problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 6
values | synthetic bool 1
class |
|---|---|---|---|---|---|---|---|---|
Richard has a four infinitely large piles of coins: a pile of pennies (worth 1 cent each), a pile of nickels (5 cents), a pile of dimes (10 cents), and a pile of quarters (25 cents). He chooses one pile at random and takes one coin from that pile. Richard then repeats this process until the sum of the values of the coi... | 1. Define \( E(x) \) as the expected number of additional cents needed to get to a multiple of 100 starting from \( x \) cents. We know that \( E(100k) = 0 \) for any integer \( k \), because if we already have a multiple of 100 cents, no additional cents are needed.
2. For \( 1 \leq x \leq 99 \), we can write the rec... | 1025 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In $\triangle ABC$, points $E$ and $F$ lie on $\overline{AC}, \overline{AB}$, respectively. Denote by $P$ the intersection of $\overline{BE}$ and $\overline{CF}$. Compute the maximum possible area of $\triangle ABC$ if $PB = 14$, $PC = 4$, $PE = 7$, $PF = 2$.
[i]Proposed by Eugene Chen[/i] | 1. **Identify the given information and the goal:**
- We are given a triangle \( \triangle ABC \) with points \( E \) and \( F \) on \( \overline{AC} \) and \( \overline{AB} \) respectively.
- \( P \) is the intersection of \( \overline{BE} \) and \( \overline{CF} \).
- We are given the lengths \( PB = 14 \), ... | 84 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
At ARML, Santa is asked to give rubber duckies to $2013$ students, one for each student. The students are conveniently numbered $1,2,\cdots,2013$, and for any integers $1 \le m < n \le 2013$, students $m$ and $n$ are friends if and only if $0 \le n-2m \le 1$.
Santa has only four different colors of duckies, but beca... | To solve this problem, we need to understand the structure of the friendships and how to color the students such that no two friends or students sharing a common friend have the same color.
1. **Understanding the Friendship Structure:**
- Students \( m \) and \( n \) are friends if and only if \( 0 \leq n - 2m \le... | 288 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the positive integer $N$ for which there exist reals $\alpha, \beta, \gamma, \theta$ which obey
\begin{align*}
0.1 &= \sin \gamma \cos \theta \sin \alpha, \\
0.2 &= \sin \gamma \sin \theta \cos \alpha, \\
0.3 &= \cos \gamma \cos \theta \sin \beta, \\
0.4 &= \cos \gamma \sin \theta \cos \beta, \\
0.5 &\ge \left\lve... | 1. We start with the given equations:
\[
0.1 = \sin \gamma \cos \theta \sin \alpha,
\]
\[
0.2 = \sin \gamma \sin \theta \cos \alpha,
\]
\[
0.3 = \cos \gamma \cos \theta \sin \beta,
\]
\[
0.4 = \cos \gamma \sin \theta \cos \beta,
\]
\[
0.5 \ge \left\lvert N-100 \cos 2\theta ... | 79 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
For every integer $n \ge 1$, the function $f_n : \left\{ 0, 1, \cdots, n \right\} \to \mathbb R$ is defined recursively by $f_n(0) = 0$, $f_n(1) = 1$ and \[ (n-k) f_n(k-1) + kf_n(k+1) = nf_n(k) \] for each $1 \le k < n$. Let $S_N = f_{N+1}(1) + f_{N+2}(2) + \cdots + f_{2N} (N)$. Find the remainder when $\left\lfloor S... | 1. **Define the function \( f_n(k) \) recursively:**
Given the recursive definition:
\[
f_n(0) = 0, \quad f_n(1) = 1, \quad \text{and} \quad (n-k) f_n(k-1) + k f_n(k+1) = n f_n(k) \quad \text{for} \quad 1 \le k < n
\]
We need to find a general form for \( f_n(k) \).
2. **Identify a pattern:**
By exam... | 26 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Jacob and Aaron are playing a game in which Aaron is trying to guess the outcome of an unfair coin which shows heads $\tfrac{2}{3}$ of the time. Aaron randomly guesses ``heads'' $\tfrac{2}{3}$ of the time, and guesses ``tails'' the other $\tfrac{1}{3}$ of the time. If the probability that Aaron guesses correctly is $... | 1. Let's denote the probability of the coin showing heads as \( P(H) = \frac{2}{3} \) and the probability of the coin showing tails as \( P(T) = 1 - P(H) = \frac{1}{3} \).
2. Aaron guesses heads with probability \( P(G_H) = \frac{2}{3} \) and guesses tails with probability \( P(G_T) = 1 - P(G_H) = \frac{1}{3} \).
3. ... | 5000 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of the real roots of the polynomial \[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \][i]Proposed by Evan Chen[/i] | To find the sum of the real roots of the polynomial
\[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right), \]
we need to analyze the quadratic factors \( x^2 - 11x + k \) for \( k = 1, 2, \ldots, 100 \).
1. **Determine the conditions for real roo... | 330 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A point $(a,b)$ in the plane is called [i]sparkling[/i] if it also lies on the line $ax+by=1$. Find the maximum possible distance between two sparkling points.
[i]Proposed by Evan Chen[/i] | 1. **Identify the condition for a point \((a, b)\) to be sparkling:**
A point \((a, b)\) is sparkling if it lies on the line \(ax + by = 1\). Substituting \((a, b)\) into the equation, we get:
\[
a \cdot a + b \cdot b = 1 \implies a^2 + b^2 = 1
\]
This implies that the point \((a, b)\) must lie on the un... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Circle $\omega_1$ and $\omega_2$ have centers $(0,6)$ and $(20,0)$, respectively. Both circles have radius $30$, and intersect at two points $X$ and $Y$. The line through $X$ and $Y$ can be written in the form $y = mx+b$. Compute $100m+b$.
[i]Proposed by Evan Chen[/i] | 1. We start with the equations of the two circles. The first circle $\omega_1$ has center $(0,6)$ and radius $30$, so its equation is:
\[
x^2 + (y - 6)^2 = 900
\]
The second circle $\omega_2$ has center $(20,0)$ and radius $30$, so its equation is:
\[
(x - 20)^2 + y^2 = 900
\]
2. To find the line ... | 303 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A pair of positive integers $(m,n)$ is called [i]compatible[/i] if $m \ge \tfrac{1}{2} n + 7$ and $n \ge \tfrac{1}{2} m + 7$. A positive integer $k \ge 1$ is called [i]lonely[/i] if $(k,\ell)$ is not compatible for any integer $\ell \ge 1$. Find the sum of all lonely integers.
[i]Proposed by Evan Chen[/i] | To determine the sum of all lonely integers, we need to analyze the conditions under which a pair \((m, n)\) is compatible. The conditions given are:
\[ m \ge \frac{1}{2} n + 7 \]
\[ n \ge \frac{1}{2} m + 7 \]
A positive integer \( k \ge 1 \) is called lonely if \((k, \ell)\) is not compatible for any integer \(\ell \... | 91 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find $100m+n$ if $m$ and $n$ are relatively prime positive integers such that \[ \sum_{\substack{i,j \ge 0 \\ i+j \text{ odd}}} \frac{1}{2^i3^j} = \frac{m}{n}. \][i]Proposed by Aaron Lin[/i] | To solve the problem, we need to evaluate the sum
\[
\sum_{\substack{i,j \ge 0 \\ i+j \text{ odd}}} \frac{1}{2^i3^j}
\]
and express it in the form \(\frac{m}{n}\) where \(m\) and \(n\) are relatively prime positive integers. We will then find \(100m + n\).
1. **Separate the sum based on the parity of \(i\) and \(j\... | 504 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
In $\triangle ABC$, $AB = 40$, $BC = 60$, and $CA = 50$. The angle bisector of $\angle A$ intersects the circumcircle of $\triangle ABC$ at $A$ and $P$. Find $BP$.
