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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[b]p1.[/b] Twelve tables are set up in a row for a Millenium party. You want to put $2000$ cupcakes on the tables so that the numbers of cupcakes on adjacent tables always differ by one (for example, if the $5$th table has $20$ cupcakes, then the $4$th table has either $19$ or $21$ cupcakes, and the $6$th table has eit... | To solve this problem, we will use coordinate geometry to find the value of \( |AB| \).
1. **Assign coordinates to points:**
Let \( C = (0, 0) \), \( B = (3b, 0) \), and \( A = (0, 3a) \). This setup places \( C \) at the origin, \( B \) on the x-axis, and \( A \) on the y-axis.
2. **Determine coordinates of \( P ... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find all natural numbers $n$ such that when we multiply all divisors of $n$, we will obtain $10^9$. Prove that your number(s) $n$ works and that there are no other such numbers.
([i]Note[/i]: A natural number $n$ is a positive integer; i.e., $n$ is among the counting numbers $1, 2, 3, \dots$. A [i]divisor[/i] of $n$ is... | To solve the problem, we need to find all natural numbers \( n \) such that the product of all its divisors equals \( 10^9 \).
1. **Understanding the Product of Divisors**:
Let \( n \) have the prime factorization \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \). The number of divisors of \( n \) is given by:
\[
... | 100 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Exhibit a $13$-digit integer $N$ that is an integer multiple of $2^{13}$ and whose digits consist of only $8$s and $9$s. | To find a 13-digit integer \( N \) that is an integer multiple of \( 2^{13} \) and whose digits consist only of \( 8 \)s and \( 9 \)s, we need to ensure that \( N \) is divisible by \( 8192 \) (since \( 2^{13} = 8192 \)).
1. **Last Digit Analysis**:
- For \( N \) to be divisible by \( 2 \), its last digit must be e... | 8888888888888 | Number Theory | other | Yes | Yes | aops_forum | false |
In a convex pentagon $ABCDE$ the sides have lengths $1,2,3,4,$ and $5$, though not necessarily in that order. Let $F,G,H,$ and $I$ be the midpoints of the sides $AB$, $BC$, $CD$, and $DE$, respectively. Let $X$ be the midpoint of segment $FH$, and $Y$ be the midpoint of segment $GI$. The length of segment $XY$ is an... | 1. **Assign coordinates to the vertices of the pentagon:**
Let \( A \equiv (0,0) \), \( B \equiv (4\alpha_1, 4\beta_1) \), \( C \equiv (4\alpha_2, 4\beta_2) \), \( D \equiv (4\alpha_3, 4\beta_3) \), and \( E \equiv (4\alpha_4, 4\beta_4) \).
2. **Find the coordinates of the midpoints:**
- \( F \) is the midpoint ... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The number $N$ consists of $1999$ digits such that if each pair of consecutive digits in $N$ were viewed as a two-digit number, then that number would either be a multiple of $17$ or a multiple of $23$. THe sum of the digits of $N$ is $9599$. Determine the rightmost ten digits of $N$. | 1. **Identify the two-digit multiples of 17 and 23:**
- Multiples of 17: \(17, 34, 51, 68, 85\)
- Multiples of 23: \(23, 46, 69, 92\)
2. **Analyze the constraints:**
- Each pair of consecutive digits in \(N\) must be a multiple of 17 or 23.
- The sum of the digits of \(N\) is 9599.
- \(N\) consists of 1... | 3469234685 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The Fibonacci numbers are defined by $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n>2$. It is well-known that the sum of any $10$ consecutive Fibonacci numbers is divisible by $11$. Determine the smallest integer $N$ so that the sum of any $N$ consecutive Fibonacci numbers is divisible by $12$. | 1. **Define the Fibonacci sequence modulo 12:**
The Fibonacci sequence is defined as:
\[
F_1 = 1, \quad F_2 = 1, \quad F_n = F_{n-1} + F_{n-2} \quad \text{for} \quad n > 2
\]
We need to consider this sequence modulo 12.
2. **Compute the first few terms of the Fibonacci sequence modulo 12:**
\[
\be... | 24 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The integer $n$, between 10000 and 99999, is $abcde$ when written in decimal notation. The digit $a$ is the remainder when $n$ is divided by 2, the digit $b$ is the remainder when $n$ is divided by 3, the digit $c$ is the remainder when $n$ is divided by 4, the digit $d$ is the remainder when $n$ is divied by 5, and t... | 1. **Identify the constraints for each digit:**
- \( a \) is the remainder when \( n \) is divided by 2, so \( a \in \{0, 1\} \).
- \( b \) is the remainder when \( n \) is divided by 3, so \( b \in \{0, 1, 2\} \).
- \( c \) is the remainder when \( n \) is divided by 4, so \( c \in \{0, 1, 2, 3\} \).
- \( ... | 11311 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Each member of the sequence $112002, 11210, 1121, 117, 46, 34,\ldots$ is obtained by adding five times the rightmost digit to the number formed by omitting that digit. Determine the billionth ($10^9$th) member of this sequence. | 1. **Understanding the sequence generation rule:**
Each member of the sequence is obtained by adding five times the rightmost digit to the number formed by omitting that digit. Let's denote the sequence as \(a_n\). The rule can be written as:
\[
a_{n+1} = a_n + 5 \times (a_n \mod 10) - 10 \times \left\lfloor \... | 16 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Determine with proof the number of positive integers $n$ such that a convex regular polygon with $n$ sides has interior angles whose measures, in degrees, are integers. | 1. The measure of each interior angle of a regular $n$-gon can be calculated using the formula:
\[
\text{Interior angle} = \frac{(n-2) \cdot 180^\circ}{n}
\]
Simplifying this, we get:
\[
\text{Interior angle} = 180^\circ - \frac{360^\circ}{n}
\]
2. For the interior angle to be an integer, $\frac{3... | 22 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An increasing arithmetic sequence with infinitely many terms is determined as follows. A single die is thrown and the number that appears is taken as the first term. The die is thrown again and the second number that appears is taken as the common difference between each pair of consecutive terms. Determine with proo... | 1. **Understanding the Problem:**
We need to determine how many of the 36 possible sequences formed by rolling a die twice contain at least one perfect square. The first roll determines the first term \(a\) of the arithmetic sequence, and the second roll determines the common difference \(d\).
2. **Analyzing the Co... | 27 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation \[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \] are plotted on the complex plane (i.e. using the real part of each root as the x-coordinate
and the imaginary part as the y-coordinate). Show that the area of the convex quadrilatera... | 1. We start with the given quartic equation:
\[
r^4z^4 + (10r^6 - 2r^2)z^2 - 16r^5z + (9r^8 + 10r^4 + 1) = 0
\]
2. We factor the quartic polynomial. Notice that the constant term can be factored as:
\[
9r^8 + 10r^4 + 1 = (9r^4 + 1)(r^4 + 1)
\]
3. Assume the quartic can be factored into two quadratic... | 8 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A group of people is lined up in [i]almost-order[/i] if, whenever person $A$ is to the left of person $B$ in the line, $A$ is not more than $8$ centimeters taller than $B$. For example, five people with heights $160, 165, 170, 175$, and $180$ centimeters could line up in almost-order with heights (from left-to-right) o... | 1. **Define the problem and the sequence:**
Let \( a_n \) be the number of ways to line up \( n \) people in almost-order such that the height of the shortest person is 140 cm and the difference in height between two consecutive people is 5 cm. We want to find \( a_{10} \).
2. **Establish the recurrence relation:**... | 89 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A point $P$ in the interior of a convex polyhedron in Euclidean space is called a [i]pivot point[/i] of the polyhedron if every line through $P$ contains exactly $0$ or $2$ vertices of the polyhedron. Determine, with proof, the maximum number of pivot points that a polyhedron can contain. | 1. **Assume for contradiction that a convex polyhedron $\Pi$ has two or more pivot points, and let two of them be $P$ and $Q$.**
2. **Construct a graph $G$ where the vertices of $G$ represent the vertices of $\Pi$. Connect two vertices with a red edge if they are collinear with $P$ and a blue edge if they are collinea... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ that satisfies the following:
We can color each positive integer with one of $n$ colors such that the equation $w + 6x = 2y + 3z$ has no solutions in positive integers with all of $w, x, y$ and $z$ having the same color. (Note that $w, x, y$ and $z$ need not be distinct.) | 1. **Prove that \( n > 3 \):**
- Assume for the sake of contradiction that there are at most 3 colors, say \( c_1 \), \( c_2 \), and \( c_3 \).
