problem stringlengths 15 4.7k | solution stringlengths 2 11.9k | answer stringclasses 51
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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 |
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 |
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 |
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 |
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 |
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 |
[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 |
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 |
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 |
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 |
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 |
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 |
[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]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 |
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 |
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 |
[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 |
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 |
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 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 |
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 |
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 |
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 |
[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] 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] 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 |
[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] 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] 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] 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 |
[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 |
[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 |
[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 |
[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 |
[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 |
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 |
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 |
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 |
$25.$ 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 |
[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 |
[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 |
[u]Round 5[/u]
[b]p13.[/b] Sally is at the special glasses shop, where there are many different optical lenses that distort what she sees and cause her to see things strangely. Whenever she looks at a shape through lens $A$, she sees a shape with $2$ more sides than the original (so a square would look like a hexagon... | 1. Let the people be \( P_1, P_2, P_3, \ldots, P_{2015} \), who retrieve their hats in that order.
2. Define \( x_1, x_2, x_3, \ldots, x_{2015} \) such that, for all \( i \in \{1,2,3,\ldots,2015\} \), \( x_i=0 \) if \( P_i \) does not get their hat and \( x_i=1 \) if \( P_i \) does get their hat.
3. We wish to find \( ... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Albert rolls a fair six-sided die thirteen times. For each time he rolls a number that is strictly greater than the previous number he rolled, he gains a point, where his first roll does not gain him a point. Find the expected number of points that Albert receives.
[i]Proposed by Nathan Ramesh | 1. **Define the problem**: Albert rolls a fair six-sided die thirteen times. For each roll after the first, he gains a point if the number rolled is strictly greater than the previous number. We need to find the expected number of points Albert receives.
2. **Calculate the probability of gaining a point on a single ro... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A single elimination tournament is held with $2016$ participants. In each round, players pair up to play games with each other. There are no ties, and if there are an odd number of players remaining before a round then one person will get a bye for the round. Find the minimum number of rounds needed to determine a winn... | 1. In a single elimination tournament, each round halves the number of participants (or nearly halves if there is an odd number of participants, in which case one participant gets a bye).
2. We start with 2016 participants. After each round, the number of participants is approximately halved.
Let's calculate the numbe... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Compute $2020 \cdot \left( 2^{(0\cdot1)} + 9 - \frac{(20^1)}{8}\right)$.
[b]p2.[/b] Nathan has five distinct shirts, three distinct pairs of pants, and four distinct pairs of shoes. If an “outfit” has a shirt, pair of pants, and a pair of shoes, how many distinct outfits can Nathan make?
[b]p3.[/b] Let $... | 4. Find the units digit of \(2019^{2019}\).
Step-by-step solution:
1. Determine the units digit of \(2019\), which is \(9\).
2. Observe the pattern in the units digits of powers of \(9\):
\[
9^1 = 9, \quad 9^2 = 81, \quad 9^3 = 729, \quad 9^4 = 6561, \quad \ldots
\]
The units digits repeat every 2 powers: ... | 9 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Compute the maximum integer value of $k$ such that $2^k$ divides $3^{2n+3}+40n-27$ for any positive integer $n$. | To find the maximum integer value of \( k \) such that \( 2^k \) divides \( 3^{2n+3} + 40n - 27 \) for any positive integer \( n \), we will use modular arithmetic and analyze the expression modulo powers of 2.
1. **Rewrite the expression:**
\[
3^{2n+3} + 40n - 27
\]
We can factor out common terms:
\[
... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Farmer Boso has a busy farm with lots of animals. He tends to $5b$ cows, $5a +7$ chickens, and $b^{a-5}$ insects. Note that each insect has $6$ legs. The number of cows is equal to the number of insects. The total number of legs present amongst his animals can be expressed as $\overline{LLL }+1$, where $L$ stands for a... | 1. We are given that the number of cows is equal to the number of insects. This can be expressed as:
\[
5b = b^{a-5}
\]
Dividing both sides by \( b \) (assuming \( b \neq 0 \)):
\[
5 = b^{a-6}
\]
Since \( a \) and \( b \) are integers, we need to find integer values of \( a \) and \( b \) that s... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest possible distance between any two points inside or along the perimeter of an equilateral triangle with side length $2$.