[i]Proposed by Eugene Chen[/i] | 1. **Scale the Triangle:**
Given the sides of $\triangle ABC$ are $AB = 40$, $BC = 60$, and $CA = 50$, we can scale the triangle down to a simpler ratio. Let's scale it down by a factor of 10, resulting in a triangle with sides $AB = 4$, $BC = 6$, and $CA = 5$.
2. **Angle Bisector Theorem:**
Let $D$ be the point... | 40 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In trapezoid $ABCD$, $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$. Compute $AB^2$ given that $AB \cdot \tan(\angle BCD) = 20$ and $CD = 13$.
[i]Proposed by Lewis Chen[/i] | 1. Given that in trapezoid \(ABCD\), \(AD \parallel BC\) and \(\angle ABC + \angle CDA = 270^\circ\). We need to compute \(AB^2\) given that \(AB \cdot \tan(\angle BCD) = 20\) and \(CD = 13\).
2. Since \(\angle ABC + \angle CDA = 270^\circ\), and knowing that \(\angle ABC > 90^\circ\) because \(\angle CDA < 180^\circ\... | 260 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $p$, $q$, and $r$ be primes satisfying \[ pqr = 189999999999999999999999999999999999999999999999999999962.
\] Compute $S(p) + S(q) + S(r) - S(pqr)$, where $S(n)$ denote the sum of the decimals digits of $n$.
[i]Proposed by Evan Chen[/i] | 1. **Identify the prime factors of \( pqr \):**
Given \( pqr = 189999999999999999999999999999999999999999999999999999962 \), we need to find the prime factors \( p \), \( q \), and \( r \).
2. **Factorize the number:**
Notice that the number \( 189999999999999999999999999999999999999999999999999999962 \) is very... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$, $b$, $c$, $d$, $e$ be positive reals satisfying \begin{align*} a + b &= c \\ a + b + c &= d \\ a + b + c + d &= e.\end{align*} If $c=5$, compute $a+b+c+d+e$.
[i]Proposed by Evan Chen[/i] | 1. We start with the given equations:
\[
a + b = c
\]
\[
a + b + c = d
\]
\[
a + b + c + d = e
\]
and the given value \( c = 5 \).
2. From the first equation, we substitute \( c \) with 5:
\[
a + b = 5
\]
3. Using the second equation, we substitute \( a + b \) and \( c \):
\[... | 40 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $N$ has $20$ digits when written in base $9$ and $13$ digits when written in base $27$. How many digits does $N$ have when written in base $3$?
[i]Proposed by Aaron Lin[/i] | 1. We start by interpreting the given conditions for the number \( N \). The first condition states that \( N \) has 20 digits when written in base 9. This implies:
\[
9^{19} \leq N < 9^{20}
\]
This is because a number with \( d \) digits in base \( b \) satisfies \( b^{d-1} \leq N < b^d \).
2. The second ... | 39 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The diagonals of convex quadrilateral $BSCT$ meet at the midpoint $M$ of $\overline{ST}$. Lines $BT$ and $SC$ meet at $A$, and $AB = 91$, $BC = 98$, $CA = 105$. Given that $\overline{AM} \perp \overline{BC}$, find the positive difference between the areas of $\triangle SMC$ and $\triangle BMT$.
[i]Proposed by Evan C... | 1. **Identify the given information and setup the problem:**
- The diagonals of convex quadrilateral \( BSCT \) meet at the midpoint \( M \) of \( \overline{ST} \).
- Lines \( BT \) and \( SC \) meet at \( A \).
- Given lengths: \( AB = 91 \), \( BC = 98 \), \( CA = 105 \).
- \( \overline{AM} \perp \overlin... | 336 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Dragon selects three positive real numbers with sum $100$, uniformly at random. He asks Cat to copy them down, but Cat gets lazy and rounds them all to the nearest tenth during transcription. If the probability the three new numbers still sum to $100$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive... | 1. Let the three numbers be \(a, b, c\) such that \(a + b + c = 100\). We can express each number as \(a = a' + a''\), \(b = b' + b''\), and \(c = c' + c''\), where \(a', b', c'\) are integers and \(a'', b'', c''\) are the fractional parts satisfying \(|a''| \leq 0.5\), \(|b''| \leq 0.5\), and \(|c''| \leq 0.5\).
2. T... | 704 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(n)=\varphi(n^3)^{-1}$, where $\varphi(n)$ denotes the number of positive integers not greater than $n$ that are relatively prime to $n$. Suppose
\[ \frac{f(1)+f(3)+f(5)+\dots}{f(2)+f(4)+f(6)+\dots} = \frac{m}{n} \]
where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
[i]Proposed by Lewis ... | 1. **Define the function and simplify the problem:**
Let \( f(n) = \varphi(n^3)^{-1} \), where \(\varphi(n)\) denotes the Euler's totient function, which counts the number of positive integers up to \( n \) that are relatively prime to \( n \).
2. **Express the given fraction:**
We need to compute:
\[
\fra... | 704 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f$ be a non-constant polynomial such that \[ f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x} \] for all nonzero real numbers $x$. Find the sum of all possible values of $f(1)$.
[i]Proposed by Ahaan S. Rungta[/i] | 1. Let \( f \) be a polynomial of degree \( d \). We start by comparing the degrees of both sides of the given equation:
\[
f(x-1) + f(x) + f(x+1) = \frac{f(x)^2}{2013x}
\]
The left-hand side (LHS) is a sum of three polynomials each of degree \( d \), so the degree of the LHS is \( d \).
2. The right-hand ... | 6039 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Consider a set of $1001$ points in the plane, no three collinear. Compute the minimum number of segments that must be drawn so that among any four points, we can find a triangle.
[i]Proposed by Ahaan S. Rungta / Amir Hossein[/i] | To solve this problem, we need to ensure that among any four points chosen from the set of 1001 points, we can always find a triangle. This means that we need to avoid the situation where all four points are connected in such a way that they form a quadrilateral without any diagonals.
1. **Understanding the Problem:**... | 500500 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $d$ and $n$ be positive integers such that $d$ divides $n$, $n > 1000$, and $n$ is not a perfect square. The minimum possible value of $\left\lvert d - \sqrt{n} \right\rvert$ can be written in the form $a\sqrt{b} + c$, where $b$ is a positive integer not divisible by the square of any prime, and $a$ and $c$ are no... | 1. We are given that \( d \) and \( n \) are positive integers such that \( d \) divides \( n \), \( n > 1000 \), and \( n \) is not a perfect square. We need to minimize \( \left\lvert d - \sqrt{n} \right\rvert \).
2. To minimize \( \left\lvert d - \sqrt{n} \right\rvert \), we want \( \sqrt{n} \) to be as close to \(... | 38 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AB = 42$, $AC = 39$, $BC = 45$. Let $E$, $F$ be on the sides $\overline{AC}$ and $\overline{AB}$ such that $AF = 21, AE = 13$. Let $\overline{CF}$ and $\overline{BE}$ intersect at $P$, and let ray $AP$ meet $\overline{BC}$ at $D$. Let $O$ denote the circumcenter of $\triangle DEF$, and $R... | To solve the problem, we need to compute \( CO^2 - R^2 \), where \( O \) is the circumcenter of \(\triangle DEF\) and \( R \) is its circumradius. We will use the concept of power of a point and properties of the circumcircle.
1. **Define the function \( f(P) \)**:
Define \( f(P) = (P, \omega_C) - (P, \omega) \), w... | 0 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Richard likes to solve problems from the IMO Shortlist. In 2013, Richard solves $5$ problems each Saturday and $7$ problems each Sunday. He has school on weekdays, so he ``only'' solves $2$, $1$, $2$, $1$, $2$ problems on each Monday, Tuesday, Wednesday, Thursday, and Friday, respectively -- with the exception of Decem... | 1. **Determine the number of weeks in 2013:**
Since 2013 is not a leap year, it has 365 days. There are 52 weeks and 1 extra day in a non-leap year.
\[
365 = 52 \times 7 + 1
\]
2. **Calculate the number of problems solved in a typical week:**
Richard solves problems as follows:
- Monday: 2 problems
... | 1099 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many integers $n$ are there such that $(n+1!)(n+2!)(n+3!)\cdots(n+2013!)$ is divisible by $210$ and $1 \le n \le 210$?