- Let 1 have color \( c_1 \) without loss of generality.
- By considering the tuple \((1,1,2,1)\), the color of 2 must be different from \( c_1 \), so let it be \( c_2... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1,a_2,a_3,...$ be a sequence of positive real numbers such that:
(i) For all positive integers $m,n$, we have $a_{mn}=a_ma_n$
(ii) There exists a positive real number $B$ such that for all positive integers $m,n$ with $m<n$, we have $a_m < Ba_n$
Find all possible values of $\log_{2015}(a_{2015}) - \log_{2014}... | 1. **Given Conditions and Initial Observations:**
- We are given a sequence of positive real numbers \(a_1, a_2, a_3, \ldots\) such that:
\[
a_{mn} = a_m a_n \quad \text{for all positive integers } m, n
\]
\[
\exists B > 0 \text{ such that } a_m < B a_n \quad \text{for all positive integers ... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Nine distinct positive integers are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of $n$ and the product of any two adjacent numbers in the circle is not a multiple of $n$, where $n$ is a fixed positive integer. Find the smallest possible value for $n$. | 1. **Define the problem and notation:**
Let the nine distinct positive integers be \(a_1, a_2, \ldots, a_9\) arranged in a circle. We need to find the smallest positive integer \(n\) such that:
- The product of any two non-adjacent numbers is a multiple of \(n\).
- The product of any two adjacent numbers is no... | 485100 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Cyclic quadrilateral $ABCD$ has $AC\perp BD$, $AB+CD=12$, and $BC+AD=13$. FInd the greatest possible area of $ABCD$. | 1. **Identify the given conditions and variables:**
- Cyclic quadrilateral \(ABCD\) with \(AC \perp BD\).
- \(AB + CD = 12\).
- \(BC + AD = 13\).
- Let the intersection of diagonals \(AC\) and \(BD\) be \(P\).
- Let \(AB = a\), \(BC = b\), \(CD = c\), \(AD = d\), \(AP = w\), \(BP = z\), \(CP = x\), \(DP ... | 36 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Hello all. Post your solutions below.
[b]Also, I think it is beneficial to everyone if you all attempt to comment on each other's solutions.[/b]
4/1/31. A group of $100$ friends stands in a circle. Initially, one person has $2019$ mangos, and
no one else has mangos. The friends split the mangos according to the follow... | 1. **Initial Setup and Problem Understanding**:
- We have 100 friends standing in a circle.
- Initially, one person has 2019 mangos, and no one else has any mangos.
- Sharing rules: pass 2 mangos to the left and 1 mango to the right.
- Eating rules: eat 1 mango and pass another mango to the right.
- A pe... | 8 | Logic and Puzzles | proof | Yes | Yes | aops_forum | false |
Princess Pear has $100$ jesters with heights $1, 2, \dots, 100$ inches. On day $n$ with $1 \leq n \leq 100$, Princess Pear holds a court with all her jesters with height at most $n$ inches, and she receives two candied cherries from every group of $6$ jesters with a median height of $n - 50$ inches. A jester can be par... | 1. **Understanding the Problem:**
- Princess Pear has 100 jesters with heights ranging from 1 to 100 inches.
- On day \( n \) (where \( 1 \leq n \leq 100 \)), she holds a court with jesters of height at most \( n \) inches.
- She receives two candied cherries from every group of 6 jesters with a median height ... | 384160000 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given a nonconstant polynomial with real coefficients $f(x),$ let $S(f)$ denote the sum of its roots. Let p and q be nonconstant polynomials with real coefficients such that $S(p) = 7,$ $S(q) = 9,$ and $S(p-q)= 11$. Find, with proof, all possible values for $S(p + q)$. | To solve the problem, we need to find the possible values of \( S(p + q) \) given that \( S(p) = 7 \), \( S(q) = 9 \), and \( S(p - q) = 11 \). We will use Vieta's formulas and properties of polynomial roots to derive the solution.
1. **Expressing the sums of roots using Vieta's formulas:**
- For a polynomial \( p(... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find, with proof, all positive integers $n$ with the following property: There are only finitely many positive multiples of $n$ which have exactly $n$ positive divisors | 1. **Define the problem and notation:**
- We need to find all positive integers \( n \) such that there are only finitely many positive multiples of \( n \) which have exactly \( n \) positive divisors.
- Let \(\sigma(n)\) denote the number of positive divisors of \( n \).
- Let \(\mathbb{P}\) be the set of al... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find, with proof, the minimum positive integer n with the following property: for
any coloring of the integers $\{1, 2, . . . , n\}$ using the colors red and blue (that is, assigning the
color “red” or “blue” to each integer in the set), there exist distinct integers a, b, c between
1 and n, inclusive, all of the same ... | To find the minimum positive integer \( n \) such that for any coloring of the integers \(\{1, 2, \ldots, n\}\) using the colors red and blue, there exist distinct integers \(a, b, c\) all of the same color such that \(2a + b = c\), we will use the concept of Ramsey theory.
1. **Claim**: The minimum \( n \) is \( \box... | 15 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a subset of $\{1, 2, . . . , 500\}$ such that no two distinct elements of S have a
product that is a perfect square. Find, with proof, the maximum possible number of elements
in $S$. | 1. **Define the problem and the set \( S \)**:
We need to find the maximum number of elements in a subset \( S \) of \(\{1, 2, \ldots, 500\}\) such that no two distinct elements of \( S \) have a product that is a perfect square.
2. **Square-free numbers**:
A square-free number is an integer which is not divisib... | 306 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Grogg and Winnie are playing a game using a deck of $50$ cards numbered $1$ through $50$. They take turns with Grogg going first. On each turn a player chooses a card from the deck—this choice is made deliberately, not at random—and then adds it to one of two piles (both piles are empty at the start of the game). After... | 1. **Introduction and Problem Restatement:**
Grogg and Winnie are playing a game with a deck of 50 cards numbered from 1 to 50. They take turns picking a card and adding it to one of two piles. Grogg goes first. After all cards are placed, Winnie wins the positive difference of the sums of the two piles in dollars. ... | 75 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Malmer Pebane's apartment uses a six-digit access code, with leading zeros allowed. He noticed that his fingers leave that reveal which digits were pressed. He decided to change his access code to provide the largest number of possible combinations for a burglar to try when the digits are known. For each number of dis... | 1. Define \( f(x) \) as the number of possible combinations for the access code given that it has \( x \) distinct digits, for all integers \( 1 \leq x \leq 6 \). Clearly, \( f(1) = 1 \).
2. We claim that \( f(n) = n^6 - \left( \sum_{i=1}^{n-1} \binom{n}{i} f(i) \right) \) for all integers \( 2 \leq n \leq 6 \).
3. T... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Len's Spanish class has four tests in the first term. Len scores $72$, $81$, and $78$ on the first three tests. If Len wants to have an 80 average for the term, what is the minimum score he needs on the last test?
[b]p2.[/b] In $1824$, the Electoral College had $261$ members. Andrew Jackson won $99$ Electo... | 8. Given \(a, b, c\) are the roots of the polynomial \(x^3 + 2x^2 + 3x + 4\), we need to find the value of \(a\#b + b\#c + c\#a\), where:
\[
a\#b = \frac{a^3 - b^3}{a - b} = a^2 + ab + b^2
\]
Therefore:
\[
a\#b + b\#c + c\#a = a^2 + ab + b^2 + b^2 + bc + c^2 + c^2 + ca + a^2
\]
\[
= 2(a^2 + b... | -1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 5[/u]
[b]5.1.[/b] A triangle has lengths such that one side is $12$ less than the sum of the other two sides, the semi-perimeter of the triangle is $21$, and the largest and smallest sides have a difference of $2$. Find the area of this triangle.