[i]Proposed by Alex Li[/i] | 1. Consider an equilateral triangle with side length \(2\). Let's denote the vertices of the triangle as \(A\), \(B\), and \(C\).
2. The distance between any two vertices of an equilateral triangle is equal to the side length of the triangle. Therefore, the distance between any two vertices \(A\) and \(B\), \(B\) and ... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Aidan rolls a pair of fair, six sided dice. Let$ n$ be the probability that the product of the two numbers at the top is prime. Given that $n$ can be written as $a/b$ , where $a$ and $b$ are relatively prime positive integers, find $a +b$.
[i]Proposed by Aidan Duncan[/i] | 1. To determine the probability that the product of the two numbers on the top of the dice is prime, we first need to understand the conditions under which the product of two numbers is prime. A product of two numbers is prime if and only if one of the numbers is 1 and the other number is a prime number. This is becaus... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of arithmetic sequences $a_1,a_2,a_3$ of three nonzero integers such that the sum of the terms in the sequence is equal to the product of the terms in the sequence.
[i]Proposed by Sammy Charney[/i] | 1. Let \( a_1 = x \), \( a_2 = x + d \), and \( a_3 = x + 2d \), where \( x, d \in \mathbb{Z} \). Given that the sum of the terms in the sequence is equal to the product of the terms in the sequence, we have:
\[
a_1 + a_2 + a_3 = a_1 a_2 a_3
\]
Substituting the values of \( a_1 \), \( a_2 \), and \( a_3 \),... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f (x)$ be a function mapping real numbers to real numbers. Given that $f (f (x)) =\frac{1}{3x}$, and $f (2) =\frac19$, find $ f\left(\frac{1}{6}\right)$.
[i]Proposed by Zachary Perry[/i] | 1. We start with the given functional equation:
\[
f(f(x)) = \frac{1}{3x}
\]
and the specific value:
\[
f(2) = \frac{1}{9}
\]
2. Substitute \( x = 2 \) into the functional equation:
\[
f(f(2)) = \frac{1}{3 \cdot 2} = \frac{1}{6}
\]
Since \( f(2) = \frac{1}{9} \), we have:
\[
f\le... | 3 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Kevin colors a ninja star on a piece of graph paper where each small square has area $1$ square inch. Find the area of the region colored, in square inches.
[img]https://cdn.artofproblemsolving.com/attachments/3/3/86f0ae7465e99d3e4bd3a816201383b98dc429.png[/img] | 1. **Identify the outer triangles:**
The ninja star can be divided into 4 outer triangles. Each of these triangles has a base of 2 inches and a height of 2 inches.
2. **Calculate the area of one outer triangle:**
The area \( A \) of a triangle is given by:
\[
A = \frac{1}{2} \times \text{base} \times \text... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The equation of line $\ell_1$ is $24x-7y = 319$ and the equation of line $\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\ell_1$ and $\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$.... | 1. **Intersection of Lines**:
We start by finding the intersection of the lines $\ell_1$ and $\ell_2$. The equations are:
\[
24x - 7y = 319
\]
\[
12x - 5y = 125
\]
To find the intersection, we solve this system of linear equations. We can use the method of elimination or substitution. Here, we w... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Eddie has a study block that lasts $1$ hour. It takes Eddie $25$ minutes to do his homework and $5$ minutes to play a game of Clash Royale. He can’t do both at the same time. How many games can he play in this study block while still completing his homework?
[i]Proposed by Edwin Zhao[/i]
[hide=Solution]
[i]Solution.[... | 1. Convert the study block duration from hours to minutes:
\[
1 \text{ hour} = 60 \text{ minutes}
\]
2. Subtract the time taken to do homework from the total study block time:
\[
60 \text{ minutes} - 25 \text{ minutes} = 35 \text{ minutes}
\]
3. Determine the number of games Eddie can play in the re... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Sam Wang decides to evaluate an expression of the form $x +2 \cdot 2+ y$. However, he unfortunately reads each ’plus’ as a ’times’ and reads each ’times’ as a ’plus’. Surprisingly, he still gets the problem correct. Find $x + y$.
