Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2025 SSMO Speed Round Problems"

m (→‎Problem 4)
m (→‎Problem 7)
 
Line 37: Line 37:
 
==Problem 7==
 
==Problem 7==
  
−
Positive integers <math>a</math> and <math>b</math> satisfy <math>63a = 40b</math>. The sum of all possible values of <math>\tfrac{\varphi(a)}{\varphi(b)}</math> is <math>\frac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
+
Positive integers <math>a</math> and <math>b</math> satisfy <math>63a = 40b</math>. The sum of all possible values of <math>\tfrac{\varphi(a)}{\varphi(b)}</math> is <math>\tfrac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
  
 
[[2025 SSMO Speed Round Problems/Problem 7|Solution]]
 
[[2025 SSMO Speed Round Problems/Problem 7|Solution]]

Latest revision as of 15:15, 10 September 2025

Problem 1

Define \[f(x) = \begin{cases} x + 2 & \text{if } x < 0 \\ x^2 + 1 & \text{if } x \geq 0 \end{cases}\] and let $x_0 = -3$. Define $x_{n+1} = f(x_n)$. Find the least $n$ such that $x_n > 100$.

Solution

Problem 2

Let $A$ and $B$ be points such that $AB = 50$. Points $M$ and $N$ lie on $\overline{AB}$ such that $M$ lies between points $A$ and $N$ and $N$ lies between points $B$ and $M$. Given that $MN = 20$ and $AN \cdot BM$ is maximized, find the length of $AM$.

Solution

Problem 3

Anna is buying different types of cheese from the local supermarket. Let $x,$ $y,$ and $z$ be the number of pieces of blue, cheddar, and mozzarella cheese, respectively, that Anna buys. She can buy any nonnegative integer number of each type, but the total number of pieces must be at most 12. How many different combinations $(x, y, z)$ of cheese can Anna buy? (Anna is allowed to buy 0 pieces of cheese.)

Solution

Problem 4

In rectangle $ABCD,$ let $AB = 8,BC = 15,\omega$ be the circumcircle of $ABCD$, $\ell$ be the line through $B$ parallel to $AC,$ and $X \neq B$ be the intersection of $\ell$ and $\omega$. Suppose the value of $BX$ can be expressed as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 5

Let $N = 101112\cdots9899$ be the number formed when all the two-digit positive integers are concatenated in increasing order. How many ordered triples of digits $(a,b,c)$ are there such that $a,$ $b,$ and $c$ appear as consecutive digits (in that order) in the decimal representation of $N$?

Solution

Problem 6

The centroid $G$ of $\triangle{ABC}$ has distances $21,$ $60,$ and $28$ from sides $AB,$ $BC,$ and $CA,$ respectively. Find the perimeter of $\triangle{ABC}$.

Solution

Problem 7

Positive integers $a$ and $b$ satisfy $63a = 40b$. The sum of all possible values of $\tfrac{\varphi(a)}{\varphi(b)}$ is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 8

Let $S$ be the set of all ordered pairs $(P,Q),$ where $P$ and $Q$ are subsets of $\{1,2,\dots, 25\}$ satisfying $|P\cup Q| = 17$ and $|(P\cap Q)\cap \{20,25\}|\ge 1$. If an ordered pair $(A,B)$ is chosen randomly from $S,$ the expected value of $|A\cap B|$ is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 9

Let $ABC$ be a triangle. The point $P$ lies on side $BC,$ the point $Q$ lies on side $AB,$ and the point $R$ lies on side $AC$ such that $PQ = BQ,$ $CR = PR,$ and $\angle APB < 90^\circ$. Let $H$ be the foot of the altitude from $A$ to $BC$. Given that $BP = 3,$ $CP = 5,$ and $[AQPR] = \tfrac{3}{7} \cdot [ABC],$ the value of $BH \cdot CH$ can be expressed in the form $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 10

Let $p$ be a quadratic with a positive leading coefficient, and let $r$ be a real number satisfying $r < 1 < \tfrac{5}{2r} < 5$. Given that $p(p(x)) = x$ for $x \in \{ r,1, \tfrac{5}{2r} , 5\}$, find $p(12)$.

Solution

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2025_SSMO_Speed_Round_Problems&oldid=260315"