[b]5.2.[/b] A rhombus has side length $85$ and diagonals of in... | 1. Let the sides of the triangle be \(a\), \(b\), and \(c\) such that \(a \leq b \leq c\). Given that one side is 12 less than the sum of the other two sides, we can assume without loss of generality that \(c = a + b - 12\).
2. The semi-perimeter \(s\) of the triangle is given as 21. Therefore, we have:
\[
s = \f... | 84 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Compute the greatest integer less than or equal to $$\frac{10 + 12 + 14 + 16 + 18 + 20}{21}$$
[b]p2.[/b] Let$ A = 1$.$B = 2$, $C = 3$, $...$, $Z = 26$. Find $A + B +M + C$.
[b]p3.[/b] In Mr. M's farm, there are $10$ cows, $8$ chickens, and $4$ spiders. How many legs are there (including Mr. M's legs)?
... | To solve the problem, we need to find the linear function \( f(x) = ax + b \) given the conditions \( f(1) = 2017 \) and \( f(2) = 2018 \).
1. **Set up the equations based on the given conditions:**
\[
f(1) = a \cdot 1 + b = 2017 \implies a + b = 2017
\]
\[
f(2) = a \cdot 2 + b = 2018 \implies 2a + b = ... | 4035 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] How many lines of symmetry does a square have?
[b]p2.[/b] Compute$ 1/2 + 1/6 + 1/12 + 1/4$.
[b]p3.[/b] What is the maximum possible area of a rectangle with integer side lengths and perimeter $8$?
[b]p4.[/b] Given that $1$ printer weighs $400000$ pennies, and $80$ pennies weighs $2$ books, what is the ... | To solve this problem, we need to find the area of the pentagon \(ABOPC\) which is formed by inscribing a square \(ABMC\) in circle \(F\) and an equilateral triangle \(MOP\) in circle \(G\). The circles \(F\) and \(G\) are externally tangent and share the vertex \(M\).
1. **Calculate the side length of the square \(AB... | 295 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Fun facts! We know that $1008^2-1007^2 = 1008+1007$ and $1009^2-1008^2 = 1009+1008$. Now compute the following: $$1010^2 - 1009^2 - 1.$$
[b]p2.[/b] Let $m$ be the smallest positive multiple of $2018$ such that the fraction $m/2019$ can be simplified. What is the number $m$?
[b]p3.[/b] Given that $n$ sati... | **p9.** In $\triangle ABC$, $AB = 12$ and $AC = 15$. Alex draws the angle bisector of $\angle BAC$, $AD$, such that $D$ is on $BC$. If $CD = 10$, then the area of $\triangle ABC$ can be expressed in the form $\frac{m \sqrt{n}}{p}$ where $m, p$ are relatively prime and $n$ is not divisible by the square of any prime. Fi... | 146 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Fluffy the Dog is an extremely fluffy dog. Because of his extreme fluffiness, children always love petting Fluffy anywhere. Given that Fluffy likes being petted $1/4$ of the time, out of $120$ random people who each pet Fluffy once, what is the expected number of times Fluffy will enjoy being petted?
[b]p2... | 1. Let the side length of the original square garden be \( x \). The original area of the garden is \( x^2 \).
2. When the length is increased by 2 and the width is increased by 3, the new dimensions of the garden are \( x + 2 \) and \( x + 3 \) respectively.
3. The new area of the garden is given by:
\[
(x + 2... | 121 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] The remainder of a number when divided by $7$ is $5$. If I multiply the number by $32$ and add $18$ to the product, what is the new remainder when divided by $7$?
[b]p2.[/b] If a fair coin is flipped $15$ times, what is the probability that there are more heads than tails?
[b]p3.[/b] Let $-\frac{\sqrt{p}... | 1. We start with the given equation involving the smallest nonzero real number \( x \) such that the reciprocal of the number is equal to the number minus the square root of the square of the number. Mathematically, this can be written as:
\[
\frac{1}{x} = x - \sqrt{x^2}
\]
2. Since \(\sqrt{x^2} = |x|\), the e... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Let $a$ be an integer. How many fractions $\frac{a}{100}$ are greater than $\frac17$ and less than $\frac13$ ?.
[b]p2.[/b] Justin Bieber invited Justin Timberlake and Justin Shan to eat sushi. There were $5$ different kinds of fish, $3$ different rice colors, and $11$ different sauces. Justin Shan insisted... | To solve this problem, we need to find the area of a three-way Venn Diagram composed of three circles of radius 1, where each circle passes through the center of the other two circles. The area of the entire Venn Diagram is given in the form \(\frac{a}{b}\pi + \sqrt{c}\) for positive integers \(a\), \(b\), and \(c\), w... | 24 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 5[/u]
[b]5.1.[/b] Quadrilateral $ABCD$ is such that $\angle ABC = \angle ADC = 90^o$ , $\angle BAD = 150^o$ , $AD = 3$, and $AB = \sqrt3$. The area of $ABCD$ can be expressed as $p\sqrt{q}$ for positive integers $p, q$ where $q$ is not divisible by the square of any prime. Find $p + q$.
[b]5.2.[/b] Neetin wa... | To find the sum of the fourth powers of the roots of the polynomial \( P(x) = x^2 + 2x + 3 \), we will use the properties of the roots and symmetric functions.
1. **Identify the roots and their properties:**
Let the roots of the polynomial \( P(x) = x^2 + 2x + 3 \) be \( r_1 \) and \( r_2 \). By Vieta's formulas, w... | -14 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] James has $8$ Instagram accounts, $3$ Facebook accounts, $4$ QQ accounts, and $3$ YouTube accounts. If each Instagram account has $19$ pictures, each Facebook account has $5$ pictures and $9$ videos, each QQ account has a total of $17$ pictures, and each YouTube account has $13$ videos and no pictures, how m... | To solve the equation \(\frac{1}{2006} = \frac{1}{a} + \frac{1}{b}\) where \(a\) is a 4-digit positive integer and \(b\) is a 6-digit positive integer, we start by clearing the denominators:
1. Multiply both sides by \(2006ab\):
\[
ab = 2006a + 2006b
\]
2. Rearrange the equation:
\[
ab - 2006a - 2006b ... | 120360 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] If $a \diamond b = ab - a + b$, find $(3 \diamond 4) \diamond 5$
[b]p2.[/b] If $5$ chickens lay $5$ eggs in $5$ days, how many chickens are needed to lay $10$ eggs in $10$ days?
[b]p3.[/b] As Alissa left her house to go to work one hour away, she noticed that her odometer read $16261$ miles. This number ... | To solve the problem, we need to find the number of trailing zeroes in \(6!\) and then find the number of trailing zeroes in that result.
1. **Calculate \(z(6!)\):**
The number of trailing zeroes in \(n!\) is given by the formula:
\[
z(n!) = \left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{25} \... | 0 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Martin’s car insurance costed $\$6000$ before he switched to Geico, when he saved $15\%$ on car insurance. When Mayhem switched to Allstate, he, a safe driver, saved $40\%$ on car insurance. If Mayhem and Martin are now paying the same amount for car insurance, how much was Mayhem paying before he switched t... | 1. By the Angle Bisector Theorem, we know that the ratio of the segments created by the angle bisector is equal to the ratio of the other two sides of the triangle. Therefore, we have:
\[
\frac{BD}{DC} = \frac{AB}{AC} = \frac{4}{7}
\]
2. Given that \(\frac{AP}{PD} = \frac{3}{1}\), we can infer that the area r... | 23 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 5[/u]
[b]5.1[/b] A circle with a radius of $1$ is inscribed in a regular hexagon. This hexagon is inscribed in a larger circle. If the area that is outside the hexagon but inside the larger circle can be expressed as $\frac{a\pi}{b} - c\sqrt{d}$, where $a, b, c, d$ are positive integers, $a, b$ are relativel... | To solve the problem, we need to find the sum of all positive integers less than \(2022\) that are relatively prime to \(1011\). We will use the Principle of Inclusion/Exclusion (PIE) to achieve this.
