problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
values | question_type stringclasses 4
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Triangle $ABC$ is so that $AB = 15$, $BC = 22$, and $AC = 20$. Let $D, E, F$ lie on $BC$, $AC$, and $AB$, respectively, so $AD$, $BE$, $CF$ all contain a point $K$. Let $L$ be the second intersection of the circumcircles of $BFK$ and $CEK$. Suppose that $\frac{AK}{KD} = \frac{11}{7}$ , and $BD = 6$. If $KL^2 =\frac{a}... | 1. **Using Menelaus' Theorem:**
Menelaus' theorem states that for a triangle \( \triangle ABC \) with a transversal line intersecting \( BC, CA, \) and \( AB \) at points \( D, E, \) and \( F \) respectively, the following relation holds:
\[
\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = 1
\]
Gi... | 497 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $P$ be a $10$-degree monic polynomial with roots $r_1, r_2, . . . , r_{10} \ne $ and let $Q$ be a $45$-degree monic polynomial with roots $\frac{1}{r_i}+\frac{1}{r_j}-\frac{1}{r_ir_j}$ where $i < j$ and $i, j \in \{1, ... , 10\}$. If $P(0) = Q(1) = 2$, then $\log_2 (|P(1)|)$ can be written as $a/b$ for relatively p... | 1. **Given Information:**
- \( P \) is a 10-degree monic polynomial with roots \( r_1, r_2, \ldots, r_{10} \).
- \( Q \) is a 45-degree monic polynomial with roots \( \frac{1}{r_i} + \frac{1}{r_j} - \frac{1}{r_i r_j} \) where \( i < j \) and \( i, j \in \{1, \ldots, 10\} \).
- \( P(0) = 2 \).
- \( Q(1) = 2 ... | 19 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The function $f(x) = x^2 + (2a + 3)x + (a^2 + 1)$ only has real zeroes. Suppose the smallest possible value of $a$ can be written in the form $p/q$, where $p, q$ are relatively prime integers. Find $|p| + |q|$. | 1. To determine the conditions under which the quadratic function \( f(x) = x^2 + (2a + 3)x + (a^2 + 1) \) has real zeros, we need to ensure that its discriminant is non-negative. The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by:
\[
\Delta = b^2 - 4ac
\]
For the given fu... | 17 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Princeton has an endowment of $5$ million dollars and wants to invest it into improving campus life. The university has three options: it can either invest in improving the dorms, campus parties or dining hall food quality. If they invest $a$ million dollars in the dorms, the students will spend an additional $5a$ hour... | 1. **Define the variables and constraints:**
- Let \( a \) be the amount invested in dorms (in million dollars).
- Let \( b \) be the amount invested in better food (in million dollars).
- Let \( c \) be the amount invested in parties (in million dollars).
- The total investment is constrained by \( a + b +... | 34 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
How many ordered triples of nonzero integers $(a, b, c)$ satisfy $2abc = a + b + c + 4$? | To solve the problem, we need to find all ordered triples of nonzero integers \((a, b, c)\) that satisfy the equation:
\[ 2abc = a + b + c + 4. \]
1. **Rearrange the equation:**
\[ 2abc - a - b - c = 4. \]
2. **Solve for \(c\):**
\[ 2abc - a - b - c = 4 \]
\[ c(2ab - 1) = a + b + 4 \]
\[ c = \frac{a + b +... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The number $2021$ leaves a remainder of $11$ when divided by a positive integer. Find the smallest such integer. | 1. We are given that the number \(2021\) leaves a remainder of \(11\) when divided by a positive integer \(d\). This can be written as:
\[
2021 \equiv 11 \pmod{d}
\]
This implies:
\[
2021 = kd + 11 \quad \text{for some integer } k
\]
Rearranging, we get:
\[
2021 - 11 = kd \implies 2010 = k... | 15 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Cassidy has string of $n$ bits, where $n$ is a positive integer, which initially are all $0$s or $1$s. Every second, Cassidy may choose to do one of two things:
1. Change the first bit (so the first bit changes from a $0$ to a $1$, or vice versa)
2. Change the first bit after the first $1$.
Let $M$ be the minimum num... | 1. **Constructing the new binary string \( T \) from \( S \):**
- We define the \( i \)-th bit of \( T \) to be 1 if and only if the suffix of \( S \) consisting of the \( i \)-th to \( n \)-th bits (inclusive) contains an odd number of 1's.
- This transformation ensures that the operations on \( S \) translate t... | 6826 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, . . . , 2020$. The students are initially divided up into $20$ groups of $101$, with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance to the s... | 1. **Restate the problem**: We need to find the expected value of the champion's number in a tournament where 2020 students are divided into 20 groups of 101, and the lowest-numbered student in each group wins. The champion is then chosen uniformly at random from these 20 winners. The expected value of the champion's n... | 2123 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The roots of the polynomial $f(x) = x^8 +x^7 -x^5 -x^4 -x^3 +x+ 1 $ are all roots of unity. We say that a real number $r \in [0, 1)$ is nice if $e^{2i \pi r} = \cos 2\pi r + i \sin 2\pi r$ is a root of the polynomial $f$ and if $e^{2i \pi r}$ has positive imaginary part. Let $S$ be the sum of the values of nice real n... | 1. **Identify the polynomial and its roots**:
The given polynomial is \( f(x) = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1 \). We are told that all roots of this polynomial are roots of unity.
2. **Factor the polynomial**:
We need to find a way to factor \( f(x) \). Notice that multiplying \( f(x) \) by \( x^2 - x + 1 ... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $AX$ be a diameter of a circle $\Omega$ with radius $10$, and suppose that $C$ lies on $\Omega$ so that $AC = 16$. Let $D$ be the other point on $\Omega$ so $CX = CD$. From here, define $D'$ to be the reflection of $D$ across the midpoint of $AC$, and $X'$ to be the reflection of $X$ across the midpoint of $CD$. ... | 1. **Identify the given elements and their properties:**
- Circle \(\Omega\) with radius \(10\).
- \(AX\) is a diameter, so \(AX = 20\).
- Point \(C\) on \(\Omega\) such that \(AC = 16\).
- Point \(D\) on \(\Omega\) such that \(CX = CD\).
2. **Determine the coordinates of the points:**
- Place the circl... | 96 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A circle is inscribed in a regular octagon with area $2024$. A second regular octagon is inscribed in the circle, and its area can be expressed as $a + b\sqrt{c}$, where $a, b, c$ are integers and $c$ is square-free. Compute $a + b + c$.
| 1. **Determine the side length of the first octagon:**
- Let the radius of the circle inscribed in the first octagon be \( r \).
- The area \( A \) of a regular octagon with side length \( s \) can be given by the formula:
\[
A = 2(1 + \sqrt{2})s^2
\]
- Since the octagon is inscribed in a circle... | 1520 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider a circle centered at $O$. Parallel chords $AB$ of length $8$ and $CD$ of length $10$ are of distance $2$ apart such that $AC < AD$. We can write $\tan \angle BOD =\frac{a}{b}$ , where $a, b$ are positive integers such that gcd $(a, b) = 1$. Compute $a + b$. | 1. **Identify the given information and set up the problem:**
- We have a circle centered at \( O \).
- Chords \( AB \) and \( CD \) are parallel with lengths 8 and 10, respectively.
- The distance between the chords is 2 units.
- We need to find \(\tan \angle BOD\) in the form \(\frac{a}{b}\) where \(a\) a... | 113 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $BCDE$ be a trapezoid with $BE\parallel CD$, $BE = 20$, $BC = 2\sqrt{34}$, $CD = 8$, $DE = 2\sqrt{10}$. Draw a line through $E$ parallel to $BD$ and a line through $B$ perpendicular to $BE$, and let $A$ be the intersection of these two lines. Let $M$ be the intersection of diagonals $BD$ and $CE$, and let $X$ be th... | 1. **Determine the height of the trapezoid:**
- Drop perpendiculars from \(C\) and \(D\) to \(BE\), denoted as \(CN\) and \(DK\) respectively.