[i]Proposed by Edwin Zhao[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{4}$
We have $x+2*... | 1. The original expression given is \( x + 2 \cdot 2 + y \).
2. Sam reads each 'plus' as 'times' and each 'times' as 'plus'. Therefore, the expression Sam evaluates is \( x \cdot 2 + 2 + y \).
3. We need to simplify both expressions and set them equal to each other since Sam gets the problem correct.
4. Simplify the... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Blue rolls a fair $n$-sided die that has sides its numbered with the integers from $1$ to $n$, and then he flips a coin. Blue knows that the coin is weighted to land heads either $\dfrac{1}{3}$ or $\dfrac{2}{3}$ of the time. Given that the probability of both rolling a $7$ and flipping heads is $\dfrac{1}{15}$, find $n... | 1. Let's denote the probability of rolling any specific number on an $n$-sided die as $\frac{1}{n}$, since the die is fair.
2. The probability of flipping heads with the weighted coin is either $\frac{1}{3}$ or $\frac{2}{3}$.
3. We are given that the probability of both rolling a $7$ and flipping heads is $\frac{1}{15}... | 10 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $k$ such that when $\frac{k}{2023}$ is written in simplest form, the sum of the numerator and denominator is divisible by $7$.
[i]Proposed byMuztaba Syed[/i] | 1. We need to find the least positive integer \( k \) such that when \(\frac{k}{2023}\) is written in simplest form, the sum of the numerator and denominator is divisible by \( 7 \).
2. First, note that \( 2023 \) is not divisible by \( 7 \). We can verify this by checking the divisibility rule for \( 7 \):
\[
2... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given that the base-$17$ integer $\overline{8323a02421_{17}}$ (where a is a base-$17$ digit) is divisible by $\overline{16_{10}}$, find $a$. Express your answer in base $10$.
[i]Proposed by Jonathan Liu[/i] | To determine the value of \( a \) such that the base-17 integer \(\overline{8323a02421_{17}}\) is divisible by \(16_{10}\), we can use the property that a number is divisible by 16 if the sum of its digits is divisible by 16.
1. **Convert the base-17 number to a sum of its digits:**
\[
\overline{8323a02421_{17}}... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Andrew writes down all of the prime numbers less than $50$. How many times does he write the digit $2$? | 1. First, list all the prime numbers less than $50$. These are:
\[
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
\]
2. Identify the prime numbers that contain the digit $2$:
- The number $2$ itself contains the digit $2$.
- The number $23$ contains the digit $2$.
- The number $29$ contains t... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given that $2x + 5 - 3x + 7 = 8$, what is the value of $x$?
$\textbf{(A) }{-}4\qquad\textbf{(B) }{-}2\qquad\textbf{(C) }0\qquad\textbf{(D) }2\qquad\textbf{(E) }4$ | 1. Start with the given equation:
\[
2x + 5 - 3x + 7 = 8
\]
2. Combine like terms on the left-hand side:
\[
(2x - 3x) + (5 + 7) = 8
\]
\[
-x + 12 = 8
\]
3. Isolate the variable \(x\) by subtracting 12 from both sides:
\[
-x + 12 - 12 = 8 - 12
\]
\[
-x = -4
\]
4. Solve for... | 4 | Algebra | MCQ | Yes | Yes | aops_forum | false |
An integer $N$ which satisfies exactly three of the four following conditions is called [i]two-good[/i].
$~$
[center]
(I) $N$ is divisible by $2$
(II) $N$ is divisible by $4$
(III) $N$ is divisible by $8$
(IV) $N$ is divisible by $16$
[/center]$~$
How many integers between $1$ and $100$, inclusive, are [i]two-good[/i]?... | To solve the problem, we need to identify integers \( N \) between 1 and 100 that satisfy exactly three of the four given conditions:
1. \( N \) is divisible by 2.
2. \( N \) is divisible by 4.
