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 Accuracy Round Problems"

(→‎Problem 8)
(→‎Problem 10)
 
Line 88: Line 88:
 
==Problem 10==
 
==Problem 10==
  
−
Let <math>ABCDE</math> be a convex pentagon with <math>\angle{BAC} = \angle{CAD} = \angle{DAE}</math> and <math>\angle{ABC} = \angle{ACD} = \angle{ADE}</math>. Let <math>BD</math> and <math>CE</math> meet at <math>P</math>. Given that <math>BC = 6</math>, <math>\sin{\angle{BAC}} = \tfrac{3}{5}</math>, and <math>\tfrac{AC}{AB} = 5</math>, the length of <math>AP</math> can be expressed as <math>\frac{m}{\sqrt{n}},</math> where <math>m</math> and <math>n</math> are positive integers such that <math>n</math> is square-free. Find <math>m+n</math>.
+
Let <math>ABCDE</math> be a convex pentagon with <math>\angle{BAC} = \angle{CAD} = \angle{DAE}</math> and <math>\angle{ABC} = \angle{ACD} = \angle{ADE}</math>. Let <math>BD</math> and <math>CE</math> meet at <math>P</math>. Given that <math>BC = 6</math>, <math>\sin{\angle{BAC}} = \tfrac{3}{5}</math>, and <math>\tfrac{AC}{AB} = 5</math>, the length of <math>AP</math> can be expressed as <math>\tfrac{m}{\sqrt{n}},</math> where <math>m</math> and <math>n</math> are positive integers such that <math>n</math> is square-free. Find <math>m+n</math>.
  
 
[[2025 SSMO Accuracy Round Problems/Problem 10|Solution]]
 
[[2025 SSMO Accuracy Round Problems/Problem 10|Solution]]

Latest revision as of 02:45, 11 September 2025

Problem 1

An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies \[a_n = a_1+a_2+\dots+a_{n-1}\] for all positive integers $n > 1$. Given that $a_{20}=25,$ find $a_{25}$.

Solution

Problem 2

Let $ABC$ be a triangle with circumcircle $\omega$. The midpoint of $AB$ is $M,$ and the line $CM$ intersects $\omega$ again at $P$. Given $\angle BMC = 120^\circ,$ $\triangle BMC$ is isosceles, and $BC = 20,$ the length of $PM$ can be written as $\tfrac{a\sqrt{b}}{c},$ where $a,$ $b,$ and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is square-free. Find $a+b+c$.

Solution

Problem 3

Nonnegative real numbers $x,y,$ and $z$ satisfy \[\frac{\sqrt{x}+13}{y} = \frac{\sqrt{y}+29}{z} = \frac{\sqrt{z} + 46}{x} = 2\] and \[\frac{\sqrt{x} + \sqrt{y}+\sqrt{z}}{x+y+z} = \frac{6}{25}.\] Find the value of $x+y+z$.

Solution

Problem 4

Let $a,$ $b,$ and $c$ be the roots of the polynomial $x^3+4x^2-3x-4$. Suppose that \[\left|\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right|=\frac{m}{n},\] for relatively prime positive integers $m$ and $n$. Find $m+n$.

Solution

Problem 5

$ABC$ is an isosceles triangle with base $BC = 6$ and $AB = AC$. Point $M$ is the midpoint of $BC$ such that $AM = 9$. Circle $\omega_1$ is the circumcircle of $ABC$ with radius $R,$ and $\omega_2$ is a circle passing through $B$ and $C$ with radius $2R$ and center on the opposite side of $BC$ as $A$. Segment $AM$ intersects $\omega_2$ at point $X$ and $\omega_1$ at point $Y,$ where $X$ lies between $A$ and $Y$. The length $XY$ can be expressed as $m - \sqrt{n},$ where $m$ and $n$ are positive integers. Find $m+n$.

Solution

Problem 6

Andy the ant starts at the square labeled $1$. On each move Andy moves to any orthogonal square (a square with which his current square shares a side). What is the expected number of moves before Andy is in the square labeled $2$?

[asy] unitsize(1cm); int[][] grid = { {0, 0, 0}, {0, 0, 0}, {0, 0, 0} }; grid[0][0] = 1; grid[2][2] = 2; for (int i = 0; i <= 3; ++i) { draw((0,i)--(3,i),black); draw((i,0)--(i,3),black); } for (int i = 0; i < 3; ++i) { for (int j = 0; j < 3; ++j) { if (grid[i][j] != 0) { } } } label("$1$",(0.5,0.5)); label("$2$",(2.5,2.5)); [/asy]

Solution

Problem 7

There is a unique ordered triple of positive reals $(a,b,c)$ satisfying the system of equations \begin{align*} a^2 + 9 &= (b-8\sqrt{3})^2 + 4 \\ b^2 + 4 &= (c-8\sqrt{3})^2 + 49 \\ c^2 + 49 &= (a-8\sqrt{3})^2 + 9. \end{align*} The value of $100a+10b+c$ can be expressed as $m\sqrt{n},$ where $m$ and $n$ are positive integers such that $n$ is square-free. Find $m+n$.

Solution

Problem 8

We say that a permutation $(a_1, a_2, \dots ,a_{10})$ of the integers $1$ through $10$ inclusive is peaked if there do not exist three integers $1\le i < j < k \le 10$ such that $a_i > a_j$ and $a_j< a_k$. Let $\mathcal{S}$ be the set of all peaked permutations. If $a_p = 9$ and $a_q = 4$, the expected value of $|p-q|$ over all permutations in $\mathcal{S}$ can be written as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is the value of $m+n$?

Solution

Problem 9

For a positive integer $n,$ let $r(n)$ denote the value of the binary number obtained by reading the binary representation of $n$ from right to left. For example, $r(6) = r(110_2) = 011_2 = 3$. If $k$ is the smallest positive integer such that the equation $n+r(n)=2k$ has at least ten positive integer solutions $n,$ find $k$.

Solution

Problem 10

Let $ABCDE$ be a convex pentagon with $\angle{BAC} = \angle{CAD} = \angle{DAE}$ and $\angle{ABC} = \angle{ACD} = \angle{ADE}$. Let $BD$ and $CE$ meet at $P$. Given that $BC = 6$, $\sin{\angle{BAC}} = \tfrac{3}{5}$, and $\tfrac{AC}{AB} = 5$, the length of $AP$ can be expressed as $\tfrac{m}{\sqrt{n}},$ where $m$ and $n$ are positive integers such that $n$ is square-free. Find $m+n$.

Solution

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2025_SSMO_Accuracy_Round_Problems&oldid=260342"