- Let \(BN = x\) and \(EK = 12 - x\).
- Using the Pythagorean theorem in triangles \(BCN\) and \(DEK\):
\[
BC^2 = BN^2 + CN^2 \implies (2\sqrt{34})^2 = x^2 + ... | 203 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A pentagon has vertices labelled $A, B, C, D, E$ in that order counterclockwise, such that $AB$, $ED$ are parallel and $\angle EAB = \angle ABD = \angle ACD = \angle CDA$. Furthermore, suppose that$ AB = 8$, $AC = 12$, $AE = 10$. If the area of triangle $CDE$ can be expressed as $\frac{a \sqrt{b}}{c}$, where $a, b, c$ ... | 1. **Claim that the five points lie on the same circle:**
- Given that \(AB \parallel ED\) and \(\angle EAB = \angle ABD = \angle ACD = \angle CDA\), we can infer that \(ABED\) is an isosceles trapezoid.
- Since \(AB \parallel ED\) and \(\angle EAB = \angle ABD\), it follows that \(A, B, D, E\) lie on the same ci... | 264 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Parallelogram $ABCD$ is given such that $\angle ABC$ equals $30^o$ . Let $X$ be the foot of the perpendicular from $A$ onto $BC$, and $Y$ the foot of the perpendicular from $C$ to $AB$. If $AX = 20$ and $CY = 22$, find the area of the parallelogram.
| 1. Given that $\angle ABC = 30^\circ$ in parallelogram $ABCD$, we need to find the area of the parallelogram.
2. Let $X$ be the foot of the perpendicular from $A$ onto $BC$, and $Y$ be the foot of the perpendicular from $C$ onto $AB$.
3. We are given $AX = 20$ and $CY = 22$.
4. In $\triangle CYB$, we know that $\angle ... | 880 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A right cylinder is given with a height of $20$ and a circular base of radius $5$. A vertical planar cut is made into this base of radius $5$. A vertical planar cut, perpendicular to the base, is made into this cylinder, splitting the cylinder into two pieces. Suppose the area the cut leaves behind on one of the piece... | 1. **Identify the given parameters and the problem setup:**
- Height of the cylinder, \( h = 20 \)
- Radius of the base, \( r = 5 \)
- Area of the cut left behind on one of the pieces, \( 100\sqrt{2} \)
2. **Determine the length of the chord \( AB \) formed by the cut:**
- The area of the cut is given as \... | 625 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Compute the sum of all real numbers x which satisfy the following equation $$\frac {8^x - 19 \cdot 4^x}{16 - 25 \cdot 2^x}= 2$$ | 1. **Substitution**: Let \( a = 2^x \). Then the given equation transforms as follows:
\[
\frac{8^x - 19 \cdot 4^x}{16 - 25 \cdot 2^x} = 2
\]
Since \( 8^x = (2^3)^x = (2^x)^3 = a^3 \) and \( 4^x = (2^2)^x = (2^x)^2 = a^2 \), the equation becomes:
\[
\frac{a^3 - 19a^2}{16 - 25a} = 2
\]
2. **Simplif... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For a bijective function $g : R \to R$, we say that a function $f : R \to R$ is its superinverse if it satisfies the following identity $(f \circ g)(x) = g^{-1}(x)$, where $g^{-1}$ is the inverse of $g$. Given $g(x) = x^3 + 9x^2 + 27x + 81$ and $f$ is its superinverse, find $|f(-289)|$. | 1. Given the function \( g(x) = x^3 + 9x^2 + 27x + 81 \), we need to find its inverse \( g^{-1}(x) \). Notice that:
\[
g(x) = (x+3)^3 + 54
\]
To find the inverse, we solve for \( x \) in terms of \( y \):
\[
y = (x+3)^3 + 54
\]
Subtract 54 from both sides:
\[
y - 54 = (x+3)^3
\]
Take... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4$ and let $\zeta = e^{2\pi i/5} = \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}$. Find the value of the following expression: $$f(\zeta)f(\zeta^2)f(\zeta^3)f(\zeta^4).$$
| 1. Define the function \( f(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4 \).
2. Let \(\zeta = e^{2\pi i/5} = \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}\), which is a primitive 5th root of unity.
3. Consider the polynomial \( g(x) = x(1 + x + x^2 + x^3 + x^4) \). Notice that \( f(x) = g'(x) \).
4. The polynomial \( 1 + x + x^2 + x... | 125 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The roots of a monic cubic polynomial $p$ are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to $10$. Under these conditions, the largest possible value of $|p(-1)|$ can be written as $\frac{m}{n}$, where $m$, $n$ are relatively prime integers. Find $m + n$. | 1. **Identify the roots and their properties:**
Let the roots of the monic cubic polynomial \( p(x) \) be \( a \), \( ar \), and \( ar^2 \), where \( a \) is the first term and \( r \) is the common ratio of the geometric sequence. Since the roots are positive real numbers and form a geometric sequence, we have:
... | 2224 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Consider the sum $$S =\sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|.$$
The value of $S$ can be written as $\tan \left( \frac{c\pi}{d} \right)$ for some relatively prime positive integers $c, d$, satisfying $2c < d$. Find the value of $c + d$. | 1. Consider the sum \( S = \sum_{j=1}^{2021} \left|\sin \frac{2\pi j}{2021}\right| \). Notice that \(\sin \frac{2\pi j}{2021} = 0\) when \(j = 2021\), so we can ignore this term.
2. Due to the absolute value, we can rewrite the sum as:
\[
S = 2 \sum_{j=1}^{1010} \left|\sin \frac{2\pi j}{2021}\right|
\]
This... | 3031 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Kris is asked to compute $\log_{10} (x^y)$, where $y$ is a positive integer and $x$ is a positive real number. However, they misread this as $(\log_{10} x)^y$ , and compute this value. Despite the reading error, Kris still got the right answer. Given that $x > 10^{1.5}$ , determine the largest possible value of $y$. | 1. We start with the given problem: Kris is asked to compute $\log_{10} (x^y)$, but instead computes $(\log_{10} x)^y$. Despite the error, Kris gets the correct answer. We need to determine the largest possible value of $y$ given that $x > 10^{1.5}$.
2. Let $a = x^y$. Then, $\log_{10} (x^y) = \log_{10} (a)$. By the pr... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A [i]substring [/i] of a number $n$ is a number formed by removing some digits from the beginning and end of $n$ (possibly a different number of digits is removed from each side). Find the sum of all prime numbers $p$ that have the property that any substring of $p$ is also prime.
| To solve this problem, we need to find all prime numbers \( p \) such that any substring of \( p \) is also prime. Let's break down the solution step by step.
1. **Identify the digits that can form prime numbers:**
The digits that can form prime numbers are \(2, 3, 5,\) and \(7\). This is because any other digit wi... | 576 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Andrew has a four-digit number whose last digit is $2$. Given that this number is divisible by $9$, determine the number of possible values for this number that Andrew could have. Note that leading zeros are not allowed. | To determine the number of possible four-digit numbers ending in 2 that are divisible by 9, we need to use the property that a number is divisible by 9 if and only if the sum of its digits is divisible by 9.
Let the four-digit number be represented as \( A_1A_2A_3 2 \). The sum of the digits of this number is \( A_1 +... | 489 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The smallest three positive proper divisors of an integer n are $d_1 < d_2 < d_3$ and they satisfy $d_1 + d_2 + d_3 = 57$. Find the sum of the possible values of $d_2$. | 1. Given that the smallest three positive proper divisors of an integer \( n \) are \( d_1 < d_2 < d_3 \) and they satisfy \( d_1 + d_2 + d_3 = 57 \).
