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2024 SSMO Accuracy Round Problems

Problem 1

Let a time of day be three-full if exactly three of its digits are $3$s when displayed on a $12$-hour clock in the $hh:mm:ss$ format. How many seconds of the day are three-full?

Solution

Problem 2

Equilateral triangle $N$ is inscribed within circle $O$. A smaller equilateral triangle $P$ is inscribed within $N$ such that the vertices of $P$ lie on the midpoints of $N$. The ratio of the areas between $O$ and $P$ can be expressed as $\frac{a\pi\sqrt{b}}{c},$ for relatively prime positive integers $a,c$ and squarefree $b.$ Find $a+b+c$.

Solution

Problem 3

Three distinct random integers $a$, $b$, and $c$ are selected so that $1 \le a, b, c \le 10$. Let the probability that $5|a!+b!+c!$ be $\frac{m}{n}$, where $\gcd(m,n)=1$. Find $m+n$.

Solution

Problem 4

Right triangle $ABC$ has a right angle at $C$ and hypotenuse $1$. Let points $D$ and $E$ lie on $AC$ such that $\angle BDC=\angle BEC=45^{\circ}$. $A,D,C,$ and $E$ are colinear in that order. Given that $EA=13DA$, the area of $\triangle ABC$ can be expressed as $\frac{m}{n}$ for relatively prime $m$ and $n$. Find $m+n$.

Solution

Problem 5

Let $ABCD$ be a convex quadrilateral such that $\angle ABC = 120^\circ$ and $\angle ADC = 60^\circ$. If $AB = BC = CD = 5$, the area of $ABCD$ can be expressed as $a+b\sqrt{c},$ where \(a,b,\) and \(c\) are positive integers and \(c\) is squarefree. Find $a+b+c.$

Solution

Problem 6

Three six-sided dice are rolled. Then, the product of the three numbers on the top faces in calculated. If the probability of getting the product that is not a perfect square can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, what is $m+n$?

Solution

Problem 7

Find the value of $a+b+c+d$ given \begin{align*} a^2+d^2 &= b^2+c^2 = 361,\\ ac+bd &= 247,\text{ and }\\ ab+cd &= 570.\\ \end{align*}

Solution

Problem 8

$ABCD$ is a convex cyclic quadrilateral with $AB = 2, BC = 5, CD = 10,$ and $AD = 11.$ Let $W, Y, X,$ and $Z$ be the midpoints of sides $AB, BC, CD,$ and $DA$ respectively. If $|(WX^2-YZ^2)|$ can be expressed as $\frac{m}{n},$ for relatively prime positive integers $m$ and $n,$ find $m+n.$

Solution

Problem 9

In the game of $S$ - set, there are $3^{12}$ unique cards, each card containing a 12 traits of variants $A,B,$ and $C.$ A full house in the game of Sset is defined to be a hand of cards in which for each trait, all cards in the hand either share the same variant or have all different variants. The number of hands that can be considered a full house in the game of set can be expressed as $a^b+c^d+e^f,$ where $a,b,c,d,e,$ and $f$ are positive integers and $a+c+e$ is minimized. Find $a+b+c+d+e+f.$

Solution

Problem 10

Bobby is spinning a rigged wheel with three sections labeled $A,B,$ and $C.$ Two integers $0\leq x,y\leq 50$ are chosen randomly and independently, such that there is a $x\%$ chance the wheel lands on $A$ and a $y\%$ chance the wheel lands on $B.$ Given that the wheel lands on $C$ the first time, the probability that it will land on $C$ if Bobby spins it again can be expressed as $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$

Solution

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