[i]Proposed by Lewis Chen[/i] | To determine how many integers \( n \) satisfy the condition that \((n+1!)(n+2!)(n+3!)\cdots(n+2013!)\) is divisible by \( 210 \) and \( 1 \le n \le 210 \), we need to analyze the prime factorization of \( 210 \).
The prime factorization of \( 210 \) is:
\[ 210 = 2 \times 3 \times 5 \times 7 \]
For the product \((n+... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
At Stanford in 1988, human calculator Shakuntala Devi was asked to compute $m = \sqrt[3]{61{,}629{,}875}$ and $n = \sqrt[7]{170{,}859{,}375}$. Given that $m$ and $n$ are both integers, compute $100m+n$.
[i]Proposed by Evan Chen[/i] | 1. **Compute \( m = \sqrt[3]{61,629,875} \):**
- We need to find an integer \( m \) such that \( m^3 = 61,629,875 \).
- Estimate \( m \) by noting that \( 61,629,875 \approx 62 \times 10^6 \).
- Since \( 400^3 = 64,000,000 \) and \( 300^3 = 27,000,000 \), \( m \) should be close to 400.
- The units digit of... | 39515 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S = \{1,2,\cdots,2013\}$. Let $N$ denote the number $9$-tuples of sets $(S_1, S_2, \dots, S_9)$ such that $S_{2n-1}, S_{2n+1} \subseteq S_{2n} \subseteq S$ for $n=1,2,3,4$. Find the remainder when $N$ is divided by $1000$.
[i]Proposed by Lewis Chen[/i] | 1. **Understanding the Problem:**
We need to find the number of 9-tuples of sets \((S_1, S_2, \dots, S_9)\) such that \(S_{2n-1}, S_{2n+1} \subseteq S_{2n} \subseteq S\) for \(n=1,2,3,4\), where \(S = \{1, 2, \cdots, 2013\}\).
2. **Analyzing the Constraints:**
For each element \(x \in S\), we need to determine h... | 72 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a certain game, Auntie Hall has four boxes $B_1$, $B_2$, $B_3$, $B_4$, exactly one of which contains a valuable gemstone; the other three contain cups of yogurt. You are told the probability the gemstone lies in box $B_n$ is $\frac{n}{10}$ for $n=1,2,3,4$.
Initially you may select any of the four boxes; Auntie Hal... | 1. **Initial Setup and Probabilities:**
- We have four boxes \( B_1, B_2, B_3, B_4 \).
- The probability that the gemstone is in box \( B_n \) is \( \frac{n}{10} \) for \( n = 1, 2, 3, 4 \).
2. **Initial Selection:**
- We initially select box \( B_1 \).
3. **Auntie Hall's Action:**
- Auntie Hall opens one... | 1930 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The number $\frac{1}{2}$ is written on a blackboard. For a real number $c$ with $0 < c < 1$, a [i]$c$-splay[/i] is an operation in which every number $x$ on the board is erased and replaced by the two numbers $cx$ and $1-c(1-x)$. A [i]splay-sequence[/i] $C = (c_1,c_2,c_3,c_4)$ is an application of a $c_i$-splay for $i=... | 1. **Understanding the Problem:**
- We start with the number $\frac{1}{2}$ on the blackboard.
- A $c$-splay operation replaces each number $x$ with two numbers: $cx$ and $1 - c(1 - x)$.
- A splay-sequence $C = (c_1, c_2, c_3, c_4)$ involves applying $c_i$-splays in sequence.
- The power of a splay-sequence ... | 4817 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ denote the number of ordered 2011-tuples of positive integers $(a_1,a_2,\ldots,a_{2011})$ with $1\le a_1,a_2,\ldots,a_{2011} \le 2011^2$ such that there exists a polynomial $f$ of degree $4019$ satisfying the following three properties:
[list] [*] $f(n)$ is an integer for every integer $n$; [*] $2011^2 \mid f(i... | 1. **Understanding the Problem:**
We need to find the number of ordered 2011-tuples of positive integers \((a_1, a_2, \ldots, a_{2011})\) such that there exists a polynomial \(f\) of degree 4019 satisfying:
- \(f(n)\) is an integer for every integer \(n\),
- \(2011^2 \mid f(i) - a_i\) for \(i = 1, 2, \ldots, 2... | 281 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$\omega$ is a complex number such that $\omega^{2013} = 1$ and $\omega^m \neq 1$ for $m=1,2,\ldots,2012$. Find the number of ordered pairs of integers $(a,b)$ with $1 \le a, b \le 2013$ such that \[ \frac{(1 + \omega + \cdots + \omega^a)(1 + \omega + \cdots + \omega^b)}{3} \] is the root of some polynomial with integer... | 1. **Understanding the Problem:**
We are given a complex number \(\omega\) such that \(\omega^{2013} = 1\) and \(\omega^m \neq 1\) for \(m = 1, 2, \ldots, 2012\). This means \(\omega\) is a primitive 2013th root of unity. We need to find the number of ordered pairs \((a, b)\) with \(1 \le a, b \le 2013\) such that
... | 4029 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $\angle B - \angle C = 30^{\circ}$. Let $D$ be the point where the $A$-excircle touches line $BC$, $O$ the circumcenter of triangle $ABC$, and $X,Y$ the intersections of the altitude from $A$ with the incircle with $X$ in between $A$ and $Y$. Suppose points $A$, $O$ and $D$ are collinear. I... | 1. **Given Information and Initial Setup:**
- We have a triangle \(ABC\) with \(\angle B - \angle C = 30^\circ\).
- \(D\) is the point where the \(A\)-excircle touches line \(BC\).
- \(O\) is the circumcenter of \(\triangle ABC\).
- \(X\) and \(Y\) are the intersections of the altitude from \(A\) with the i... | 64 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The number $123454321$ is written on a blackboard. Evan walks by and erases some (but not all) of the digits, and notices that the resulting number (when spaces are removed) is divisible by $9$. What is the fewest number of digits he could have erased?
[i]Ray Li[/i] | 1. **Calculate the sum of the digits of the original number:**
The number is \(123454321\). The sum of its digits is:
\[
1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25
\]
2. **Determine the condition for divisibility by 9:**
A number is divisible by 9 if the sum of its digits is divisible by 9. Therefore, we ne... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Three lines $m$, $n$, and $\ell$ lie in a plane such that no two are parallel. Lines $m$ and $n$ meet at an acute angle of $14^{\circ}$, and lines $m$ and $\ell$ meet at an acute angle of $20^{\circ}$. Find, in degrees, the sum of all possible acute angles formed by lines $n$ and $\ell$.
[i]Ray Li[/i] | 1. Given three lines \( m \), \( n \), and \( \ell \) in a plane such that no two are parallel. Lines \( m \) and \( n \) meet at an acute angle of \( 14^\circ \), and lines \( m \) and \( \ell \) meet at an acute angle of \( 20^\circ \).
2. We need to find the sum of all possible acute angles formed by lines \( n \) ... | 40 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For how many ordered pairs of positive integers $(a,b)$ with $a,b<1000$ is it true that $a$ times $b$ is equal to $b^2$ divided by $a$? For example, $3$ times $9$ is equal to $9^2$ divided by $3$.