2. Since \( d_1 \) is the smallest positive proper divisor, it must be \( d_1 = 1 \).
3. Therefore, we have \( 1 + d_2 + d_3 = 57 \), which simplifies to \( d_2 + d_3 = ... | 42 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A triangle $\vartriangle A_0A_1A_2$ in the plane has sidelengths $A_0A_1 = 7$,$A_1A_2 = 8$,$A_2A_0 = 9$. For $i \ge 0$, given $\vartriangle A_iA_{i+1}A_{i+2}$, let $A_{i+3}$ be the midpoint of $A_iA_{i+1}$ and let Gi be the centroid of $\vartriangle A_iA_{i+1}A_{i+2}$. Let point $G$ be the limit of the sequence of poin... | 1. **Define the initial points and centroids:**
- Given the triangle $\triangle A_0A_1A_2$ with side lengths $A_0A_1 = 7$, $A_1A_2 = 8$, and $A_2A_0 = 9$.
- For $i \ge 0$, define $A_{i+3}$ as the midpoint of $A_iA_{i+1}$.
- Let $G_i$ be the centroid of $\triangle A_iA_{i+1}A_{i+2}$.
2. **Calculate the coordin... | 422 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\vartriangle ABC$ be a triangle. Let $Q$ be a point in the interior of $\vartriangle ABC$, and let $X, Y,Z$ denote the feet of the altitudes from $Q$ to sides $BC$, $CA$, $AB$, respectively. Suppose that $BC = 15$, $\angle ABC = 60^o$, $BZ = 8$, $ZQ = 6$, and $\angle QCA = 30^o$. Let line $QX$ intersect the circu... | 1. **Identify the given information and setup the problem:**
- We have a triangle $\triangle ABC$ with $BC = 15$, $\angle ABC = 60^\circ$, $BZ = 8$, $ZQ = 6$, and $\angle QCA = 30^\circ$.
- $Q$ is a point inside $\triangle ABC$ and $X, Y, Z$ are the feet of the perpendiculars from $Q$ to $BC$, $CA$, and $AB$ resp... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\vartriangle ABC$ be an equilateral triangle. Points $D,E, F$ are drawn on sides $AB$,$BC$, and $CA$ respectively such that $[ADF] = [BED] + [CEF]$ and $\vartriangle ADF \sim \vartriangle BED \sim \vartriangle CEF$. The ratio $\frac{[ABC]}{[DEF]}$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ , where $a$, $b$, $c$,... | 1. **Understanding the Problem:**
We are given an equilateral triangle \( \triangle ABC \) with points \( D, E, F \) on sides \( AB, BC, \) and \( CA \) respectively such that \( [ADF] = [BED] + [CEF] \) and \( \triangle ADF \sim \triangle BED \sim \triangle CEF \). We need to find the ratio \( \frac{[ABC]}{[DEF]} \... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$2$. What is the smallest positive number $k$ such that there are real number satisfying $a+b=k$ and $ab=k$ | 1. We start with the given equations:
\[
a + b = k \quad \text{and} \quad ab = k
\]
2. We can rewrite the second equation as:
\[
ab = a + b
\]
3. Rearrange the equation:
\[
ab - a - b = 0
\]
4. Add 1 to both sides:
\[
ab - a - b + 1 = 1
\]
5. Factor the left-hand side:
\[
(a-1)... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
$4.$ Harry, Hermione, and Ron go to Diagon Alley to buy chocolate frogs. If Harry and Hermione spent one-fourth of their own money, they would spend $3$ galleons in total. If Harry and Ron spent one-fifth of their own money, they would spend $24$ galleons in total. Everyone has a whole number of galleons, and the numbe... | 1. Let \( H \), \( He \), and \( R \) represent the number of galleons Harry, Hermione, and Ron have, respectively.
2. According to the problem, if Harry and Hermione spent one-fourth of their own money, they would spend 3 galleons in total. This can be written as:
\[
\frac{1}{4}H + \frac{1}{4}He = 3
\]
Sim... | 6 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] You are given a number, and round it to the nearest thousandth, round this result to nearest hundredth, and round this result to the nearest tenth. If the final result is $.7$, what is the smallest number you could have been given? As is customary, $5$’s are always rounded up. Give the answer as a decimal.
... | 1. We start with the expression \(4x^4 + 1\) and factor it using the difference of squares:
\[
4x^4 + 1 = (2x^2 + 1)^2 - (2x)^2
\]
2. Applying the difference of squares formula \(a^2 - b^2 = (a + b)(a - b)\), we get:
\[
4x^4 + 1 = (2x^2 + 1 + 2x)(2x^2 + 1 - 2x)
\]
3. Simplifying the factors, we have:
... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The temperatures $ f^\circ \text{F}$ and $ c^\circ \text{C}$ are equal when $ f \equal{} \frac {9}{5}c \plus{} 32$. What temperature is the same in both $ ^\circ \text{F}$ and $ ^\circ \text{C}$? | 1. Let \( x \) be the temperature that is the same in both Fahrenheit and Celsius. Therefore, we have:
\[
x^\circ \text{F} = x^\circ \text{C}
\]
2. According to the given formula for converting Celsius to Fahrenheit:
\[
f = \frac{9}{5}c + 32
\]
Since \( f = x \) and \( c = x \), we substitute \( x ... | -40 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Tickets for the football game are \$10 for students and \$15 for non-students. If 3000 fans attend and pay \$36250, how many students went? | 1. Let \( s \) represent the number of students and \( n \) represent the number of non-students. We are given two equations based on the problem statement:
\[
s + n = 3000
\]
\[
10s + 15n = 36250
\]
2. To eliminate one of the variables, we can manipulate these equations. First, multiply the first eq... | 1750 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find $ [\sqrt{19992000}]$ where $ [x]$ is the greatest integer less than or equal to $ x$. | To find the greatest integer less than or equal to $\sqrt{19992000}$, we will follow these steps:
1. **Estimate the square root**:
- Notice that $19992000$ is close to $20000000$.
- We know that $4500^2 = 20250000$.
- Therefore, $\sqrt{19992000}$ should be slightly less than $4500$.
2. **Set up the equation*... | 4471 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Edward's formula for the stock market predicts correctly that the price of HMMT is directly proportional to a secret quantity $ x$ and inversely proportional to $ y$, the number of hours he slept the night before. If the price of HMMT is $ \$12$ when $ x\equal{}8$ and $ y\equal{}4$, how many dollars does it cost when $... | 1. Let the price of HMMT be denoted by \( h \). According to the problem, \( h \) is directly proportional to \( x \) and inversely proportional to \( y \). This relationship can be expressed as:
\[
h = k \frac{x}{y}
\]
where \( k \) is a constant of proportionality.
2. We are given that when \( x = 8 \) a... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Joe bikes $x$ miles East at $20$ mph to his friend’s house. He then turns South and bikes $x$ miles at $20$ mph to the store. Then, Joe turns East again and goes to his grandma’s house at $14$ mph. On this last leg, he has to carry flour he bought for her at the store. Her house is $2$ more miles from the store than Joe... | 1. Joe bikes $x$ miles East at $20$ mph to his friend’s house. The time taken for this leg of the journey is:
\[
t_1 = \frac{x}{20}
\]
2. Joe then turns South and bikes $x$ miles at $20$ mph to the store. The time taken for this leg of the journey is:
\[
t_2 = \frac{x}{20}
\]
3. Joe then turns East ... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
You use a lock with four dials, each of which is set to a number between 0 and 9 (inclusive). You can never remember your code, so normally you just leave the lock with each dial one higher than the correct value. Unfortunately, last night someone changed all the values to 5. All you remember about your code is that no... | 1. Let the combination be $\overline{abcd}$, where $a, b, c,$ and $d$ are the digits of the lock combination.
2. From the problem, we know that none of the digits are prime, 0, or 1. The non-prime digits between 0 and 9 are $\{4, 6, 8, 9\}$.