1. **Prime Factorization of \(1011\)**:
\[
1011 = 3 \times 337
\]
2. **Sum of all positive integers less tha... | 672 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Let $X = 2022 + 022 + 22 + 2$. When $X$ is divided by $22$, there is a remainder of $R$. What is the value of $R$?
[b]p2.[/b] When Amy makes paper airplanes, her airplanes fly $75\%$ of the time. If her airplane flies, there is a $\frac56$ chance that it won’t fly straight. Given that she makes $80$ airpla... | 1. **Determine the probability that an airplane flies straight:**
- The probability that an airplane flies is \(0.75\) or \(\frac{3}{4}\).
- Given that an airplane flies, the probability that it won't fly straight is \(\frac{5}{6}\).
- Therefore, the probability that it flies straight is \(1 - \frac{5}{6} = \... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Calculate $A \cdot B +M \cdot C$, where $A = 1$, $B = 2$, $C = 3$, $M = 13$.
[b]p2.[/b] What is the remainder of $\frac{2022\cdot2023}{10}$ ?
[b]p3.[/b] Daniel and Bryan are rolling fair $7$-sided dice. If the probability that the sum of the numbers that Daniel and Bryan roll is greater than $11$ can be ... | Given the problem \(35_a = 42_b\), where \(a\) and \(b\) are bases, we need to find the minimum possible value of the sum \(a + b\) in base 10.
1. **Convert \(35_a\) to base 10:**
\[
35_a = 3a + 5
\]
2. **Convert \(42_b\) to base 10:**
\[
42_b = 4b + 2
\]
3. **Set the two expressions equal to each ... | 13 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] If $A = 0$, $B = 1$, $C = 2$, $...$, $Z = 25$, then what is the sum of $A + B + M+ C$?
[b]p2.[/b] Eric is playing Tetris against Bryan. If Eric wins one-fifth of the games he plays and he plays $15$ games, find the expected number of games Eric will win.
[b]p3.[/b] What is the sum of the measures of the ... | **p1.** If \( A = 0 \), \( B = 1 \), \( C = 2 \), \( \ldots \), \( Z = 25 \), then what is the sum of \( A + B + M + C \)?
1. Assign the values to the letters:
- \( A = 0 \)
- \( B = 1 \)
- \( M = 12 \)
- \( C = 2 \)
2. Sum the values:
\[
A + B + M + C = 0 + 1 + 12 + 2 = 15
\]
The final answer i... | 16 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 1[/u]
[b]1.1.[/b] A classroom has $29$ students. A teacher needs to split up the students into groups of at most $4$. What is the minimum number of groups needed?
[b]1.2.[/b] On his history map quiz, Eric recalls that Sweden, Norway and Finland are adjacent countries, but he has
forgotten which is which, so ... | 1. **Problem 1.1:**
To find the minimum number of groups needed to split 29 students into groups of at most 4, we use the ceiling function to determine the smallest integer greater than or equal to the division of 29 by 4.
\[
\left\lceil \frac{29}{4} \right\rceil = \left\lceil 7.25 \right\rceil = 8
\]
Th... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Find $$2^{\left(0^{\left(2^3\right)}\right)}$$
[b]p2.[/b] Amy likes to spin pencils. She has an $n\%$ probability of dropping the $n$th pencil. If she makes $100$ attempts, the expected number of pencils Amy will drop is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$... | 1. We start by noting that \(0 < \sqrt{7} - \sqrt{3} < 1\). Therefore, raising this expression to the 6th power, we have:
\[
0 < (\sqrt{7} - \sqrt{3})^6 < 1
\]
2. Next, we use the Binomial Theorem to expand \((\sqrt{7} + \sqrt{3})^6\) and \((\sqrt{7} - \sqrt{3})^6\). The Binomial Theorem states:
\[
(a + ... | 7039 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Roll two dice. What is the probability that the sum of the rolls is prime?
[b]p2. [/b]Compute the sum of the first $20$ squares.
[b]p3.[/b] How many integers between $0$ and $999$ are not divisible by $7, 11$, or $13$?
[b]p4.[/b] Compute the number of ways to make $50$ cents using only pennies, nickels... | To solve the problem of finding how many integers between $0$ and $999$ are not divisible by $7$, $11$, or $13$, we will use the principle of inclusion-exclusion.
1. **Define the sets:**
Let $A_7$, $A_{11}$, and $A_{13}$ represent the sets of integers between $0$ and $999$ that are divisible by $7$, $11$, and $13$,... | 719 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Compute $17^2 + 17 \cdot 7 + 7^2$.
[b]p2.[/b] You have $\$1.17$ in the minimum number of quarters, dimes, nickels, and pennies required to make exact change for all amounts up to $\$1.17$. How many coins do you have?
[b]p3.[/b] Suppose that there is a $40\%$ chance it will rain today, and a $20\%$ chance... | 1. We are given the cubic function \( f(x) = x^3 + x + 2014 \) and a line that intersects this cubic at three points, two of which have \( x \)-coordinates 20 and 14. We need to find the \( x \)-coordinate of the third intersection point.
2. First, we substitute \( x = 20 \) and \( x = 14 \) into the cubic function to... | -34 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] What is the units digit of $1 + 9 + 9^2 +... + 9^{2015}$ ?
[b]p2.[/b] In Fourtown, every person must have a car and therefore a license plate. Every license plate must be a $4$-digit number where each digit is a value between $0$ and $9$ inclusive. However $0000$ is not a valid license plate. What is the m... | To solve the problem, we need to find a three-digit number \( n \) that is a multiple of 35 and whose digits sum to 15.
1. **Express \( n \) in terms of its digits:**
\[
n = 100a + 10b + c
\]
where \( a, b, \) and \( c \) are the digits of \( n \).
2. **Sum of the digits:**
\[
a + b + c = 15
\]
... | 735 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] David is taking a $50$-question test, and he needs to answer at least $70\%$ of the questions correctly in order to pass the test. What is the minimum number of questions he must answer correctly in order to pass the test?
[b]p2.[/b] You decide to flip a coin some number of times, and record each of the re... | ### Problem 1
David needs to answer at least 70% of the 50 questions correctly to pass the test.
1. Calculate 70% of 50:
\[
0.7 \times 50 = 35
\]
Conclusion:
David must answer at least 35 questions correctly to pass the test.
The final answer is \(\boxed{40}\) | 40 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] It’s currently $6:00$ on a $12$ hour clock. What time will be shown on the clock $100$ hours from now? Express your answer in the form hh : mm.
[b]p2.[/b] A tub originally contains $10$ gallons of water. Alex adds some water, increasing the amount of water by 20%. Barbara, unhappy with Alex’s decision, dec... | To solve the problem, we need to find the smallest integer \( x \) greater than 1 such that \( x^2 \) is one more than a multiple of 7. Mathematically, we need to find \( x \) such that:
\[ x^2 \equiv 1 \pmod{7} \]
We will check the squares of integers modulo 7 starting from \( x = 2 \).
1. Calculate \( 2^2 \mod 7 \... | 6 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Consider a $4 \times 4$ lattice on the coordinate plane. At $(0,0)$ is Mori’s house, and at $(4,4)$ is Mori’s workplace. Every morning, Mori goes to work by choosing a path going up and right along the roads on the lattice. Recently, the intersection at $(2, 2)$ was closed. How many ways are there now for M... | ### p1
1. **Total Paths Calculation**:
- A path from \((0,0)\) to \((4,4)\) can be represented as a sequence of 4 'U' (up) moves and 4 'R' (right) moves.
- The total number of such sequences is given by the binomial coefficient:
\[
\binom{8}{4} = \frac{8!}{4!4!} = 70
\]
2. **Paths Through \((2,2)\... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] If $x$ is a real number that satisfies $\frac{48}{x} = 16$, find the value of $x$.