[i]Ray Li[/i] | 1. We start with the given equation:
\[
ab = \frac{b^2}{a}
\]
2. Multiply both sides by \(a\) to eliminate the fraction:
\[
a^2 b = b^2
\]
3. Assuming \(b \neq 0\), we can divide both sides by \(b\):
\[
a^2 = b
\]
4. We need to find the number of ordered pairs \((a, b)\) such that \(a\) and \... | 31 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
At the Mountain School, Micchell is assigned a [i]submissiveness rating[/i] of $3.0$ or $4.0$ for each class he takes. His [i]college potential[/i] is then defined as the average of his submissiveness ratings over all classes taken. After taking 40 classes, Micchell has a college potential of $3.975$. Unfortunately, he... | 1. Let \( x \) be the number of additional classes Micchell needs to take.
2. The total submissiveness rating for the 40 classes he has already taken is \( 40 \times 3.975 \).
3. The total submissiveness rating for the additional \( x \) classes, assuming he receives a rating of 4.0 in each, is \( 4x \).
4. The new tot... | 160 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Jacob's analog clock has 12 equally spaced tick marks on the perimeter, but all the digits have been erased, so he doesn't know which tick mark corresponds to which hour. Jacob takes an arbitrary tick mark and measures clockwise to the hour hand and minute hand. He measures that the minute hand is 300 degrees clockwise... | 1. **Understanding the problem:**
- Jacob's clock has 12 tick marks, each representing an hour.
- The minute hand is 300 degrees clockwise from an arbitrary tick mark.
- The hour hand is 70 degrees clockwise from the same tick mark.
- We need to determine the time in minutes past midnight.
2. **Convert deg... | 110 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
David has a collection of 40 rocks, 30 stones, 20 minerals and 10 gemstones. An operation consists of removing three objects, no two of the same type. What is the maximum number of operations he can possibly perform?
[i]Ray Li[/i] | To solve this problem, we need to determine the maximum number of operations where each operation consists of removing three objects, no two of the same type, from David's collection. Let's break down the steps:
1. **Initial Counts**:
- Rocks: 40
- Stones: 30
- Minerals: 20
- Gemstones: 10
2. **Operation ... | 30 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
At certain store, a package of 3 apples and 12 oranges costs 5 dollars, and a package of 20 apples and 5 oranges costs 13 dollars. Given that apples and oranges can only be bought in these two packages, what is the minimum nonzero amount of dollars that must be spent to have an equal number of apples and oranges?
[i]R... | 1. Let us denote the first package as \( P_1 \) and the second package as \( P_2 \).
- \( P_1 \) contains 3 apples and 12 oranges and costs 5 dollars.
- \( P_2 \) contains 20 apples and 5 oranges and costs 13 dollars.
2. We need to find the minimum nonzero amount of dollars that must be spent to have an equal nu... | 64 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
There are 25 ants on a number line; five at each of the coordinates $1$, $2$, $3$, $4$, and $5$. Each minute, one ant moves from its current position to a position one unit away. What is the minimum number of minutes that must pass before it is possible for no two ants to be on the same coordinate?
[i]Ray Li[/i] | 1. We start by noting that there are 25 ants, with 5 ants initially positioned at each of the coordinates \(1, 2, 3, 4,\) and \(5\).
2. The goal is to move the ants such that no two ants occupy the same coordinate. This means we need to distribute the ants to 25 unique positions on the number line.
3. To achieve this, ... | 250 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are three flies of negligible size that start at the same position on a circular track with circumference 1000 meters. They fly clockwise at speeds of 2, 6, and $k$ meters per second, respectively, where $k$ is some positive integer with $7\le k \le 2013$. Suppose that at some point in time, all three flies meet ... | 1. We need to find the values of \( k \) such that the three flies meet at a location different from their starting point. The flies travel at speeds of 2, 6, and \( k \) meters per second, respectively, on a circular track with a circumference of 1000 meters.
2. Let \( t \) be the time in seconds when the flies meet ... | 501 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What is the smallest perfect square larger than $1$ with a perfect square number of positive integer factors?
[i]Ray Li[/i] | 1. We need to find the smallest perfect square larger than $1$ that has a perfect square number of positive integer factors.
2. Let's list the perfect squares and count their factors:
- $4 = 2^2$: The factors are $1, 2, 4$. There are $3$ factors.
- $9 = 3^2$: The factors are $1, 3, 9$. There are $3$ factors.
-... | 36 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.
(You may use the fact that $\sin{n^\circ}$ is irrational for positive in... | 1. We need to determine the absolute value of the sum
\[
\lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor,
\]
where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.
2. First, note that $\sin{0^\circ} = 0$, $\sin... | 178 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$A,B,C$ are points in the plane such that $\angle ABC=90^\circ$. Circles with diameters $BA$ and $BC$ meet at $D$. If $BA=20$ and $BC=21$, then the length of segment $BD$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
[i]Ray Li[/i] | 1. Given that $\angle ABC = 90^\circ$, we know that $A$, $B$, and $C$ form a right triangle with $B$ as the right angle. Therefore, $AB$ and $BC$ are perpendicular to each other.
2. The circles with diameters $BA$ and $BC$ meet at point $D$. Since $D$ lies on both circles, it must satisfy the properties of both circle... | 449 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Dirock has a very neat rectangular backyard that can be represented as a $32\times 32$ grid of unit squares. The rows and columns are each numbered $1,2,\ldots, 32$. Dirock is very fond of rocks, and places a rock in every grid square whose row and column number are both divisible by $3$. Dirock would like to build a r... | 1. **Identify the grid and rocks placement:**
- The backyard is a \(32 \times 32\) grid.
- Rocks are placed in every grid square where both the row and column numbers are divisible by 3.
- Therefore, rocks are placed at coordinates \((3i, 3j)\) where \(i, j \in \{1, 2, \ldots, 10\}\).
2. **Determine the numbe... | 1920 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, $AB = 28$, $AC = 36$, and $BC = 32$. Let $D$ be the point on segment $BC$ satisfying $\angle BAD = \angle DAC$, and let $E$ be the unique point such that $DE \parallel AB$ and line $AE$ is tangent to the circumcircle of $ABC$. Find the length of segment $AE$.
[i]Ray Li[/i] | 1. **Identify the given lengths and properties:**
- In triangle \(ABC\), we have \(AB = 28\), \(AC = 36\), and \(BC = 32\).
- Point \(D\) on segment \(BC\) satisfies \(\angle BAD = \angle DAC\), making \(AD\) the angle bisector of \(\angle BAC\).
2. **Use the Angle Bisector Theorem:**
- The Angle Bisector The... | 18 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A set of 10 distinct integers $S$ is chosen. Let $M$ be the number of nonempty subsets of $S$ whose elements have an even sum. What is the minimum possible value of $M$?
[hide="Clarifications"]
[list]
[*] $S$ is the ``set of 10 distinct integers'' from the first sentence.[/list][/hide]
[i]Ray Li[/i] | 1. Let the set be \( S = \{a_1, a_2, \ldots, a_{10}\} \) and define the polynomial
\[
P(x) = (x^{a_1} + 1)(x^{a_2} + 1) \cdots (x^{a_{10}} + 1) - 1.
\]
This polynomial represents the sum of \( x^{\sigma(T)} \) for all non-empty subsets \( T \subset S \), where \( \sigma(T) \) denotes the sum of the element... | 511 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a permutation $\pi$ of the integers from 1 to 10, define
\[ S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), \]
where $\pi (i)$ denotes the $i$th element of the permutation. Suppose that $M$ is the maximum possible value of $S(\pi)$ over all permutations $\pi$ of the integers from 1 to 10. ... | To solve the problem, we need to analyze the expression for \( S(\pi) \) and determine the maximum possible value of \( S(\pi) \) over all permutations of the integers from 1 to 10. Then, we need to count the number of permutations that achieve this maximum value.