3. We are also given that the average value of the digits is 5. Therefore, the... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Peter is randomly filling boxes with candy. If he has 10 pieces of candy and 5 boxes in a row labeled A, B, C, D, and E, how many ways can he distribute the candy so that no two adjacent boxes are empty? | To solve this problem, we need to distribute 10 pieces of candy into 5 boxes labeled A, B, C, D, and E such that no two adjacent boxes are empty. We will use combinatorial methods to count the number of valid distributions.
1. **Total number of ways to distribute candies without any restrictions:**
The total number... | 34 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many points does one have to place on a unit square to guarantee that two of them are strictly less than 1/2 unit apart? | 1. **Understanding the problem**: We need to determine the minimum number of points that must be placed on a unit square to ensure that at least two of them are strictly less than \( \frac{1}{2} \) unit apart.
2. **Optimal placement of points**: Consider placing points in a way that maximizes the minimum distance betw... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the least $n$ such that any subset of ${1,2,\dots,100}$ with $n$ elements has 2 elements with a difference of 9. | 1. We need to find the smallest \( n \) such that any subset of \(\{1, 2, \ldots, 100\}\) with \( n \) elements has at least two elements with a difference of 9.
2. Consider the set \(\{1, 3, 5, 7, \ldots, 99\}\). This set contains all odd numbers from 1 to 99. There are 50 elements in this set.
3. Notice that no two e... | 51 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the minimum number of straight cuts needed to cut a cake in 100 pieces? The pieces do not need to be the same size or shape but cannot be rearranged between cuts. You may assume that the cake is a large cube and may be cut from any direction. | 1. **Understanding the Problem:**
We need to determine the minimum number of straight cuts required to divide a cube-shaped cake into exactly 100 pieces. The pieces do not need to be of equal size or shape, and the cuts can be made in any direction.
2. **Analyzing the Cutting Process:**
Let's denote the number o... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many integers $x$, from $10$ to $99$ inclusive, have the property that the remainder of $x^2$ divided by $100$ is equal to the square of the units digit of $x$? | 1. Let \( x = 10a + b \), where \( a \) is the tens digit and \( b \) is the units digit of \( x \). Therefore, \( 10 \leq x \leq 99 \) implies \( 1 \leq a \leq 9 \) and \( 0 \leq b \leq 9 \).
2. We need to find the integers \( x \) such that the remainder of \( x^2 \) divided by 100 is equal to the square of the unit... | 26 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
After a cyclist has gone $ \frac{2}{3}$ of his route, he gets a flat tire. Finishing on foot, he spends twice as long walking as he did riding. How many times as fast does he ride as walk? | 1. Let the total distance of the route be \( D \).
2. The cyclist rides \( \frac{2}{3}D \) of the route and walks the remaining \( \frac{1}{3}D \).
3. Let \( t_r \) be the time spent riding and \( t_w \) be the time spent walking.
4. According to the problem, the time spent walking is twice the time spent riding:
\... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
After a typist has written ten letters and had addressed the ten corresponding envelopes, a careless mailing clerk inserted the letters in the envelopes at random, one letter per envelope. What is the probability that [b]exactly[/b] nine letters were inserted in the proper envelopes? | 1. Let's denote the letters as \( L_1, L_2, \ldots, L_{10} \) and the corresponding envelopes as \( E_1, E_2, \ldots, E_{10} \).
2. We are asked to find the probability that exactly nine letters are inserted into their corresponding envelopes.
3. If nine letters are correctly placed in their corresponding envelopes, th... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In a certain tournament bracket, a player must be defeated three times to be eliminated. If 512 contestants enter the tournament, what is the greatest number of games that could be played? | 1. In this tournament, each player must be defeated three times to be eliminated. Therefore, each of the 511 players who do not win will be defeated exactly three times.
2. The winner of the tournament will be defeated at most 2 times, as they are not eliminated.
3. To find the total number of games played, we calculat... | 1535 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A geometric series is one where the ratio between each two consecutive terms is constant (ex. 3,6,12,24,...). The fifth term of a geometric series is 5!, and the sixth term is 6!. What is the fourth term? | 1. Let the first term of the geometric series be \( a \) and the common ratio be \( r \).
2. The \( n \)-th term of a geometric series can be expressed as \( a \cdot r^{n-1} \).
3. Given that the fifth term is \( 5! \), we have:
\[
a \cdot r^4 = 5!
\]
4. Given that the sixth term is \( 6! \), we have:
\[
... | 20 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
An alarm clock runs 4 minutes slow every hour. It was set right $ 3 \frac{1}{2}$ hours ago. Now another clock which is correct shows noon. In how many minutes, to the nearest minute, will the alarm clock show noon? | 1. The alarm clock runs 4 minutes slow every hour. This means that for every 60 minutes of real time, the alarm clock only counts 56 minutes.
2. The clock was set correctly 3.5 hours ago. We need to convert this time into minutes:
\[
3.5 \text{ hours} = 3.5 \times 60 = 210 \text{ minutes}
\]
3. Since the alarm... | 14 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Car A is traveling at 20 miles per hour. Car B is 1 mile behind, following at 30 miles per hour. A fast fly can move at 40 miles per hour. The fly begins on the front bumper of car B, and flies back and forth between the two cars. How many miles will the fly travel before it is crushed in the collision? | 1. **Determine the time it takes for Car B to catch up to Car A:**
- Car A is traveling at 20 miles per hour.
- Car B is traveling at 30 miles per hour and is 1 mile behind Car A.
- The relative speed of Car B with respect to Car A is \(30 - 20 = 10\) miles per hour.
- The time \(t\) it takes for Car B to c... | 4 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Given a random string of 33 bits (0 or 1), how many (they can overlap) occurrences of two consecutive 0's would you expect? (i.e. "100101" has 1 occurrence, "0001" has 2 occurrences) | 1. **Determine the probability of two consecutive bits being both 0:**
- Each bit in the string can be either 0 or 1 with equal probability, i.e., \( \frac{1}{2} \).
- The probability that two consecutive bits are both 0 is:
\[
\left( \frac{1}{2} \right)^2 = \frac{1}{4}
\]
2. **Calculate the numbe... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A finite sequence of positive integers $ m_i$ for $ i\equal{}1,2,...,2006$ are defined so that $ m_1\equal{}1$ and $ m_i\equal{}10m_{i\minus{}1} \plus{}1$ for $ i>1$. How many of these integers are divisible by $ 37$? | 1. **Define the sequence**: The sequence \( m_i \) is defined such that \( m_1 = 1 \) and \( m_i = 10m_{i-1} + 1 \) for \( i > 1 \).
2. **Express the sequence in a closed form**: We can observe that the sequence can be written as:
\[
m_1 = 1
\]
\[
m_2 = 10 \cdot 1 + 1 = 11
\]
\[
m_3 = 10 \cdot ... | 668 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given $ \triangle{ABC}$, where $ A$ is at $ (0,0)$, $ B$ is at $ (20,0)$, and $ C$ is on the positive $ y$-axis. Cone $ M$ is formed when $ \triangle{ABC}$ is rotated about the $ x$-axis, and cone $ N$ is formed when $ \triangle{ABC}$ is rotated about the $ y$-axis. If the volume of cone $ M$ minus the volume of cone $... | 1. Assign point \( C \) to be \( (0, x) \). This means that \( C \) is on the positive \( y \)-axis at a height \( x \).
2. When \( \triangle ABC \) is rotated about the \( y \)-axis, it forms a cone with:
- Radius \( r = 20 \) (the distance from \( A \) to \( B \))
- Height \( h = x \) (the height of point \( C... | 29 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There exist two positive numbers $ x$ such that $ \sin(\arccos(\tan(\arcsin x)))\equal{}x$. Find the product of the two possible $ x$. | 1. Let $\arcsin x = \theta$. Then, $x = \sin \theta$ and $\theta \in [0, \frac{\pi}{2}]$ since $x$ is a positive number.
