2025 SSMO Team Round Problems

Problem 1

There are $N$ solutions $(a,b,c)$ to $a + b^2 + c = 2025,$ where $a$ and $b$ are positive integers and $c$ is a nonnegative integer. Find the number of positive factors of $N$ .

Solution

Problem 2

Nonnegative integers $a$ and $b$ satisfy $a!+b^4 = 16a+41$ . Find the sum of all possible values of $b$ .

Solution

Problem 3

A rectangle is divided as shown into nine smaller rectangles. The areas of the five smallest rectangles are $1,$ $2,$ $3,$ $4,$ and $5$ . What is the largest possible area of the original rectangle? $[asy] unitsize(1cm); draw((0,0)--(4,0)--(4,3)--(0,3)--cycle); draw((1,0)--(1,3)); draw((1.7,0)--(1.7,3)); draw((0,0.8)--(4,0.8)); draw((0,2)--(4,2)); [/asy]$

Solution

Problem 4

Let $P(x) = x^2 - ax + b$ be a quadratic with nonzero real coefficients. Given that $P(a)$ and $P(-b)$ are roots of $P(x),$ there exists a value of $c$ such that $P(c)$ is constant for all possible $P(x)$ . Find $c$ .

Solution

Problem 5

Bob rolls a fair six-sided die until he rolls an even number. What is the expected sum of all the numbers he rolled?

Solution

Problem 6

The rhombus $PQRS$ has side length $3$ . The point $X$ lies on segment $PR$ such that line $QX$ is perpendicular to line $PS$ . Given $QX=2$ , the area of $PQRS$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution

Problem 7

Let $n$ be the largest integer such that $20!^{25!} + 25!^{20!}$ is divisible by $2025^n$ . The value of $n$ can be written in the form $a\cdot b!,$ where $a$ and $b$ are positive integers and $b$ is maximized. Find $a+b$ .

Solution

Problem 8

Adam has a fair coin with a $2$ written on one side and a $3$ written on the other. What is the expected number of times Adam needs to flip the coin for the sum of all his coin flips to be a multiple of $36$ ?

Solution

Problem 9

Pairwise distinct integers $a,$ $b,$ $c,$ and $d$ satisfy the system of equations $\begin{align*} ab &= cd \\ a+d &= b+c \\ \tfrac{1}{a} + \tfrac{1}{b} - \tfrac{1}{5} &= \tfrac{1}{c} + \tfrac{1}{d} + \tfrac{1}{5}. \end{align*}$ What is the minimum possible value of $a^2+b^2+c^2+d^2$ ?

Solution

Problem 10

Anna has a three term arithmetic sequence of integers. She divides each term of her sequence by a positive integer $n>1$ , and tells Bob that the three resulting remainders are $20$ , $52$ , and $R$ , in some order. For how many values of $R$ is it possible for Bob to uniquely determine $n$ ?

Solution

Problem 11

Squares $s_1,$ $s_2,$ and $s_3$ with side lengths $9,$ $17,$ and $10,$ respectively, lie inside of $\triangle ABC$ such that:

$s_1$ has a side that lies on $\overline{AB},$ $s_2$ has a side that lies on $\overline{BC},$ and $s_3$ has a side that lies on $\overline{CA}$ ;
each square shares exactly one vertex with each of the other two squares.

Find the perimeter of $\triangle ABC$ .

Solution

Problem 12

Let $a_n=(4+3\sqrt2)^n$ for all nonnegative integers $n$ . Let $\sum_{k=0}^\infty\frac{\lfloor a_k\rfloor}{10^k}=\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution

Problem 13

Let $S$ be the set of all ordered quintuples of integers $(x_1,x_2,x_3,x_4,x_5)$ satisfying $0 \le x_1\le x_2\le x_3\le x_4 \le x_5\le 5$ . The conjugate of a quintuple in $S$ is defined as $(y_1,y_2,y_3,y_4,y_5),$ where for each integer $1\le i \le 5,$ $y_i$ is the number of indices $1\le j\le 5$ satisfying $x_j\ge i$ . A randomly chosen quintuple of $S$ is a permutation of its conjugate with probability $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution

Problem 14

Find the number of ordered triples of positive integers $(a,b,c)$ such that $a+b+c=305$ and $ab+bc+ca$ is a multiple of $305$ .

Solution

Problem 15

The circles $\omega_1$ and $\omega_2$ have radii $8$ and $7$ , respectively, and intersect at points $A$ and $B$ . A line $\ell$ passing through $A$ intersects $\omega_1$ again at $P$ and $\omega_2$ again at $Q$ , with $PQ = 24$ . There exists a point $T$ on line $AB$ , with $A$ between $T$ and $B$ , such that $PT$ is tangent to $\omega_1$ and $QT$ is tangent to $\omega_2$ . The length of $BT$ can be written as $\frac{m}{\sqrt{n}}$ , where $m$ and $n$ are positive integers such that $n$ is square-free. Find $m+n$ .