$, and $\angle XEB=\angle ADC$, then $AB=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 9

The graph $\Gamma$ has $52$ vertices labeled $A, A', B, B', \dots, Z, Z'$, such that $A$ is not connected to $A'$, $B$ is not connected to $B'$, and so on. Suppose that all the vertices other than $A'$ have different degrees (number of connections to the vertex). Find the sum of all possible values for the number of edges (connections) in $\Gamma$.

Solution

Problem 10

Let \[S=\sum^{\infty}_{k=5}2^{-10k}\dbinom{2k}{10}.\]Then the value of $S$ can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find the largest positive integer $a$ such that $2^a\mid mn$.

Solution

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