2. We need to find $\tan(\arcsin x)$. Using the right triangle representation, we have:
\[
\sin \theta = x \quad \text{and} \quad \cos \theta = \sqrt{1 - x^2}
\]
Therefore,
\[
\tan ... | 1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
The expression $16^n+4^n+1$ is equiavalent to the expression $(2^{p(n)}-1)/(2^{q(n)}-1)$ for all positive integers $n>1$ where $p(n)$ and $q(n)$ are functions and $\tfrac{p(n)}{q(n)}$ is constant. Find $p(2006)-q(2006)$. | 1. We start with the given expression \(16^n + 4^n + 1\). We need to show that this expression can be written in the form \(\frac{2^{p(n)} - 1}{2^{q(n)} - 1}\) for some functions \(p(n)\) and \(q(n)\).
2. Notice that \(16^n = (2^4)^n = 2^{4n}\) and \(4^n = (2^2)^n = 2^{2n}\). Therefore, the given expression can be rew... | 8024 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all solutions to $aabb=n^4-6n^3$, where $a$ and $b$ are non-zero digits, and $n$ is an integer. ($a$ and $b$ are not necessarily distinct.) | To find all solutions to the equation \(aabb = n^4 - 6n^3\), where \(a\) and \(b\) are non-zero digits and \(n\) is an integer, we can proceed as follows:
1. **Rewrite \(aabb\) in a more convenient form:**
\[
aabb = 1100a + 11b
\]
This is because \(aabb\) can be interpreted as \(1000a + 100a + 10b + b = 11... | 6655 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Principal Skinner is thinking of two integers $m$ and $n$ and bets Superintendent Chalmers that he will not be able to determine these integers with a single piece of information. Chalmers asks Skinner the numerical value of $mn+13m+13n-m^2-n^2$. From the value of this expression alone, he miraculously determines bot... | 1. We start with the given expression:
\[
mn + 13m + 13n - m^2 - n^2
\]
Let this expression be equal to \( Z \):
\[
mn + 13m + 13n - m^2 - n^2 = Z
\]
2. Rearrange the terms to group the quadratic and linear terms:
\[
m^2 + n^2 - mn - 13m - 13n = -Z
\]
3. Multiply the entire equation by 2... | 169 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
There are three bins: one with 30 apples, one with 30 oranges, and one with 15 of each. Each is labeled "apples," "oranges," or "mixed." Given that all three labels are wrong, how many pieces of fruit must you look at to determine the correct labels? | 1. **Identify the problem constraints:**
- There are three bins, each labeled incorrectly.
- One bin contains only apples, one contains only oranges, and one contains a mix of both.
- The labels are "apples," "oranges," and "mixed."
2. **Pick a fruit from the bin labeled "mixed":**
- Since the labels are a... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Aliens from Lumix have one head and four legs, while those from Obscra have two heads and only one leg. If 60 aliens attend a joint Lumix and Obscra interworld conference, and there are 129 legs present, how many heads are there? | 1. Let \( l \) represent the number of Lumix aliens and \( o \) represent the number of Obscra aliens.
2. We are given two pieces of information:
- The total number of aliens is 60.
- The total number of legs is 129.
These can be written as the following system of equations:
\[
l + o = 60 \quad \text... | 97 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Peter Pan and Crocodile are each getting hired for a job. Peter wants to get paid 6.4 dollars daily, but Crocodile demands to be paid 10 cents on day 1, 20 cents on day 2, 40 cents on day 3, 80 cents on day 4, and so on. After how many whole days will Crocodile's total earnings exceed that of Peter's? | To determine after how many whole days Crocodile's total earnings will exceed Peter's, we need to set up the equations for their earnings and solve for the number of days, \( n \).
1. **Calculate Peter's total earnings after \( n \) days:**
Peter earns $6.4 per day. Therefore, after \( n \) days, Peter's total earn... | 10 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A rope of length 10 [i]m[/i] is tied tautly from the top of a flagpole to the ground 6 [i]m[/i] away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 [i]cm/min[/i]. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropi... | 1. **Determine the distance traveled by the ant's shadow:**
- The ant's shadow moves at a rate of 30 cm/min.
- After 5 minutes, the distance traveled by the shadow is:
\[
30 \, \text{cm/min} \times 5 \, \text{min} = 150 \, \text{cm} = 1.5 \, \text{m}
\]
2. **Set up the problem using similar triang... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A farmer wants to build a rectangular region, using a river as one side and some fencing as the other three sides. He has 1200 feet of fence which he can arrange to different dimensions. He creates the rectangular region with length $ L$ and width $ W$ to enclose the greatest area. Find $ L\plus{}W$. | 1. **Set up the equation for the perimeter:**
Since the farmer is using the river as one side of the rectangle, he only needs to fence the other three sides. Let $L$ be the length parallel to the river and $W$ be the width perpendicular to the river. The total length of the fence used is given by:
\[
2L + W = ... | 900 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
How many rational solutions for $x$ are there to the equation $x^4+(2-p)x^3+(2-2p)x^2+(1-2p)x-p=0$ if $p$ is a prime number? | To determine the number of rational solutions for the equation
\[ x^4 + (2 - p)x^3 + (2 - 2p)x^2 + (1 - 2p)x - p = 0 \]
where \( p \) is a prime number, we can use the Rational Root Theorem. The Rational Root Theorem states that any rational solution, expressed as a fraction \(\frac{p}{q}\), must have \( p \) as a f... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many five-letter "words" can you spell using the letters $S$, $I$, and $T$, if a "word" is defines as any sequence of letters that does not contain three consecutive consonants? | 1. **Calculate the total number of five-letter "words" without restrictions:**
Each position in the five-letter word can be filled by one of the three letters \( S \), \( I \), or \( T \). Therefore, the total number of possible words is:
\[
3^5 = 243
\]
2. **Count the number of words with exactly 3 consec... | 139 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The polynomial $R(x)$ is the remainder upon dividing $x^{2007}$ by $x^2-5x+6$. $R(0)$ can be expressed as $ab(a^c-b^c)$. Find $a+c-b$. | 1. **Identify the problem**: We need to find the remainder \( R(x) \) when \( x^{2007} \) is divided by \( x^2 - 5x + 6 \). Then, we need to evaluate \( R(0) \) and express it in the form \( ab(a^c - b^c) \) to find \( a + c - b \).
2. **Factor the divisor**: The polynomial \( x^2 - 5x + 6 \) can be factored as:
\[... | 2010 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let N be the number of distinct rearrangements of the 34 letters in SUPERCALIFRAGILISTICEXPIALIDOCIOUS. How many positive factors does N have? | 1. **Identify the frequency of each letter in the word "SUPERCALIFRAGILISTICEXPIALIDOCIOUS":**
- I appears 7 times.
- S, C, A, L each appear 3 times.
- U, P, E, R, O each appear 2 times.
- F, G, T, X, D each appear 1 time.
2. **Calculate the number of distinct rearrangements of the letters:**
The formul... | 1612800 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Given that the three points where the parabola $y = bx^2 -2$ intersects the $x$-axis and $y$-axis form an equilateral triangle, compute $b$.
[b]p2.[/b] Compute the last digit of $$2^{(3^{(4^{...2014)})})}$$
[b]p3.[/b] A math tournament has a test which contains $10$ questions, each of which come from one... | 1. We start by expressing \( S_{2014} \) as the sum of a series of numbers consisting of repeated 1's:
\[
S_{2014} = 1 + 11 + 111 + 1111 + \ldots + \underbrace{111\ldots11}_{2014 \text{ ones}}
\]
2. Each term in the series can be written as:
\[
\underbrace{111\ldots11}_{k \text{ ones}} = \frac{10^k - 1}{... | 8100 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Given $x + y = 7$, find the value of x that minimizes $4x^2 + 12xy + 9y^2$.
[b]p2.[/b] There are real numbers $b$ and $c$ such that the only $x$-intercept of $8y = x^2 + bx + c$ equals its $y$-intercept. Compute $b + c$.