1. **Expression Analysis**:
\[
S(\pi) = \sum_{i=... | 40320 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of all lattice points $(x, y)$ in the plane satisfying $|x|+|y|\le 10$. Let $P_1,P_2,\ldots,P_{2013}$ be a sequence of 2013 (not necessarily distinct) points such that for every point $Q$ in $S$, there exists at least one index $i$ such that $1\le i\le 2013$ and $P_i = Q$. Suppose that the minimum po... | 1. **Understanding the Problem:**
We need to find the minimum possible value of the sum of distances between consecutive points in a sequence of 2013 points, where each point is a lattice point within the set \( S \) defined by \( |x| + |y| \leq 10 \). The distance between two points \( (x_1, y_1) \) and \( (x_2, y_... | 222 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Pairwise distinct points $P_1,P_2,\ldots, P_{16}$ lie on the perimeter of a square with side length $4$ centered at $O$ such that $\lvert P_iP_{i+1} \rvert = 1$ for $i=1,2,\ldots, 16$. (We take $P_{17}$ to be the point $P_1$.) We construct points $Q_1,Q_2,\ldots,Q_{16}$ as follows: for each $i$, a fair coin is flipped.... | 1. **Define the problem in terms of vectors:**
Each point \( P_i \) lies on the perimeter of a square with side length 4 centered at \( O \). The points \( P_i \) are such that \( |P_iP_{i+1}| = 1 \) for \( i = 1, 2, \ldots, 16 \). We need to compute the expected value of \( D^2 \), where \( D \) is the length of th... | 64 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Beyond the Point of No Return is a large lake containing 2013 islands arranged at the vertices of a regular $2013$-gon. Adjacent islands are joined with exactly two bridges. Christine starts on one of the islands with the intention of burning all the bridges. Each minute, if the island she is on has at least one bri... | 1. **Understanding the Problem:**
Christine starts on one of the 2013 islands arranged in a regular 2013-gon. Each island is connected to its adjacent islands by exactly two bridges. Christine randomly selects a bridge to cross and burns it immediately. We need to find the probability that Christine burns all the br... | 937 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. E. Chen and E. Chen play a game on the $n^2$ points of an $n \times n$ lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be marked only once. The game ends w... | 1. **Understanding the Game Rules**:
- The game is played on an \( n \times n \) lattice grid.
- Players alternately mark points on the grid.
- A point cannot be marked if it is on or inside a non-degenerate triangle formed by three previously marked points.
- Each point can be marked only once.
- The ga... | 1007 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For positive integers $n$, let $s(n)$ denote the sum of the squares of the positive integers less than or equal to $n$ that are relatively prime to $n$. Find the greatest integer less than or equal to \[ \sum_{n\mid 2013} \frac{s(n)}{n^2}, \] where the summation runs over all positive integers $n$ dividing $2013$.
[i]... | To solve the problem, we need to find the greatest integer less than or equal to
\[
\sum_{n \mid 2013} \frac{s(n)}{n^2},
\]
where \( s(n) \) denotes the sum of the squares of the positive integers less than or equal to \( n \) that are relatively prime to \( n \), and the summation runs over all positive integers \( n... | 345 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a nondegenerate isosceles trapezoid with integer side lengths such that $BC \parallel AD$ and $AB=BC=CD$. Given that the distance between the incenters of triangles $ABD$ and $ACD$ is $8!$, determine the number of possible lengths of segment $AD$.
[i]Ray Li[/i] | 1. **Identify the sides and diagonals of the trapezoid:**
Let the sides of the isosceles trapezoid \(ABCD\) be \(AB = BC = CD = a\) and \(AD = b\). Since \(BC \parallel AD\), \(ABCD\) is an isosceles trapezoid with \(AB = CD\) and \(BC = a\).
2. **Calculate the length of the diagonals:**
Using the Pythagorean th... | 337 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$.
[hide="C... | 1. Given the equation \(a^2 + b^2 + c^2 = ab + bc + ca\), we can rearrange it as:
\[
a^2 + b^2 + c^2 - ab - bc - ca = 0
\]
This can be rewritten using the identity for the square of sums:
\[
\frac{1}{2} \left( (a-b)^2 + (b-c)^2 + (c-a)^2 \right) = 0
\]
Since the sum of squares is zero, each indi... | 132 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a tennis tournament, each competitor plays against every other competitor, and there are no draws. Call a group of four tennis players ``ordered'' if there is a clear winner and a clear loser (i.e., one person who beat the other three, and one person who lost to the other three.) Find the smallest integer $n$ for wh... | 1. **Show that 8 is sufficient:**
- In a tournament of 8 people, each player plays against every other player. The total number of games played is given by the combination formula:
\[
\binom{8}{2} = \frac{8 \times 7}{2} = 28
\]
- Therefore, there are 28 wins and 28 losses in total.
- By the pige... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose tetrahedron $PABC$ has volume $420$ and satisfies $AB = 13$, $BC = 14$, and $CA = 15$. The minimum possible surface area of $PABC$ can be written as $m+n\sqrt{k}$, where $m,n,k$ are positive integers and $k$ is not divisible by the square of any prime. Compute $m+n+k$.
[i]Ray Li[/i] | 1. **Projection and Altitudes**:
Let \( P' \) be the projection of \( P \) onto triangle \( ABC \). The height from \( P \) to the plane \( ABC \) is given as \( 15 \). Let \( x, y, z \) be such that \( x + y + z = 1 \), where \( [P'BC]/[ABC] = x \), \( [P'CA]/[ABC] = y \), and \( [P'AB]/[ABC] = z \). Let \( h_A \) ... | 346 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ denote the set of words $W = w_1w_2\ldots w_n$ of any length $n\ge0$ (including the empty string $\lambda$), with each letter $w_i$ from the set $\{x,y,z\}$. Call two words $U,V$ [i]similar[/i] if we can insert a string $s\in\{xyz,yzx,zxy\}$ of three consecutive letters somewhere in $U$ (possibly at one of the ... | 1. **Reformulate the Problem**: We need to find \( p + q \) given that:
\[
\frac{p}{q} = \sum_{n \ge 0} f(n) \left( \frac{225}{8192} \right)^n
\]
where \( f(n) \) denotes the number of trivial words in \( S \) of length \( 3n \).
2. **Simplify the Problem**: We can replace the strings \( xyz, yzx, zxy \) w... | 61 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A [i]palindromic table[/i] is a $3 \times 3$ array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below.
\[ \begin{array}[h]{ccc}
O & M & O \\
N & M & N \\
O & M & O
\end{array} \]
How many palindromic tables are there that use only... | To determine the number of palindromic tables that use only the letters \( O \) and \( M \), we need to analyze the constraints imposed by the palindromic property on a \( 3 \times 3 \) array.
1. **Identify the positions in the table:**
\[
\begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i \\
\en... | 16 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a_1, a_2, a_3, \dots$ is an increasing arithmetic progression of positive integers. Given that $a_3 = 13$, compute the maximum possible value of \[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \][i]Proposed by Evan Chen[/i] | 1. Let the common difference of the arithmetic progression be \( d \). Given that \( a_3 = 13 \), we can express the first few terms of the sequence as follows:
\[
a_1 = 13 - 2d, \quad a_2 = 13 - d, \quad a_3 = 13, \quad a_4 = 13 + d, \quad a_5 = 13 + 2d
\]
2. The general formula for the \( n \)-th term of th... | 365 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A wishing well is located at the point $(11,11)$ in the $xy$-plane. Rachelle randomly selects an integer $y$ from the set $\left\{ 0, 1, \dots, 10 \right\}$. Then she randomly selects, with replacement, two integers $a,b$ from the set $\left\{ 1,2,\dots,10 \right\}$. The probability the line through $(0,y)$ and $(a,b... | 1. **Identify the coordinates of the points and the conditions for the line to pass through the well:**
- The well is located at \((11, 11)\).
- Rachelle selects an integer \(y\) from \(\{0, 1, \dots, 10\}\).
- She then selects two integers \(a\) and \(b\) from \(\{1, 2, \dots, 10\}\).
2. **Determine the equa... | 111 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of integers $n$ with $n \ge 2$ such that the remainder when $2013$ is divided by $n$ is equal to the remainder when $n$ is divided by $3$.