[b]p2.[/b] If $ABC$ is a right triangle with hypotenuse $BC$ such that $\angle ABC = 35^o$, what is $\angle BCA$ in degrees?
[img]https://cdn.artofproblemsolving.com/attachments/a/b/0f83dc34fb7934281e0e3f988ac34f653cc3f1.png... | To solve the given problem, we need to find a positive real number \( x \) such that \((x \# 7) \# x = 82\), where the operation \(\#\) is defined as \(a \# b = ab - 2a - 2b + 6\).
1. First, we need to compute \( x \# 7 \):
\[
x \# 7 = x \cdot 7 - 2x - 2 \cdot 7 + 6 = 7x - 2x - 14 + 6 = 5x - 8
\]
2. Next, we... | 6 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] If Clark wants to divide $100$ pizzas among $25$ people so that each person receives the same number of pizzas, how many pizzas should each person receive?
[b]p2.[/b] In a group of $3$ people, every pair of people shakes hands once. How many handshakes occur?
[b]p3.[/b] Dylan and Joey have $14$ costumes ... | To solve the problem, we need to count the number of valid rearrangements of the letters in "LATEX" such that the letter $T$ comes before the letter $E$ and the letter $E$ comes before the letter $X$.
1. **Determine the total number of permutations of the letters in "LATEX":**
The word "LATEX" consists of 5 distin... | 20 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Frankie the frog likes to hop. On his first hop, he hops $1$ meter. On each successive hop, he hops twice as far as he did on the previous hop. For example, on his second hop, he hops $2$ meters, and on his third hop, he hops $4$ meters. How many meters, in total, has he travelled after $6$ hops?
[b]p2.[/b... | To find the median of the positive divisors of \(9999\), we first need to determine all the divisors of \(9999\).
1. **Prime Factorization**:
\[
9999 = 3^2 \cdot 11 \cdot 101
\]
This factorization tells us that \(9999\) can be expressed as a product of these prime factors.
2. **Finding the Divisors**:
... | 100 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] What is the maximum number of points of intersection between a square and a triangle, assuming that no side of the triangle is parallel to any side of the square?
[b]p2.[/b] Two angles of an isosceles triangle measure $80^o$ and $x^o$. What is the sum of all the possible values of $x$?
[b]p3.[/b] Let $p$... | 1. **p1.** What is the maximum number of points of intersection between a square and a triangle, assuming that no side of the triangle is parallel to any side of the square?
To find the maximum number of points of intersection between a square and a triangle, we need to consider the following:
- A square has 4 sides.
... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[i]20 problems for 20 minutes.[/i]
[b]p1.[/b] Euclid eats $\frac17$ of a pie in $7$ seconds. Euler eats $\frac15$ of an identical pie in $10$ seconds. Who eats faster?
[b]p2.[/b] Given that $\pi = 3.1415926...$ , compute the circumference of a circle of radius 1. Express your answer as a decimal rounded to the near... | To find the number of odd composite integers between 0 and 50, we need to follow these steps:
1. **Identify the odd numbers between 0 and 50:**
The odd numbers between 0 and 50 are:
\[
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49
\]
There are 25 odd numb... | 10 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] An $18$oz glass of apple juice is $6\%$ sugar and a $6$oz glass of orange juice is $12\%$ sugar. The two glasses are poured together to create a cocktail. What percent of the cocktail is sugar?
[b]p2.[/b] Find the number of positive numbers that can be expressed as the difference of two integers between $-... | 1. The difference can be as low as \(1\), as high as \(2014\), and everything in between is achievable, so the answer is:
\[
\boxed{2014}
\] | 2014 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Determine the number of ways to place $4$ rooks on a $4 \times 4$ chessboard such that:
(a) no two rooks attack one another, and
(b) the main diagonal (the set of squares marked $X$ below) does not contain any rooks.
[img]https://cdn.artofproblemsolving.com/attachments/e/e/e3aa96de6c8ed468c6ef3837e66a0bce360... | 1. First, we need to determine the total number of ways to place 4 rooks on a \(4 \times 4\) chessboard such that no two rooks attack each other. This is equivalent to finding the number of permutations of 4 elements, which is given by \(4!\):
\[
4! = 24
\]
2. Next, we need to use the Principle of Inclusion/E... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[i]20 problems for 20 minutes.[/i]
[b]p1.[/b] Determine how many digits the number $10^{10}$ has.
[b]p2.[/b] Let $ABC$ be a triangle with $\angle ABC = 60^o$ and $\angle BCA = 70^o$. Compute $\angle CAB$ in degrees.
[b]p3.[/b] Given that $x : y = 2012 : 2$ and $y : z = 1 : 2013$, compute $x : z$. Express your ans... | ### Problem 2
1. Given that $\angle ABC = 60^\circ$ and $\angle BCA = 70^\circ$ in triangle $ABC$.
2. The sum of the angles in a triangle is $180^\circ$.
3. Therefore, $\angle CAB = 180^\circ - (\angle ABC + \angle BCA)$.
4. Substituting the given angles, we get:
\[
\angle CAB = 180^\circ - (60^\circ + 70^\circ) ... | 81 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 5[/u]
[b]p13.[/b] Five different schools are competing in a tournament where each pair of teams plays at most once. Four pairs of teams are randomly selected and play against each other. After these four matches, what is the probability that Chad's and Jordan's respective schools have played against each oth... | 1. Let the two positive integers on the blackboard be \(a\) and \(b\), with the greatest common divisor \(d\). We can express \(a\) and \(b\) as:
\[
a = da', \quad b = db'
\]
where \(a'\) and \(b'\) are coprime integers.
2. The sum of the greatest common divisor and the least common multiple of \(a\) and \... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 1[/u]
[b]p1.[/b] Alec rated the movie Frozen $1$ out of $5$ stars. At least how many ratings of $5$ out of $5$ stars does Eric need to collect to make the average rating for Frozen greater than or equal to $4$ out of $5$ stars?
[b]p2.[/b] Bessie shuffles a standard $52$-card deck and draws five cards with... | To solve the problem of finding the number of length ten strings consisting of only \(A\)s and \(B\)s that contain neither "BAB" nor "BBB" as a substring, we can use a combinatorial approach. We will consider different cases based on the number of \(B\)s in the string and ensure that the forbidden substrings do not app... | 343 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[i]20 problems for 25 minutes.[/i]
[b]p1.[/b] Matt has a twenty dollar bill and buys two items worth $\$7:99$ each. How much change does he receive, in dollars?
[b]p2.[/b] The sum of two distinct numbers is equal to the positive difference of the two numbers. What is the product of the two numbers?
[b]p3.[/b] Eva... | To solve the problem, we need to determine the number of ordered pairs \((x, y)\) of integers satisfying \(0 \le x, y \le 10\) such that \((x + y)^2 + (xy - 1)^2\) is a prime number.
1. Start by expanding the given expression:
\[
(x + y)^2 + (xy - 1)^2 = x^2 + 2xy + y^2 + x^2y^2 - 2xy + 1 = x^2 + y^2 + x^2y^2 + ... | 10 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 5[/u]
[b]p13.[/b] Initially, the three numbers $20$, $201$, and $2016$ are written on a blackboard. Each minute, Zhuo selects two of the numbers on the board and adds $1$ to each. Find the minimum $n$ for which Zhuo can make all three numbers equal to $n$.
[b]p14.[/b] Call a three-letter string rearrangeab... | To solve the problem, we need to determine the minimum value \( n \) such that the three numbers \( 20 \), \( 201 \), and \( 2016 \) can all be made equal by repeatedly selecting two of the numbers and adding 1 to each.
1. **Initial Setup**:
- The numbers on the board are \( 20 \), \( 201 \), and \( 2016 \).
2. **... | 747 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] A right triangle has a hypotenuse of length $25$ and a leg of length $16$. Compute the length of the other leg of this triangle.
[b]p2.[/b] Tanya has a circular necklace with $5$ evenly-spaced beads, each colored red or blue. Find the number of distinct necklaces in which no two red beads are adjacent. If ... | To find the minimum value of the function \( f(x) = \sqrt{x^2 - 4x + 5} + \sqrt{x^2 + 4x + 8} \), we can interpret the terms geometrically.