[b]p3.[/b] Consider the set of $5$ digit numbers $ABCDE$ (with $A \ne 0$) such that... | To solve the problem, we need to find the number of 5-digit numbers \(ABCDE\) such that the following conditions hold:
1. \(A \neq 0\)
2. \(A + B = C\)
3. \(B + C = D\)
4. \(C + D = E\)
Let's start by expressing \(D\) and \(E\) in terms of \(A\) and \(B\):
1. From \(A + B = C\), we have:
\[
C = A + B
\]
2. ... | 6 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $M=\{0,1,2,\dots,2022\}$ and let $f:M\times M\to M$ such that for any $a,b\in M$,
\[f(a,f(b,a))=b\]
and $f(x,x)\neq x$ for each $x\in M$. How many possible functions $f$ are there $\pmod{1000}$? | 1. Given the set \( M = \{0, 1, 2, \dots, 2022\} \) and the function \( f: M \times M \to M \) such that for any \( a, b \in M \),
\[
f(a, f(b, a)) = b
\]
and \( f(x, x) \neq x \) for each \( x \in M \).
2. Denote \( c = f(a, a) \). By substituting \( a = b \) in the given functional equation, we get:
\... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For all positive integers $m>10^{2022}$, determine the maximum number of real solutions $x>0$ of the equation $mx=\lfloor x^{11/10}\rfloor$. | 1. **Setting up the problem:**
We are given the equation \( mx = \lfloor x^{11/10} \rfloor \) and need to determine the maximum number of real solutions \( x > 0 \) for all positive integers \( m > 10^{2022} \).
2. **Initial guess and example:**
Consider \( m = 10^{2022} + 1 \). We can guess solutions of the for... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f^1(x)=x^3-3x$. Let $f^n(x)=f(f^{n-1}(x))$. Let $\mathcal{R}$ be the set of roots of $\tfrac{f^{2022}(x)}{x}$. If
\[\sum_{r\in\mathcal{R}}\frac{1}{r^2}=\frac{a^b-c}{d}\]
for positive integers $a,b,c,d$, where $b$ is as large as possible and $c$ and $d$ are relatively prime, find $a+b+c+d$. | 1. **Define the function and its iterations:**
Given \( f^1(x) = x^3 - 3x \), we define \( f^n(x) = f(f^{n-1}(x)) \). We need to find the roots of \( \frac{f^{2022}(x)}{x} \).
2. **Identify the roots of \( f(x) \):**
The roots of \( f(x) = x^3 - 3x \) are \( x = 0, \pm \sqrt{3} \). These roots will be the same f... | 4060 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The straight line $y=ax+16$ intersects the graph of $y=x^3$ at $2$ distinct points. What is the value of $a$? | 1. **Set up the equations:**
The line \( y = ax + 16 \) intersects the curve \( y = x^3 \). To find the points of intersection, set the equations equal to each other:
\[
ax + 16 = x^3
\]
Rearrange this to form a polynomial equation:
\[
x^3 - ax - 16 = 0
\]
2. **Determine the conditions for dist... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let
\[f(x)=\cos(x^3-4x^2+5x-2).\]
If we let $f^{(k)}$ denote the $k$th derivative of $f$, compute $f^{(10)}(1)$. For the sake of this problem, note that $10!=3628800$. | 1. **Substitute \( u = x - 1 \):**
We start by substituting \( u = x - 1 \) into the function \( f(x) \). This gives us:
\[
f(x) = \cos(x^3 - 4x^2 + 5x - 2) \implies f(u + 1) = \cos((u + 1)^3 - 4(u + 1)^2 + 5(u + 1) - 2)
\]
Simplifying the expression inside the cosine function:
\[
(u + 1)^3 - 4(u +... | 907200 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Call a three-digit number $\overline{ABC}$ $\textit{spicy}$ if it satisfies $\overline{ABC}=A^3+B^3+C^3$. Compute the unique $n$ for which both $n$ and $n+1$ are $\textit{spicy}$. | To solve the problem, we need to find a three-digit number \( \overline{ABC} \) such that \( \overline{ABC} = A^3 + B^3 + C^3 \). Additionally, we need to find a unique \( n \) such that both \( n \) and \( n+1 \) are spicy numbers.
1. **Formulate the equation for a spicy number:**
\[
\overline{ABC} = 100A + 10B... | 370 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathcal{A}$ be the set of finite sequences of positive integers $a_1,a_2,\dots,a_k$ such that $|a_n-a_{n-1}|=a_{n-2}$ for all $3\leqslant n\leqslant k$. If $a_1=a_2=1$, and $k=18$, determine the number of elements of $\mathcal{A}$. | 1. **Define the sequence and initial conditions:**
We are given a sequence \(a_1, a_2, \ldots, a_k\) with \(a_1 = a_2 = 1\) and \(k = 18\). The sequence must satisfy the condition \(|a_n - a_{n-1}| = a_{n-2}\) for all \(3 \leq n \leq k\).
2. **Analyze the sequence behavior:**
- For \(n = 3\), we have \(|a_3 - a_... | 1597 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For any positive integer $n$, let $f(n)$ be the maximum number of groups formed by a total of $n$ people such that the following holds: every group consists of an even number of members, and every two groups share an odd number of members. Compute $\textstyle\sum_{n=1}^{2022}f(n)\text{ mod }1000$. | 1. **Understanding the problem**: We need to find the maximum number of groups, $f(n)$, that can be formed by $n$ people such that each group has an even number of members and every two groups share an odd number of members. Then, we need to compute the sum $\sum_{n=1}^{2022} f(n) \mod 1000$.
2. **Experimentation and ... | 242 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Points $A$, $B$, $C$, and $D$ lie on a circle. Let $AC$ and $BD$ intersect at point $E$ inside the circle. If $[ABE]\cdot[CDE]=36$, what is the value of $[ADE]\cdot[BCE]$? (Given a triangle $\triangle ABC$, $[ABC]$ denotes its area.) | 1. **Claim:** For any convex quadrilateral $ABCD$, $[ABE] \cdot [CDE] = [ADE] \cdot [BCE]$.
2. **Proof:**
- Consider the cyclic quadrilateral $ABCD$ with diagonals $AC$ and $BD$ intersecting at point $E$.
- The area of a triangle can be expressed using the formula:
\[
[XYZ] = \frac{1}{2} \cdot XY \cdot... | 36 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$\triangle ABC$ has side lengths $AB=20$, $BC=15$, and $CA=7$. Let the altitudes of $\triangle ABC$ be $AD$, $BE$, and $CF$. What is the distance between the orthocenter (intersection of the altitudes) of $\triangle ABC$ and the incenter of $\triangle DEF$? | 1. **Identify the given information and the goal:**
- Given: $\triangle ABC$ with side lengths $AB = 20$, $BC = 15$, and $CA = 7$.
- Goal: Find the distance between the orthocenter of $\triangle ABC$ and the incenter of $\triangle DEF$.
2. **Calculate the area of $\triangle ABC$:**
- Using Heron's formula, fi... | 15 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The bisector of $\angle BAC$ in $\triangle ABC$ intersects $BC$ in point $L$. The external bisector of $\angle ACB$ intersects $\overrightarrow{BA}$ in point $K$. If the length of $AK$ is equal to the perimeter of $\triangle ACL$, $LB=1$, and $\angle ABC=36^\circ$, find the length of $AC$. | 1. **Define the problem and given conditions:**
- We have a triangle \( \triangle ABC \).
- The bisector of \( \angle BAC \) intersects \( BC \) at point \( L \).
- The external bisector of \( \angle ACB \) intersects \( \overrightarrow{BA} \) at point \( K \).
- The length of \( AK \) is equal to the perim... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In acute triangle $\triangle ABC$, point $R$ lies on the perpendicular bisector of $AC$ such that $\overline{CA}$ bisects $\angle BAR$. Let $Q$ be the intersection of lines $AC$ and $BR$. The circumcircle of $\triangle ARC$ intersects segment $\overline{AB}$ at $P\neq A$, with $AP=1$, $PB=5$, and $AQ=2$. Compute $AR$. | 1. **Identify the given information and setup the problem:**
- $\triangle ABC$ is an acute triangle.