[i]Proposed by Michael Kural[/i] | We need to find the number of integers \( n \) with \( n \ge 2 \) such that the remainder when \( 2013 \) is divided by \( n \) is equal to the remainder when \( n \) is divided by \( 3 \). This can be expressed as:
\[ 2013 \mod n = n \mod 3. \]
We will consider three cases based on the value of \( n \mod 3 \).
1. **... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Points $M$, $N$, $P$ are selected on sides $\overline{AB}$, $\overline{AC}$, $\overline{BC}$, respectively, of triangle $ABC$. Find the area of triangle $MNP$ given that $AM=MB=BP=15$ and $AN=NC=CP=25$.
[i]Proposed by Evan Chen[/i] | 1. **Calculate the area of triangle \( \triangle ABC \)**:
- Given \( AM = MB = BP = 15 \) and \( AN = NC = CP = 25 \), we can determine the lengths of the sides of \( \triangle ABC \):
\[
AB = AM + MB = 15 + 15 = 30
\]
\[
AC = AN + NC = 25 + 25 = 50
\]
\[
BC = BP + PC = 15 + ... | 150 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $x_1 < x_2 < \dots < x_n$ is a sequence of positive integers such that $x_k$ divides $x_{k+2}$ for each $k = 1, 2, \dots, n-2$. Given that $x_n = 1000$, what is the largest possible value of $n$?
[i]Proposed by Evan Chen[/i] | 1. **Factorize 1000**:
\[
1000 = 2^3 \times 5^3
\]
This factorization will help us understand the possible divisors of 1000.
2. **Sequence Construction**:
We need to construct a sequence \( x_1 < x_2 < \dots < x_n \) such that \( x_k \) divides \( x_{k+2} \) for each \( k = 1, 2, \dots, n-2 \) and \( ... | 13 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $AXYZB$ be a regular pentagon with area $5$ inscribed in a circle with center $O$. Let $Y'$ denote the reflection of $Y$ over $\overline{AB}$ and suppose $C$ is the center of a circle passing through $A$, $Y'$ and $B$. Compute the area of triangle $ABC$.
[i]Proposed by Evan Chen[/i] | 1. **Identify the key points and their properties:**
- $AXYZB$ is a regular pentagon inscribed in a circle with center $O$.
- The area of the pentagon is given as $5$.
- $Y'$ is the reflection of $Y$ over $\overline{AB}$.
- $C$ is the center of a circle passing through $A$, $Y'$, and $B$.
2. **Determine th... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In the rectangular table shown below, the number $1$ is written in the upper-left hand corner, and every number is the sum of the any numbers directly to its left and above. The table extends infinitely downwards and to the right.
\[
\begin{array}{cccccc}
1 & 1 & 1 & 1 & 1 & \cdots \\
1 & 2 & 3 & 4 & 5 & \c... | 1. **Understanding the Table Structure**:
The table is constructed such that each entry is the sum of the number directly above it and the number directly to its left. This structure is similar to Pascal's Triangle, where each entry is a binomial coefficient. Specifically, the entry in the \(i\)-th row and \(j\)-th ... | 19 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In the universe of Pi Zone, points are labeled with $2 \times 2$ arrays of positive reals. One can teleport from point $M$ to point $M'$ if $M$ can be obtained from $M'$ by multiplying either a row or column by some positive real. For example, one can teleport from $\left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \... | 1. **Define the problem in terms of logarithms:**
Let the grid be:
\[
M = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\]
where \(a, b, c, d\) are positive reals. We can represent the numbers in terms of their base-2 logarithms. Let:
\[
\log_2(a) = x, \quad \log_2(b) = y, \quad \log_2(c) =... | 17 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the positive integer $n$ such that \[ \underbrace{f(f(\cdots f}_{2013 \ f\text{'s}}(n)\cdots ))=2014^2+1 \] where $f(n)$ denotes the $n$th positive integer which is not a perfect square.
[i]Proposed by David Stoner[/i] | 1. We start with the given equation:
\[
\underbrace{f(f(\cdots f}_{2013 \ f\text{'s}}(n)\cdots ))=2014^2+1
\]
where \( f(n) \) denotes the \( n \)-th positive integer which is not a perfect square.
2. To solve this, we need to understand the function \( f \) and its inverse \( f^{-1} \). The function \( f ... | 6077248 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\sigma(n)$ be the number of positive divisors of $n$, and let $\operatorname{rad} n$ be the product of the distinct prime divisors of $n$. By convention, $\operatorname{rad} 1 = 1$. Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\ope... | 1. **Define the function \( f(n) \):**
\[
f(n) = \frac{\sigma(n) \sigma(n \operatorname{rad} n)}{n^2 \sigma(\operatorname{rad} n)}
\]
where \(\sigma(n)\) is the sum of the divisors of \(n\) and \(\operatorname{rad}(n)\) is the product of the distinct prime divisors of \(n\).
2. **Identify the multiplicativ... | 164 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $n$ is called [i]mythical[/i] if every divisor of $n$ is two less than a prime. Find the unique mythical number with the largest number of divisors.
[i]Proposed by Evan Chen[/i] | 1. **Initial Considerations**:
- Let \( m \) be the desired mythical number.
- A number \( n \) is mythical if every divisor of \( n \) is two less than a prime.
- We need to find \( m \) such that it has the largest number of divisors.
2. **Modulo 6 Analysis**:
- Any positive even number is two less than ... | 135 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AB = 5$, $AC = 8$, and $BC = 7$. Let $D$ be on side $AC$ such that $AD = 5$ and $CD = 3$. Let $I$ be the incenter of triangle $ABC$ and $E$ be the intersection of the perpendicular bisectors of $\overline{ID}$ and $\overline{BC}$. Suppose $DE = \frac{a\sqrt{b}}{c}$ where $a$ and $c$ are r... | 1. **Determine the angle $\angle BAC$ using the Law of Cosines:**
Given:
\[
AB = 5, \quad AC = 8, \quad BC = 7
\]
Using the Law of Cosines:
\[
\cos \angle BAC = \frac{AB^2 + AC^2 - BC^2}{2 \cdot AB \cdot AC}
\]
Substitute the given values:
\[
\cos \angle BAC = \frac{5^2 + 8^2 - 7^2}{2 ... | 13 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all integers $m$ with $1 \le m \le 300$ such that for any integer $n$ with $n \ge 2$, if $2013m$ divides $n^n-1$ then $2013m$ also divides $n-1$.
[i]Proposed by Evan Chen[/i] | 1. **Understanding the Problem:**
We need to find all integers \( m \) such that \( 1 \le m \le 300 \) and for any integer \( n \ge 2 \), if \( 2013m \) divides \( n^n - 1 \), then \( 2013m \) also divides \( n - 1 \).
2. **Analyzing the Condition:**
We claim that \( M \) (where \( M = 2013m \)) is good if for e... | 4650 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$.
[i]Proposed by Evan Chen[/i] | 1. **Identify the given information and set up the problem:**
- We have a quadrilateral \(ABCD\) with \(AD = 20\) and \(BC = 13\).
- The area of \(\triangle ABC\) is \(338\) and the area of \(\triangle DBC\) is \(212\).
2. **Introduce the feet of the altitudes:**
- Let \(X\) and \(Y\) be the feet of the altit... | 118 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Ben has a big blackboard, initially empty, and Francisco has a fair coin. Francisco flips the coin $2013$ times. On the $n^{\text{th}}$ flip (where $n=1,2,\dots,2013$), Ben does the following if the coin flips heads:
(i) If the blackboard is empty, Ben writes $n$ on the blackboard.
(ii) If the blackboard is not emp... | 1. **Understanding the Problem:**
- Ben writes or erases numbers on a blackboard based on the outcome of Francisco's coin flips.
- The coin is flipped 2013 times.
- If the coin lands heads, Ben either writes or erases a number based on specific conditions.