1. Rewrite the terms inside the square roots:
\[
\sqrt{x^2 - 4x + 5} = \sqrt{(x-2)^2 + 1^2}
\]
\[
\sqrt{x^2 + 4x + 8} = \sqrt{(x+2)^2 + 2^2}
\]
This allows us... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[i]20 problems for 25 minutes.[/i]
[b]p1.[/b] Compute the value of $2 + 20 + 201 + 2016$.
[b]p2.[/b] Gleb is making a doll, whose prototype is a cube with side length $5$ centimeters. If the density of the toy is $4$ grams per cubic centimeter, compute its mass in grams.
[b]p3.[/b] Find the sum of $20\%$ of $16$ ... | To solve the problem, we need to find the area of triangle \( DEF \) given the specific geometric conditions. Let's break down the problem step by step.
1. **Understanding the Geometry:**
- We are given an isosceles right triangle \( \triangle ABC \) with \( \angle C = 90^\circ \) and \( AB = 2 \).
- Points \( D... | 2 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 1[/u]
[b]p1.[/b] If $2m = 200 cm$ and $m \ne 0$, find $c$.
[b]p2.[/b] A right triangle has two sides of lengths $3$ and $4$. Find the smallest possible length of the third side.
[b]p3.[/b] Given that $20(x + 17) = 17(x + 20)$, determine the value of $x$.
[u]Round 2[/u]
[b]p4.[/b] According to the Egy... | 1. **Identify the number of $1 \times 1$ squares:**
The diagram is a $7 \times 7$ grid of points, which means there are $6 \times 6 = 36$ $1 \times 1$ squares. However, the diagram also includes additional points around the eyes and other areas, which contribute to more $1 \times 1$ squares. By carefully counting a... | 59 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] On SeaBay, green herring costs $\$2.50$ per pound, blue herring costs $\$4.00$ per pound, and red herring costs $\$5,85$ per pound. What must Farmer James pay for $12$ pounds of green herring and $7$ pounds of blue herring, in dollars?
[b]p2.[/b] A triangle has side lengths $3$, $4$, and $6$. A second tria... | ### Problem 1
1. Calculate the cost for 12 pounds of green herring:
\[
12 \text{ pounds} \times \$2.50 \text{ per pound} = 12 \times 2.50 = \$30.00
\]
2. Calculate the cost for 7 pounds of blue herring:
\[
7 \text{ pounds} \times \$4.00 \text{ per pound} = 7 \times 4.00 = \$28.00
\]
3. Add the costs t... | 0 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[i]20 problems for 25 minutes.[/i]
[b]p1.[/b] What is $2018 - 3018 + 4018$?
[b]p2.[/b] What is the smallest integer greater than $100$ that is a multiple of both $6$ and $8$?
[b]p3.[/b] What positive real number can be expressed as both $\frac{b}{a}$ and $a:b$ in base $10$ for nonzero digits $a$ and $b$? Express ... | To solve the given problem, we start with the equation:
\[ 9xy = p(p + 3x + 6y) \]
1. **Divisibility by 9:**
Since \(9\) divides the left-hand side (LHS), it must also divide the right-hand side (RHS). This implies that \(9\) divides \(p(p + 3x + 6y)\).
2. **Prime factor considerations:**
Since \(p\) is a prim... | 29 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Three positive integers sum to $16$. What is the least possible value of the sum of their squares?
[b]p2.[/b] Ben is thinking of an odd positive integer less than $1000$. Ben subtracts $ 1$ from his number and divides by $2$, resulting in another number. If his number is still odd, Ben repeats this procedu... | 1. To minimize the sum of the squares of three positive integers that sum to 16, we need to find integers \(a\), \(b\), and \(c\) such that \(a + b + c = 16\) and the sum \(a^2 + b^2 + c^2\) is minimized.
2. By the method of Lagrange multipliers or by simple trial and error, we can see that the integers should be as c... | 86 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 1[/u]
[b]p1.[/b] What is the smallest number equal to its cube?
[b]p2.[/b] Fhomas has $5$ red spaghetti and $5$ blue spaghetti, where spaghetti are indistinguishable except for color. In how many different ways can Fhomas eat $6$ spaghetti, one after the other? (Two ways are considered the same if the sequ... | To determine the number of different ways Fhomas can eat 6 spaghetti, we need to consider the different combinations of red and blue spaghetti he can eat. We will use combinations to count the sequences.
1. **Case 1: Eating 5 of one color and 1 of the other color**
- If Fhomas eats 5 red and 1 blue, the number of w... | 62 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] A shape made by joining four identical regular hexagons side-to-side is called a hexo. Two hexos are considered the same if one can be rotated / reflected to match the other. Find the number of different hexos.
[b]p2.[/b] The sequence $1, 2, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6,... $ consists of numbers written in... | To find the 2019th term of the sequence \(1, 2, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6, \ldots\), we need to understand the pattern of the sequence. The sequence is constructed such that every even number \(2n\) appears once, and every odd number \(2n+1\) appears \(2n+1\) times.
1. **Identify the pattern:**
- Even numbers \(... | 87 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 1[/u]
[b]p1.[/b] What is the remainder when $2021$ is divided by $102$?
[b]p2.[/b] Brian has $2$ left shoes and $2$ right shoes. Given that he randomly picks $2$ of the $4$ shoes, the probability he will get a left shoe and a right shoe is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive int... | ### Problem 1
1. We need to find the remainder when \(2021\) is divided by \(102\).
2. Perform the division:
\[
2021 \div 102 \approx 19.814
\]
3. The integer part of the division is \(19\), so:
\[
2021 = 102 \times 19 + r
\]
4. Calculate the remainder \(r\):
\[
r = 2021 - 102 \times 19 = 2021 ... | 8 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 1[/u]
[b]p1.[/b] Let $ABCDEF$ be a regular hexagon. How many acute triangles have all their vertices among the vertices of $ABCDEF$?
[b]p2.[/b] A rectangle has a diagonal of length $20$. If the width of the rectangle is doubled, the length of the diagonal becomes $22$. Given that the width of the original ... | 1. Let the width of the original rectangle be \( w \) and the length be \( l \). According to the Pythagorean theorem, the diagonal of the rectangle can be expressed as:
\[
w^2 + l^2 = 20^2
\]
Simplifying, we get:
\[
w^2 + l^2 = 400
\]
2. When the width of the rectangle is doubled, the new width b... | 28 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 5[/u]
[b]p13.[/b] Find the number of six-digit positive integers that satisfy all of the following conditions:
(i) Each digit does not exceed $3$.
(ii) The number $1$ cannot appear in two consecutive digits.
(iii) The number $2$ cannot appear in two consecutive digits.
[b]p14.[/b] Find the sum of all disti... | 1. We start with the function \( f(x) = x^2 - 2 \) and need to compute \( f(f(f(f(f(f(f(2.5))))))) \).
2. First, compute \( f(2.5) \):
\[
f(2.5) = (2.5)^2 - 2 = 6.25 - 2 = 4.25
\]
3. Notice that \( 2.5 = 2 + \frac{1}{2} \). We can generalize this form for further computations:
\[
2.5 = 2^1 + \frac{1}{2^1... | 128 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 1[/u]
[b]p1. [/b] What is the sum of the digits in the binary representation of $2023$?
[b]p2.[/b] Jack is buying fruits at the EMCCmart. Three apples and two bananas cost $\$11.00$. Five apples and four bananas cost $\$19.00$. In cents, how much more does an apple cost than a banana?