- Point $R$ lies on the perpendicular bisector of $AC$.
- $\overline{CA}$ bisects $\angle BAR$.
- $Q$ is the intersection of lines $AC$ and $BR$.
- The circumcircle of $\triangle ARC$ intersects segment $... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Nathan has discovered a new way to construct chocolate bars, but it’s expensive! He starts with a single $1\times1$ square of chocolate and then adds more rows and columns from there. If his current bar has dimensions $w\times h$ ($w$ columns and $h$ rows), then it costs $w^2$ dollars to add another row and $h^2$ dolla... | To find the minimum cost to construct a $20 \times 20$ chocolate bar, we need to consider the cost of adding rows and columns. The cost to add a row is $w^2$ dollars, and the cost to add a column is $h^2$ dollars, where $w$ is the current width and $h$ is the current height of the chocolate bar.
We will compare two st... | 5339 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $p(x),q(x)$ are monic polynomials with nonnegative integer coefficients such that
\[\frac{1}{5x}\ge\frac{1}{q(x)}-\frac{1}{p(x)}\ge\frac{1}{3x^2}\]
for all integers $x\ge2$. Compute the minimum possible value of $p(1)\cdot q(1)$. | Given the inequality:
\[
\frac{1}{5x} \ge \frac{1}{q(x)} - \frac{1}{p(x)} \ge \frac{1}{3x^2}
\]
for all integers \( x \ge 2 \), we need to find the minimum possible value of \( p(1) \cdot q(1) \).
1. **Rewriting the Inequality**:
The inequality can be rewritten as:
\[
\frac{1}{5x} \ge \frac{p(x) - q(x)}{p(x)q... | 3 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
For all positive integers $n > 1$, let $f(n)$ denote the largest odd proper divisor of $n$ (a proper divisor of $n$ is a positive divisor of $n$ except for $n$ itself). Given that $N=20^{23}\cdot23^{20}$, compute
\[\frac{f(N)}{f(f(f(N)))}.\] | 1. **Identify the largest odd proper divisor of \( N \):**
Given \( N = 20^{23} \cdot 23^{20} \), we first express 20 in terms of its prime factors:
\[
20 = 2^2 \cdot 5
\]
Therefore,
\[
N = (2^2 \cdot 5)^{23} \cdot 23^{20} = 2^{46} \cdot 5^{23} \cdot 23^{20}
\]
The largest odd proper divisor ... | 25 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Michelle is drawing segments in the plane. She begins from the origin facing up the $y$-axis and draws a segment of length $1$. Now, she rotates her direction by $120^\circ$, with equal probability clockwise or counterclockwise, and draws another segment of length $1$ beginning from the end of the previous segment. She... | 1. **Understanding the Problem:**
Michelle starts at the origin and draws segments of length 1. After each segment, she rotates by $120^\circ$ either clockwise or counterclockwise with equal probability. We need to find the expected number of segments she draws before hitting an already drawn segment.
2. **Initial ... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
We say that an integer $x\in\{1,\dots,102\}$ is $\textit{square-ish}$ if there exists some integer $n$ such that $x\equiv n^2+n\pmod{103}$. Compute the product of all $\textit{square-ish}$ integers modulo $103$. | 1. **Understanding the Problem:**
We need to find the product of all integers \( x \in \{1, \dots, 102\} \) that can be expressed as \( x \equiv n^2 + n \pmod{103} \) for some integer \( n \).
2. **Analyzing the Condition:**
The condition \( x \equiv n^2 + n \pmod{103} \) can be rewritten as:
\[
x \equiv n... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the number of bijective functions $f:\{0,1,\dots,288\}\rightarrow\{0,1,\dots,288\}$ such that $f((m+n)\pmod{17})$ is divisible by $17$ if and only if $f(m)+f(n)$ is divisible by $17$. Compute the largest positive integer $n$ such that $2^n$ divides $S$. | 1. **Understanding the problem**: We need to find the number of bijective functions \( f: \{0, 1, \dots, 288\} \rightarrow \{0, 1, \dots, 288\} \) such that \( f((m+n) \pmod{17}) \) is divisible by 17 if and only if \( f(m) + f(n) \) is divisible by 17. We then need to determine the largest positive integer \( n \) suc... | 270 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a$ and $b$ are positive integers with a curious property: $(a^3 - 3ab +\tfrac{1}{2})^n + (b^3 +\tfrac{1}{2})^n$ is an integer for at least $3$, but at most finitely many different choices of positive integers $n$. What is the least possible value of $a+b$? | 1. We start with the given expression \((a^3 - 3ab + \tfrac{1}{2})^n + (b^3 + \tfrac{1}{2})^n\) and note that it must be an integer for at least 3, but at most finitely many different choices of positive integers \(n\).
2. To simplify the problem, we multiply the entire expression by 2 to clear the fractions:
\[
... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Colin has a peculiar $12$-sided dice: it is made up of two regular hexagonal pyramids. Colin wants to paint each face one of three colors so that no two adjacent faces on the same pyramid have the same color. How many ways can he do this? Two paintings are considered identical if there is a way to rotate or flip the di... | 1. **Understanding the Problem:**
Colin has a 12-sided die made up of two regular hexagonal pyramids. He wants to paint each face one of three colors such that no two adjacent faces on the same pyramid have the same color. We need to find the number of ways to do this, considering that two paintings are identical if... | 405 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p4.[/b] For how many three-digit multiples of $11$ in the form $\underline{abc}$ does the quadratic $ax^2 + bx + c$ have real roots?
[b]p5.[/b] William draws a triangle $\vartriangle ABC$ with $AB =\sqrt3$, $BC = 1$, and $AC = 2$ on a piece of paper and cuts out $\vartriangle ABC$. Let the angle bisector of $\angl... | To solve the problem, we need to compute the sum of the products \((1)(2)(3) + (2)(3)(4) + \ldots + (18)(19)(20)\).
1. **Express the general term**:
Each term in the sum can be written as \((n-1)n(n+1)\). This is a product of three consecutive integers.
2. **Simplify the general term**:
Notice that \((n-1)n(n+1... | 35910 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p7.[/b] An ant starts at the point $(0, 0)$. It travels along the integer lattice, at each lattice point choosing the positive $x$ or $y$ direction with equal probability. If the ant reaches $(20, 23)$, what is the probability it did not pass through $(20, 20)$?
[b]p8.[/b] Let $a_0 = 2023$ and $a_n$ be the sum of ... | To solve the problem, we need to compute the sum of the prime numbers that divide \(a_3\). We start with \(a_0 = 2023\) and use the given recurrence relation \(a_n\) which is the sum of all divisors of \(a_{n-1}\).
1. **Compute \(a_1\):**
\[
a_0 = 2023 = 7 \cdot 17^2
\]
The sum of the divisors of \(2023\) ... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p10.[/b] Three rectangles of dimension $X \times 2$ and four rectangles of dimension $Y \times 1$ are the pieces that form a rectangle of area $3XY$ where $X$ and $Y$ are positive, integer values. What is the sum of all possible values of $X$?
[b]p11.[/b] Suppose we have a polynomial $p(x) = x^2 + ax + b$ with re... | 1. We start with the given equation for the area of the rectangle formed by the pieces:
\[
3 \cdot (X \times 2) + 4 \cdot (Y \times 1) = 3XY
\]
Simplifying, we get:
\[
6X + 4Y = 3XY
\]
2. Rearrange the equation to isolate terms involving \(X\) and \(Y\):
\[
3XY - 6X - 4Y = 0
\]
Add 8 to... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p13.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $1$. Let the unit circles centered at $A$, $B$, and $C$ be $\Omega_A$, $\Omega_B$, and $\Omega_C$, respectively. Then, let $\Omega_A$ and $\Omega_C$ intersect again at point $D$, and $\Omega_B$ and $\Omega_C$ intersect again at point $E$. Li... | 1. To find the probability that the second largest number in the list of the largest numbers from each row is 2023, we need to consider the placement of the numbers 2023, 2024, and 2025 in the grid.
