2024 SSMO Speed Round Problems

Problem 1

Find the sum of the distinct prime factors of $2024^2 - 1$ .

Problem 2

Gracie's students play with some toys. When 4 or 5 students are present, the toys can be equally distributed to everyone. However, when there are only 3 students, there is one toy leftover after giving everyone the same number of toys. What is the least possible number of toys that Gracie could have?

Solution

Problem 3

The polynomial $x^3 - 15x^2 + 4x + 4$ has distinct real roots $r$ , $s$ , and $t$ . Find the value of $\left|(r^2 + s^2 + t^2)(rst)\right|.$

Solution

Problem 4

Sam wants to read the \textit{Harry Potter} and \textit{Warriors} books. There are 7 \textit{Harry Potter }books that must be read in a specific order, and there are 6 \textit{Warriors} books that also must be read in a specific order; however, he can read the two series at the same time. For example, he could read the first three \textit{Harry Potter} books, then the first five \textit{Warriors} books, then the remaining \textit{Harry Potter} books, and finally the last \textit{Warriors} book. In how many unique orders can Sam read the books?

Solution

Problem 5

Let $\triangle ABC$ and $\triangle ADC$ be right triangles, such that $\angle ABC = \angle ADC = 90^\circ$ . Given that $\angle ACB = 30^\circ$ and $BC = 3\sqrt{3}$ , find the maximum possible length of $BD$ .

Solution

Problem 6

There are $4$ people and $4$ houses. Each person independently randomly chooses a house to live in. The expected number of inhabited houses can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution

Problem 7

Let $S$ denote the set of all ellipses centered at the origin and with axes $AB$ and $CD$ where $A=(-x,0),B=(x,0),C=(0,-y),$ and $D=(0,y),$ for $2 \mid x+y$ and $0\le x,y \le 10.$ Let $T$ denote the set of similar ellipses centered at the origin and passing through $(x,y)$ for $2 \nmid x+y$ and $0\le x,y,\le 10.$ If the positive difference between the sum of the areas of all ellipses in $T$ and the sum of the areas of all the ellipses in $S$ is $m\pi,$ find $m.$

Solution

Problem 8

Bob has two coins; one is fair, and one lands on heads with a probability of $\frac{2}{3}.$ Bob chooses a random coin and flips it twice. Alice watches the two coin flips and guesses whether Bob flipped the fair or rigged coin. Given that Alice is a good mathematician and guesses the more likely option (guessing randomly when they are equally likely), the probability she guesses right can be expressed as $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$