- We need to find the probability that the blackboa... | 1336 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ denote the product of the first $2013$ primes. Find the sum of all primes $p$ with $20 \le p \le 150$ such that
(i) $\frac{p+1}{2}$ is even but is not a power of $2$, and
(ii) there exist pairwise distinct positive integers $a,b,c$ for which \[ a^n(a-b)(a-c) + b^n(b-c)(b-a) + c^n(c-a)(c-b) \] is divisible by ... | 1. **Identify the primes \( p \) within the given range that satisfy condition (i):**
- Condition (i) states that \(\frac{p+1}{2}\) is even but not a power of 2.
- This implies \( p+1 \) is divisible by 4 but not by any higher power of 2.
- We list the primes between 20 and 150 and check which ones satisfy thi... | 431 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Kevin has $255$ cookies, each labeled with a unique nonempty subset of $\{1,2,3,4,5,6,7,8\}$. Each day, he chooses one cookie uniformly at random out of the cookies not yet eaten. Then, he eats that cookie, and all remaining cookies that are labeled with a subset of that cookie (for example, if he chooses the cookie la... | 1. **Define the Problem and Variables:**
Kevin has \(255\) cookies, each labeled with a unique nonempty subset of \(\{1,2,3,4,5,6,7,8\}\). Each day, he chooses one cookie uniformly at random from the remaining cookies, eats it, and also eats all cookies labeled with subsets of the chosen cookie. We need to find the ... | 213 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(t) = t^3+27t^2+199t+432$. Suppose $a$, $b$, $c$, and $x$ are distinct positive reals such that $P(-a)=P(-b)=P(-c)=0$, and \[
\sqrt{\frac{a+b+c}{x}} = \sqrt{\frac{b+c+x}{a}} + \sqrt{\frac{c+a+x}{b}} + \sqrt{\frac{a+b+x}{c}}. \] If $x=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.... | 1. Given the polynomial \( P(t) = t^3 + 27t^2 + 199t + 432 \), we know that \( P(-a) = P(-b) = P(-c) = 0 \). This implies that \( -a, -b, -c \) are the roots of the polynomial \( P(t) \).
2. We are given the equation:
\[
\sqrt{\frac{a+b+c}{x}} = \sqrt{\frac{b+c+x}{a}} + \sqrt{\frac{c+a+x}{b}} + \sqrt{\frac{a+b+x}... | 847 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a,b,c>0$ are integers such that \[abc-bc-ac-ab+a+b+c=2013.\] Find the number of possibilities for the ordered triple $(a,b,c)$. | 1. We start with the given equation:
\[
abc - bc - ac - ab + a + b + c = 2013
\]
2. To simplify this, we add and subtract 1:
\[
abc - bc - ac - ab + a + b + c + 1 - 1 = 2013
\]
\[
(a-1)(b-1)(c-1) + 1 = 2013
\]
3. Subtract 1 from both sides:
\[
(a-1)(b-1)(c-1) = 2012
\]
4. Let \( A... | 39 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Betty Lou and Peggy Sue take turns flipping switches on a $100 \times 100$ grid. Initially, all switches are "off". Betty Lou always flips a horizontal row of switches on her turn; Peggy Sue always flips a vertical column of switches. When they finish, there is an odd number of switches turned "on'' in each row and col... | 1. **Understanding the Problem:**
- We have a $100 \times 100$ grid where initially all switches are "off".
- Betty Lou flips a horizontal row of switches on her turn.
- Peggy Sue flips a vertical column of switches on her turn.
- The goal is to have an odd number of switches turned "on" in each row and col... | 9802 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $x_1=1/20$, $x_2=1/13$, and \[x_{n+2}=\dfrac{2x_nx_{n+1}(x_n+x_{n+1})}{x_n^2+x_{n+1}^2}\] for all integers $n\geq 1$. Evaluate $\textstyle\sum_{n=1}^\infty(1/(x_n+x_{n+1}))$. | 1. Given the sequence \( x_1 = \frac{1}{20} \) and \( x_2 = \frac{1}{13} \), and the recurrence relation:
\[
x_{n+2} = \frac{2x_n x_{n+1} (x_n + x_{n+1})}{x_n^2 + x_{n+1}^2}
\]
for all integers \( n \geq 1 \).
2. We start by taking the reciprocal of both sides of the recurrence relation:
\[
\frac{1}{... | 23 | Other | other | Yes | Yes | aops_forum | false |
Find the number of pairs $(n,C)$ of positive integers such that $C\leq 100$ and $n^2+n+C$ is a perfect square. | 1. We start with the equation \( n^2 + n + C = (n + k)^2 \). Expanding the right-hand side, we get:
\[
n^2 + n + C = n^2 + 2nk + k^2
\]
Simplifying, we find:
\[
C = k^2 + n(2k - 1)
\]
2. Given \( C \leq 100 \), we substitute \( C \) from the above equation:
\[
k^2 + n(2k - 1) \leq 100
\]
... | 180 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a,b$ are nonzero integers such that two roots of $x^3+ax^2+bx+9a$ coincide, and all three roots are integers. Find $|ab|$. | 1. Let the roots of the polynomial \(x^3 + ax^2 + bx + 9a\) be \(m, m, n\). Since two roots coincide, we have:
\[
(x - m)^2 (x - n) = x^3 + ax^2 + bx + 9a
\]
2. By expanding \((x - m)^2 (x - n)\), we get:
\[
(x - m)^2 (x - n) = (x^2 - 2mx + m^2)(x - n) = x^3 - nx^2 - 2mx^2 + mnx + m^2x - m^2n
\]
S... | 96 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
We construct three circles: $O$ with diameter $AB$ and area $12+2x$, $P$ with diameter $AC$ and area $24+x$, and $Q$ with diameter $BC$ and area $108-x$. Given that $C$ is on circle $O$, compute $x$. | 1. **Identify the relationship between the circles and their areas:**
- Circle \( O \) has diameter \( AB \) and area \( 12 + 2x \).
- Circle \( P \) has diameter \( AC \) and area \( 24 + x \).
- Circle \( Q \) has diameter \( BC \) and area \( 108 - x \).
2. **Use the fact that \( C \) is on circle \( O \):... | 60 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider the shape formed from taking equilateral triangle $ABC$ with side length $6$ and tracing out the arc $BC$ with center $A$. Set the shape down on line $l$ so that segment $AB$ is perpendicular to $l$, and $B$ touches $l$. Beginning from arc $BC$ touching $l$, we roll $ABC$ along $l$ until both points $A$ and ... | 1. **Understanding the Problem:**
- We have an equilateral triangle \(ABC\) with side length \(6\).
- We trace out the arc \(BC\) with center \(A\).
- The triangle is placed on line \(l\) such that \(AB\) is perpendicular to \(l\) and \(B\) touches \(l\).
- We roll the triangle along \(l\) until both points... | 21 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose you have a sphere tangent to the $xy$-plane with its center having positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\tfrac pq$ where $p,q$ are integers, find $p+q$. | 1. **Identify the properties of the sphere and the point \( P \):**
- The sphere is tangent to the \( xy \)-plane.
- The center of the sphere has a positive \( z \)-coordinate.
- The point \( P = (0, b, a) \) projects the sphere onto the \( xy \)-plane, forming the conic section \( y = x^2 \).
2. **Understand... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
On a circle, points $A,B,C,D$ lie counterclockwise in this order. Let the orthocenters of $ABC,BCD,CDA,DAB$ be $H,I,J,K$ respectively. Let $HI=2$, $IJ=3$, $JK=4$, $KH=5$. Find the value of $13(BD)^2$. | 1. Given that points \(A, B, C, D\) lie counterclockwise on a circle, and the orthocenters of triangles \(ABC, BCD, CDA, DAB\) are \(H, I, J, K\) respectively. We are given the distances \(HI = 2\), \(IJ = 3\), \(JK = 4\), and \(KH = 5\).