2. The desired result will occur when 2024 and 2025 are in the same row, but 2023 is in a different row from those two. T... | 2970 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p19.[/b] $A_1A_2...A_{12}$ is a regular dodecagon with side length $1$ and center at point $O$. What is the area of the region covered by circles $(A_1A_2O)$, $(A_3A_4O)$, $(A_5A_6O)$, $(A_7A_8O)$, $(A_9A_{10}O)$, and $(A_{11}A_{12}O)$?
$(ABC)$ denotes the circle passing through points $A,B$, and $C$.
[b]p20.[/b] ... | 1. Let \( N' = 200\ldots0023 \), where there are \( 2020 \) zeroes. We need to find \( x \) such that \( N = N' + x \cdot 10^{2000} \) is divisible by \( 13 \).
2. First, we compute \( N' \mod 13 \):
\[
N' = 2 \cdot 10^{2022} + 23
\]
Since \( 10^3 \equiv -1 \pmod{13} \), we have:
\[
10^{2022} = (10^3... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p22.[/b] Consider the series $\{A_n\}^{\infty}_{n=0}$, where $A_0 = 1$ and for every $n > 0$, $$A_n = A_{\left[ \frac{n}{2023}\right]} + A_{\left[ \frac{n}{2023^2}\right]}+A_{\left[ \frac{n}{2023^3}\right]},$$ where $[x]$ denotes the largest integer value smaller than or equal to $x$. Find the $(2023^{3^2}+20)$-th ... | To solve the problem, we need to find the $(2023^{3^2} + 20)$-th element of the series $\{A_n\}_{n=0}^{\infty}$, where $A_0 = 1$ and for $n > 0$,
\[ A_n = A_{\left\lfloor \frac{n}{2023} \right\rfloor} + A_{\left\lfloor \frac{n}{2023^2} \right\rfloor} + A_{\left\lfloor \frac{n}{2023^3} \right\rfloor}. \]
1. **Understan... | 653 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $S$ is a series of real numbers between $2$ and $8$ inclusive, and that for any two elements $y > x$ in $S,$
$$
98y - 102x - xy \ge 4.
$$
What is the maximum possible size for the set $S?$
$$
\mathrm a. ~ 12\qquad \mathrm b.~14\qquad \mathrm c. ~16 \qquad \mathrm d. ~18 \qquad \mathrm e. 20
$$ | 1. We are given a set \( S \) of real numbers between \( 2 \) and \( 8 \) inclusive, and for any two elements \( y > x \) in \( S \), the inequality \( 98y - 102x - xy \ge 4 \) holds. We need to find the maximum possible size of the set \( S \).
2. To solve this, we first consider the inequality \( 98y - 102x - xy \ge... | 16 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
A sequence $a_1, a_2, \ldots$ satisfies $a_1 = \dfrac 52$ and $a_{n + 1} = {a_n}^2 - 2$ for all $n \ge 1.$ Let $M$ be the integer which is closest to $a_{2023}.$ The last digit of $M$ equals
$$
\mathrm a. ~ 0\qquad \mathrm b.~2\qquad \mathrm c. ~4 \qquad \mathrm d. ~6 \qquad \mathrm e. ~8
$$ | 1. We start with the given sequence \(a_1, a_2, \ldots\) defined by \(a_1 = \frac{5}{2}\) and \(a_{n+1} = a_n^2 - 2\) for all \(n \geq 1\).
2. We observe that the sequence can be transformed using the substitution \(a_n = u_n + \frac{1}{u_n}\). This substitution simplifies the recurrence relation. Let's verify this:
... | 4 | Other | MCQ | Yes | Yes | aops_forum | false |
How many ordered triples of integers $(a, b, c)$ satisfy the following system?
$$
\begin{cases} ab + c &= 17 \\ a + bc &= 19 \end{cases}
$$
$$
\mathrm a. ~ 2\qquad \mathrm b.~3\qquad \mathrm c. ~4 \qquad \mathrm d. ~5 \qquad \mathrm e. ~6
$$ | To solve the system of equations:
\[
\begin{cases}
ab + c &= 17 \\
a + bc &= 19
\end{cases}
\]
1. **Subtract the two equations:**
\[
(ab + c) - (a + bc) = 17 - 19
\]
Simplifying, we get:
\[
ab + c - a - bc = -2
\]
Rearrange terms:
\[
a(b-1) - c(b-1) = -2
\]
Factor out \((b-1)\):
... | 3 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Three positive real numbers $a, b, c$ satisfy $a^b = 343, b^c = 10, a^c = 7.$ Find $b^b.$
$$
\mathrm a. ~ 1000\qquad \mathrm b.~900\qquad \mathrm c. ~1200 \qquad \mathrm d. ~4000 \qquad \mathrm e. ~100
$$ | 1. Given the equations:
\[
a^b = 343, \quad b^c = 10, \quad a^c = 7
\]
We start by expressing \(a\) in terms of \(b\) and \(c\).
2. From \(a^b = 343\), we can write:
\[
a = 343^{1/b} = (7^3)^{1/b} = 7^{3/b}
\]
3. From \(a^c = 7\), we can write:
\[
a = 7^{1/c}
\]
4. Equating the two expr... | 1000 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many integers between $123$ and $789$ have at least two identical digits, when written in base $10?$
$$
\mathrm a. ~ 180\qquad \mathrm b.~184\qquad \mathrm c. ~186 \qquad \mathrm d. ~189 \qquad \mathrm e. ~191
$$ | To determine how many integers between $123$ and $789$ have at least two identical digits, we can follow these steps:
1. **Calculate the total number of integers between $123$ and $789$:**
\[
789 - 123 + 1 = 667
\]
2. **Calculate the number of integers between $123$ and $789$ with all distinct digits:**
-... | 180 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
What is the least positive integer $m$ such that the following is true?
[i]Given $\it m$ integers between $\it1$ and $\it{2023},$ inclusive, there must exist two of them $\it a, b$ such that $1 < \frac ab \le 2.$ [/i] \[\mathrm a. ~ 10\qquad \mathrm b.~11\qquad \mathrm c. ~12 \qquad \mathrm d. ~13 \qquad \mathrm e. ~... | To solve this problem, we need to find the smallest positive integer \( m \) such that among any \( m \) integers chosen from the set \(\{1, 2, \ldots, 2023\}\), there exist two integers \( a \) and \( b \) satisfying \( 1 < \frac{a}{b} \leq 2 \).
1. **Understanding the Problem:**
- We need to ensure that for any \... | 12 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
There are $100$ people in a room. Some are [i]wise[/i] and some are [i]optimists[/i].
$\quad \bullet~$ A [i]wise[/i] person can look at someone and know if they are wise or if they are an optimist.
$\quad \bullet~$ An [i]optimist[/i] thinks everyone is wise (including themselves).
Everyone in the room writes down wha... | 1. Let \( x \) be the number of wise people in the room. Therefore, the number of optimists is \( 100 - x \).
2. Each wise person knows exactly how many wise people are in the room, so each wise person writes down \( x \).
3. Each optimist thinks everyone is wise, so each optimist writes down \( 100 \).
4. To find t... | 75 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
An Indian raga has two kinds of notes: a short note, which lasts for $1$ beat and a long note, which lasts for $2$ beats. For example, there are $3$ ragas which are $3$ beats long; $3$ short notes, a short note followed by a long note, and a long note followed by a short note. How many Indian ragas are 11 beats long? | To solve the problem of finding the number of Indian ragas that are 11 beats long, we need to consider the combinations of short notes (1 beat) and long notes (2 beats) that sum up to 11 beats.
1. **Formulate the equation:**
Let \( x \) be the number of short notes and \( y \) be the number of long notes. The equa... | 144 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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