[b]p3.[/b] Define $... | 1. **Problem 1:**
To find the sum of the digits in the binary representation of \(2023\), we first convert \(2023\) to binary.
\[
2023_{10} = 11111100111_2
\]
Next, we sum the digits of the binary number:
\[
1 + 1 + 1 + 1 + 1 + 1 + 0 + 0 + 1 + 1 + 1 = 9
\]
Therefore, the sum of the digits... | 25 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Yao Ming is $7\text{ ft and }5\text{ in}$ tall. His basketball hoop is $10$ feet from the ground. Given that there are
$12$ inches in a foot, how many inches must Yao jump to touch the hoop with his head? | 1. Convert Yao Ming's height from feet and inches to inches:
\[
7 \text{ ft} + 5 \text{ in} = 7 \times 12 \text{ in/ft} + 5 \text{ in} = 84 \text{ in} + 5 \text{ in} = 89 \text{ in}
\]
2. Convert the height of the basketball hoop from feet to inches:
\[
10 \text{ ft} = 10 \times 12 \text{ in/ft} = 120 \... | 31 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
How many positive whole numbers less than $100$ are divisible by $3$, but not by $2$? | 1. **Identify numbers divisible by 3:**
We need to find the positive whole numbers less than 100 that are divisible by 3. These numbers form an arithmetic sequence: \(3, 6, 9, \ldots, 99\).
The general term of this sequence can be written as:
\[
a_n = 3n
\]
where \(a_n < 100\). To find the largest \(... | 17 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Two positive whole numbers differ by $3$. The sum of their squares is $117$. Find the larger of the two
numbers. | 1. Let the two positive whole numbers be \( a \) and \( b \). Given that they differ by 3, we can write:
\[
a - b = 3
\]
Without loss of generality, assume \( a > b \). Therefore, we can write:
\[
a = b + 3
\]
2. We are also given that the sum of their squares is 117:
\[
a^2 + b^2 = 117
\... | 9 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The ages of Mr. and Mrs. Fibonacci are both two-digit numbers. If Mr. Fibonacci’s age can be formed
by reversing the digits of Mrs. Fibonacci’s age, find the smallest possible positive difference between
their ages.
| 1. Let Mr. Fibonacci's age be represented by the two-digit number \(10a + b\), where \(a\) and \(b\) are the tens and units digits, respectively.
2. Let Mrs. Fibonacci's age be represented by the two-digit number \(10b + a\), where \(b\) and \(a\) are the tens and units digits, respectively.
3. We need to find the smal... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Tim is thinking of a positive integer between $2$ and $15,$ inclusive, and Ted is trying to guess the integer. Tim tells Ted how many factors his integer has, and Ted is then able to be certain of what Tim's integer is. What is Tim's integer? | 1. We need to determine the number of factors for each integer between $2$ and $15$.
2. The number of factors of a number $n$ can be determined by its prime factorization. If $n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}$, then the number of factors of $n$ is given by $(e_1 + 1)(e_2 + 1) \cdots (e_k + 1)$.
Let's calculate... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Al travels for $20$ miles per hour rolling down a hill in his chair for two hours, then four miles per hour climbing a hill for six hours. What is his average speed, in miles per hour? | 1. Calculate the total distance Al travels:
- Rolling down the hill:
\[
20 \text{ miles per hour} \times 2 \text{ hours} = 40 \text{ miles}
\]
- Climbing the hill:
\[
4 \text{ miles per hour} \times 6 \text{ hours} = 24 \text{ miles}
\]
- Total distance:
\[
D_{\text{tot... | 8 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A trapezoid has bases with lengths equal to $5$ and $15$ and legs with lengths equal to $13$ and $13.$ Determine the area of the trapezoid. | 1. **Identify the given values:**
- The lengths of the bases of the trapezoid are \( a = 5 \) and \( b = 15 \).
- The lengths of the legs of the trapezoid are both \( c = 13 \).
2. **Determine the height of the trapezoid:**
- Since the legs are equal, the trapezoid is isosceles.
- Drop perpendiculars from ... | 120 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Carl, James, Saif, and Ted play several games of two-player For The Win on the Art of Problem Solving website. If, among these games, Carl wins $5$ and loses $0,$ James wins $4$ and loses $2,$ Saif wins $1$ and loses $6,$ and Ted wins $4,$ how many games does Ted lose? | 1. We start by noting that each game has exactly one winner and one loser. Therefore, the total number of wins must equal the total number of losses.
2. We are given the number of wins for each player:
- Carl wins 5 games.
- James wins 4 games.
- Saif wins 1 game.
- Ted wins 4 games.
3. Summing these, the... | 6 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
$a,b,c,d,e$ are equal to $1,2,3,4,5$ in some order, such that no two of $a,b,c,d,e$ are equal to the same integer. Given that $b \leq d, c \geq a,a \leq e,b \geq e,$ and that $d\neq5,$ determine the value of $a^b+c^d+e.$ | 1. We are given the conditions:
\[
b \leq d, \quad c \geq a, \quad a \leq e, \quad b \geq e, \quad \text{and} \quad d \neq 5
\]
and the fact that \(a, b, c, d, e\) are distinct integers from the set \(\{1, 2, 3, 4, 5\}\).
2. From the inequalities \(a \leq e\) and \(b \geq e\), we can deduce:
\[
a \le... | 628 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Seongcheol has $3$ red shirts and $2$ green shirts, such that he cannot tell the difference between his three red shirts and he similarly cannot tell the difference between his two green shirts. In how many ways can he hang them in a row in his closet, given that he does not want the two green shirts next to each other... | 1. **Calculate the total number of ways to arrange the shirts without any restrictions:**
Seongcheol has 3 red shirts (R) and 2 green shirts (G). The total number of shirts is 5. Since the red shirts are indistinguishable among themselves and the green shirts are indistinguishable among themselves, the number of wa... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$ | 1. Given the function \( f(x) = x^2 - 2x + 1 \), we can rewrite it in a factored form:
\[
f(x) = (x-1)^2
\]
2. We are given that for some constant \( k \), the function \( f(x+k) \) is equal to \( x^2 + 2x + 1 \) for all real numbers \( x \). Therefore, we need to find \( k \) such that:
\[
f(x+k) = x^2... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In how many ways can six marbles be placed in the squares of a $6$-by-$6$ grid such that no two marbles lie in the same row or column? | 1. We need to place six marbles in a $6$-by-$6$ grid such that no two marbles lie in the same row or column. This is equivalent to finding the number of ways to arrange six marbles in six rows and six columns, where each row and each column contains exactly one marble.
2. Let's denote the marbles as $M_1, M_2, \ldots,... | 720 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$25.$ Let $C$ be the answer to Problem $27.$ What is the $C$-th smallest positive integer with exactly four positive factors?
$26.$ Let $A$ be the answer to Problem $25.$ Determine the absolute value of the difference between the two positive integer roots of the quadratic equation $x^2-Ax+48=0$
$27.$ Let $B$ be the... | To solve the given set of problems, we need to follow the sequence of dependencies between the problems. Let's break down each problem step-by-step.
### Problem 25:
We need to find the \( C \)-th smallest positive integer with exactly four positive factors.
A positive integer \( n \) has exactly four positive factor... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Two knights placed on distinct square of an $8\times8$ chessboard, whose squares are unit squares, are said to attack each other if the distance between the centers of the squares on which the knights lie is $\sqrt{5}.$ In how many ways can two identical knights be placed on distinct squares of an $8\times8$ chessboard... | 1. **Calculate the total number of ways to place two knights on distinct squares of an $8 \times 8$ chessboard:**
The total number of squares on an $8 \times 8$ chessboard is $64$. The number of ways to choose 2 distinct squares out of 64 is given by the binomial coefficient:
\[
\binom{64}{2} = \frac{64 \time... | 1848 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] I open my $2010$-page dictionary, whose pages are numbered $ 1$ to $2010$ starting on page $ 1$ on the right side of the spine when opened, and ending with page $2010$ on the left. If I open to a random page, what is the probability that the two page numbers showing sum to a multiple of $6$?
[b]p2.[/b] Let... | 1. To determine the number of triangles that can be formed using some three of the 8 points, we start by calculating the total number of ways to choose 3 points out of 8. This is given by the binomial coefficient:
\[
\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56
\]
2... | 46 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Cheetahs in my dresser,
Cheetahs in my hair.