2. It is a known result in geometry that the quadrilateral formed by the orthoce... | 169 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Three chords of a sphere, each having length $5,6,7$, intersect at a single point inside the sphere and are pairwise perpendicular. For $R$ the maximum possible radius of this sphere, find $R^2$. | 1. Suppose the sphere is centered at the origin with radius \( R \). Let the intersection of the chords be at the point \((x, y, z)\), and assume the directions of the chords are parallel to the coordinate axes. This means the chords are along the \(x\)-axis, \(y\)-axis, and \(z\)-axis.
2. The lengths of the chords ar... | 15 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Including the original, how many ways are there to rearrange the letters in PRINCETON so that no two vowels (I, E, O) are consecutive and no three consonants (P, R, N, C, T, N) are consecutive? | To solve the problem of rearranging the letters in "PRINCETON" such that no two vowels (I, E, O) are consecutive and no three consonants (P, R, N, C, T, N) are consecutive, we need to consider the constraints carefully and use combinatorial methods.
1. **Identify the letters and constraints:**
- Vowels: I, E, O
... | 17280 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The number of positive integer pairs $(a,b)$ that have $a$ dividing $b$ and $b$ dividing $2013^{2014}$ can be written as $2013n+k$, where $n$ and $k$ are integers and $0\leq k<2013$. What is $k$? Recall $2013=3\cdot 11\cdot 61$. | 1. We start by expressing \( b \) in terms of its prime factors. Given \( 2013 = 3 \cdot 11 \cdot 61 \), we can write \( b \) as:
\[
b = 3^x \cdot 11^y \cdot 61^z
\]
where \( 0 \leq x, y, z \leq 2014 \).
2. For \( a \) to divide \( b \), \( a \) must also be of the form:
\[
a = 3^i \cdot 11^j \cdot 6... | 27 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A regular pentagon can have the line segments forming its boundary extended to lines, giving an arrangement of lines that intersect at ten points. How many ways are there to choose five points of these ten so that no three of the points are collinear? | To solve this problem, we need to consider the arrangement of points formed by extending the sides of a regular pentagon. These extensions intersect at ten points, and we need to choose five points such that no three of them are collinear.
We will use casework based on the number of points chosen from the outer pentag... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many ways are there to color the edges of a hexagon orange and black if we assume that two hexagons are indistinguishable if one can be rotated into the other? Note that we are saying the colorings OOBBOB and BOBBOO are distinct; we ignore flips. | To solve this problem, we will use Burnside's Lemma, which is a tool in group theory for counting the number of distinct objects under group actions. Here, the group action is the rotation of the hexagon.
1. **Identify the group of symmetries:**
The group of symmetries of a hexagon under rotation is the cyclic grou... | 14 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Chris's pet tiger travels by jumping north and east. Chris wants to ride his tiger from Fine Hall to McCosh, which is $3$ jumps east and $10$ jumps north. However, Chris wants to avoid the horde of PUMaC competitors eating lunch at Frist, located $2$ jumps east and $4$ jumps north of Fine Hall. How many ways can he get... | 1. **Identify the coordinates and the problem setup:**
- Fine Hall is at the origin \((0,0)\).
- McCosh is at \((3,10)\).
- Frist is at \((2,4)\).
2. **Calculate the total number of paths from Fine Hall to McCosh:**
- To get from \((0,0)\) to \((3,10)\), Chris needs to make a total of \(3 + 10 = 13\) jumps... | 181 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many tuples of integers $(a_0,a_1,a_2,a_3,a_4)$ are there, with $1\leq a_i\leq 5$ for each $i$, so that $a_0<a_1>a_2<a_3>a_4$? | To solve the problem of finding the number of tuples \((a_0, a_1, a_2, a_3, a_4)\) such that \(1 \leq a_i \leq 5\) for each \(i\) and \(a_0 < a_1 > a_2 < a_3 > a_4\), we need to consider all possible values for each \(a_i\) while maintaining the given inequalities.
1. **Identify the constraints:**
- \(1 \leq a_i \l... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
You roll three fair six-sided dice. Given that the highest number you rolled is a $5$, the expected value of the sum of the three dice can be written as $\tfrac ab$ in simplest form. Find $a+b$. | To solve this problem, we need to find the expected value of the sum of three dice rolls given that the highest number rolled is 5. We will consider all possible cases where the highest number is 5 and calculate the expected value accordingly.
1. **Identify the possible outcomes:**
- The highest number rolled is 5,... | 706 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A sequence of vertices $v_1,v_2,\ldots,v_k$ in a graph, where $v_i=v_j$ only if $i=j$ and $k$ can be any positive integer, is called a $\textit{cycle}$ if $v_1$ is attached by an edge to $v_2$, $v_2$ to $v_3$, and so on to $v_k$ connected to $v_1$. Rotations and reflections are distinct: $A,B,C$ is distinct from $A,C,... | 1. **Understanding the Cyclomatic Number**:
The cyclomatic number (or circuit rank, or nullity) of a graph \( G \) is given by:
\[
r = m - n + c
\]
where \( m \) is the number of edges, \( n \) is the number of vertices, and \( c \) is the number of connected components. This number represents the minimu... | 1001 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The Miami Heat and the San Antonio Spurs are playing a best-of-five series basketball championship, in which the team that first wins three games wins the whole series. Assume that the probability that the Heat wins a given game is $x$ (there are no ties). The expected value for the total number of games played can be... | 1. **Determine the probabilities for the series lasting 3, 4, and 5 games:**
- The probability that the series lasts exactly 3 games:
\[
P(\text{3 games}) = x^3 + (1-x)^3
\]
This is because either the Heat wins all three games, or the Spurs win all three games.
- The probability that the ser... | 21 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the smallest positive integer $n$ such that $2013^n$ ends in $001$ (i.e. the rightmost three digits of $2013^n$ are $001$? | 1. We need to find the smallest positive integer \( n \) such that \( 2013^n \) ends in \( 001 \). This can be written as:
\[
2013^n \equiv 1 \pmod{1000}
\]
2. Using the Chinese Remainder Theorem, we can break this down into two congruences:
\[
2013^n \equiv 1 \pmod{8} \quad \text{and} \quad 2013^n \equ... | 100 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the smallest integer $n\geq 4$ such that $\textstyle\binom n4$ ends in $4$ or more zeroes (i.e. the rightmost four digits of $\textstyle\binom n4$ are $0000$). | 1. We need to find the smallest integer \( n \geq 4 \) such that \( \binom{n}{4} \) ends in 4 or more zeroes. This means \( \binom{n}{4} \) must be divisible by \( 10^4 = 10000 \).
2. Recall that \( \binom{n}{4} = \frac{n(n-1)(n-2)(n-3)}{24} \). For \( \binom{n}{4} \) to be divisible by \( 10000 = 2^4 \cdot 5^4 \), the... | 8128 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ be the greatest possible value of a product of positive integers that sums to $2014$. Compute the sum of all bases and exponents in the prime factorization of $A$. For example, if $A=7\cdot 11^5$, the answer would be $7+11+5=23$. | 1. To maximize the product of positive integers that sum to 2014, we need to consider the properties of numbers. Specifically, the product of numbers is maximized when the numbers are as close to each other as possible. For positive integers, this means using as many 3's as possible, since $3 \times 3 = 9$ is greater t... | 677 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $d$ be the greatest common divisor of $2^{30^{10}}-2$ and $2^{30^{45}}-2$. Find the remainder when $d$ is divided by $2013$. | 1. We start with the given problem: finding the greatest common divisor (gcd) of \(2^{30^{10}} - 2\) and \(2^{30^{45}} - 2\). We denote this gcd by \(d\).
2. Using the property of gcd for numbers of the form \(a^m - a\) and \(a^n - a\), we have:
\[
\gcd(a^m - a, a^n - a) = a^{\gcd(m, n)} - a
\]
Here, \(a =... | 2012 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.