Cheetahs in my pants,
Cheetahs everywhere!
-A poem by J. Samuel Trabucco, Esq.
[b]p1.[/b] J has several cheetahs in his dresser, which has $7$ drawers, such that each drawer has the same number of cheetahs. He notices that he can take out one drawer, and redistribute all of... | ### Problem 1:
1. Let \( x \) be the number of cheetahs in each drawer. Since there are 7 drawers, the total number of cheetahs is \( 7x \).
2. When one drawer is removed, there are \( 6x \) cheetahs left.
3. These \( 6x \) cheetahs are redistributed into the remaining 6 drawers, so each drawer now has \( x \) cheetahs... | 41 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Laser beams are known for reflecting off solid objects. Whenever a laser beam hits a straight, solid wall, it reflects off in the opposite direction, at an angle to the wall that is equal to the angle at which it hits, as shown.
[img]https://cdn.artofproblemsolving.com/attachments/8/2/e56b311d1cea61b999d36cbe9189b84586... | 1. **Problem 6:**
- Given a square \(ABCD\) with \(AB = 1\), and midpoints \(M\) and \(N\) of \(AB\) and \(BC\) respectively.
- A laser beam is shot from \(M\) to \(N\), reflects off \(BC\), \(CD\), \(DA\), and returns to \(M\).
- By symmetry, the beam hits the midpoints of the sides before coming back to \(M\... | 80 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
All of the digits of a seven-digit positive integer are either $7$ or $8.$ If this integer is divisible by $9,$ what is the sum of its digits? | 1. Let the seven-digit number be composed of $x$ digits of $7$ and $y$ digits of $8$. Therefore, we have the equation:
\[
x + y = 7
\]
2. For the number to be divisible by $9$, the sum of its digits must be divisible by $9$. The sum of the digits can be expressed as:
\[
7x + 8y
\]
3. We need to find... | 54 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $L,E,T,M,$ and $O$ be digits that satisfy $LEET+LMT=TOOL.$
Given that $O$ has the value of $0,$ digits may be repeated, and $L\neq0,$ what is the value of the $4$-digit integer $ELMO?$ | 1. We start with the given equation:
\[
LEET + LMT = TOOL
\]
Given that \(O = 0\), we can rewrite the equation as:
\[
LEET + LMT = T00L
\]
2. Express each term in terms of digits:
\[
LEET = 1000L + 100E + 10E + T = 1000L + 110E + T
\]
\[
LMT = 100L + 10M + T
\]
\[
TOOL = 10... | 1880 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Triangle $ABC$ has side lengths $AB = 3^2$ and $BC = 4^2$. Given that $\angle ABC$ is a right angle, determine the length of $AC$.
[b]p2.[/b] Suppose $m$ and $n$ are integers such that $m^2+n^2 = 65$. Find the largest possible value of $m-n$.
[b]p3.[/b] Six middle school students are sitting in a circle,... | 1. Given that \( \gcd(a, b) = 2 \), \( \gcd(b, c) = 3 \), and \( \gcd(c, a) = 5 \), we need to find positive integers \(a\), \(b\), and \(c\) such that \( \gcd(a, b, c) = 1 \).
2. Let's start by assigning values to \(a\), \(b\), and \(c\) that satisfy the given conditions:
- Since \( \gcd(a, b) = 2 \), both \(a\) an... | 31 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Alan leaves home when the clock in his cardboard box says $7:35$ AM and his watch says $7:41$ AM. When he arrives at school, his watch says $7:47$ AM and the $7:45$ AM bell rings. Assuming the school clock, the watch, and the home clock all go at the same rate, how many minutes behind the school clock is the... | To determine how many positive integers have the same number of digits when expressed in base \(3\) as when expressed in base \(4\), we need to analyze the ranges of numbers that have \(k\) digits in each base.
1. **Range of \(k\)-digit numbers in base \(3\):**
- A number with \(k\) digits in base \(3\) ranges from... | 35 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
“[i]In order to make an apple pie from scratch, you must first create the universe.[/i]” – Carl Sagan
[b]p1.[/b] Surya decides to sell gourmet cookies at LMT. If he sells them for $25$ cents each, he sells $36$ cookies. For every $4$ cents he raises the price of each cookie, he sells $3$ fewer cookies. What is the sm... | To solve the problem, we need to find the number of "delicious" numbers less than 99. A number is defined as "delicious" if it has exactly 4 factors and is divisible by 2.
1. **Identify the forms of numbers with exactly 4 factors:**
- A number \( n \) has exactly 4 factors if it can be expressed in one of the follo... | 15 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 1[/u]
[b]p1.[/b] How many powers of $2$ are greater than $3$ but less than $2013$?
[b]p2.[/b] What number is equal to six greater than three times the answer to this question?
[b]p3.[/b] Surya Cup-a-tea-raju goes to Starbucks Coffee to sip coffee out of a styrofoam cup. The cup is a cylinder, open on on... | 1. **Finding the powers of \(2\) greater than \(3\) but less than \(2013\)**
We need to find the integer values of \(n\) such that:
\[
2^n > 3 \quad \text{and} \quad 2^n < 2013
\]
First, solve \(2^n > 3\):
\[
n > \log_2 3 \approx 1.58496
\]
So, \(n \geq 2\).
Next, solve \(2^n < 2013\):
... | 2 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[i]“You have been beaten by a Gru”[/i]
[b]p1.[/b] All minions either have $1$ or $2$ eyes, and have one of $4$ possible hairstyles. They are all thin and short, fat and short, or thin and tall. Gru doesn’t want to have any $2$ minions that look exactly alike, so what is the maximum possible amoount of minions can he ... | ### Problem 1:
1. Each minion can have either 1 or 2 eyes.
2. Each minion can have one of 4 possible hairstyles.
3. Each minion can be either thin and short, fat and short, or thin and tall.
To find the maximum number of unique minions, we multiply the number of choices for each attribute:
\[
2 \text{ (eyes)} \times 4... | 25 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[hide=Intro]The answers to each of the ten questions in this section are integers containing only the digits $ 1$ through $ 8$, inclusive. Each answer can be written into the grid on the answer sheet, starting from the cell with the problem number, and continuing across or down until the entire answer has been written.... | 1. To find the closest integer to \(6\sqrt{35}\), we start by approximating \(\sqrt{35}\).
2. We know that \( \sqrt{36} = 6 \) and \( \sqrt{25} = 5 \). Since \(35\) is closer to \(36\) than to \(25\), we can estimate that \(\sqrt{35}\) is slightly less than \(6\).
3. For a more precise approximation, we can use the f... | 36 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] What is $\sqrt[2015]{2^01^5}$?
[b]p2.[/b] What is the ratio of the area of square $ABCD$ to the area of square $ACEF$?
[b]p3.[/b] $2015$ in binary is $11111011111$, which is a palindrome. What is the last year which also had this property?
[b]p4.[/b] What is the next number in the following geometric s... | 1. To find the last year before 2015 that has a binary representation which is a palindrome, we need to find the largest palindromic binary number less than \(11111011111_2\).
2. The next largest palindrome made out of only ones and zeroes is \(11110101111_2\).
3. We convert \(11110101111_2\) to decimal:
\[
111... | 1967 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 1[/u]
[b]p1.[/b] Every angle of a regular polygon has degree measure $179.99$ degrees. How many sides does it have?
[b]p2.[/b] What is $\frac{1}{20} + \frac{1}{1}+ \frac{1}{5}$ ?
[b]p3.[/b] If the area bounded by the lines $y = 0$, $x = 0$, and $x = 3$ and the curve $y = f(x)$ is $10$ units, what is the ... | 1. We are given that \( p, q \), and \( q^2 - p^2 \) are all prime numbers. We need to find the value of \( pq \).
2. First, consider the parity of \( p \) and \( q \). If both \( p \) and \( q \) are odd, then \( q^2 - p^2 \) will be even. The only even prime number is 2. Therefore, we would have:
\[
q^2 - p^2 ... | 6 | Other | math-word-problem | Yes | Yes | aops_forum | false |
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