3. \( N \) is divisible by 8.
4. \( N \) is divisible by 16.
We will analyze each condition and determine which integers sati... | 6 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Let $x$ be a number such that $10000x+2=4$. What is the value of $5000x+1$?
$\textbf{(A) }{-}1\qquad\textbf{(B) }0\qquad\textbf{(C) }1\qquad\textbf{(D) }2\qquad\textbf{(E) }3$ | 1. Start with the given equation:
\[
10000x + 2 = 4
\]
2. Subtract 2 from both sides to isolate the term with \(x\):
\[
10000x + 2 - 2 = 4 - 2
\]
\[
10000x = 2
\]
3. Divide both sides by 10000 to solve for \(x\):
\[
x = \frac{2}{10000}
\]
\[
x = \frac{1}{5000}
\]
4. Now, ... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Real numbers $w$, $x$, $y$, and $z$ satisfy $w+x+y = 3$, $x+y+z = 4,$ and $w+x+y+z = 5$. What is the value of $x+y$?
$\textbf{(A) }-\frac{1}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }\frac{3}{2}\qquad\textbf{(D) }2\qquad\textbf{(E) }3$
| 1. We start with the given equations:
\[
w + x + y = 3
\]
\[
x + y + z = 4
\]
\[
w + x + y + z = 5
\]
2. Subtract the first equation from the third equation:
\[
(w + x + y + z) - (w + x + y) = 5 - 3
\]
Simplifying, we get:
\[
z = 2
\]
3. Substitute \(z = 2\) into the se... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
What is the maximum possible value of $5-|6x-80|$ over all integers $x$?
$\textbf{(A) }{-}1\qquad\textbf{(B) }0\qquad\textbf{(C) }1\qquad\textbf{(D) }3\qquad\textbf{(E) }5$ | 1. We start with the expression \(5 - |6x - 80|\). To maximize this expression, we need to minimize the absolute value term \(|6x - 80|\).
2. The absolute value \(|6x - 80|\) is minimized when \(6x - 80\) is as close to 0 as possible. This occurs when \(6x = 80\).
3. Solving for \(x\), we get:
\[
x = \frac{80}{6}... | 3 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
Two truth tellers and two liars are positioned in a line, where every person is distinguishable. How many ways are there to position these four people such that everyone claims that all people directly adjacent to them are liars?
$\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }12\qquad\textbf... | 1. Let the truth tellers be denoted as \( T_1 \) and \( T_2 \), and the liars be denoted as \( L_1 \) and \( L_2 \).
2. Since a liar claims that all people directly adjacent to them are liars, the people directly adjacent to a liar must actually be truth-tellers. Therefore, no liar can be next to another liar, and no t... | 8 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
An ant climbs either two inches or three inches each day. In how many ways can the ant climb twelve inches, if the order of its climbing sequence matters?
$\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }14$ | To determine the number of ways the ant can climb twelve inches, we need to consider the different combinations of 2-inch and 3-inch climbs that sum to 12 inches. We will analyze the problem by considering the number of days the ant climbs and the combinations of 2-inch and 3-inch climbs.
1. **Case 1: The ant climbs 1... | 12 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
[u]Set 1[/u]
[b]B1.[/b] Evaluate $2 + 0 - 2 \times 0$.
[b]B2.[/b] It takes four painters four hours to paint four houses. How many hours does it take forty painters to paint forty houses?
[b]B3.[/b] Let $a$ be the answer to this question. What is $\frac{1}{2-a}$?
[u]Set 2[/u]
[b]B4.[/b] Every day at Andover i... | 1. Evaluate \(2 + 0 - 2 \times 0\).
\[
2 + 0 - 2 \times 0 = 2 + 0 - 0 = 2
\]
The final answer is \(\boxed{2}\). | 2 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $(a,b)$ is an ordered pair of integers such that the three numbers $a$, $b$, and $ab$ form an arithmetic progression, in that order. Find the sum of all possible values of $a$.
[i]Proposed by Nathan Xiong[/i] | 1. Since \(a, b, ab\) are in arithmetic progression, the common difference between consecutive terms must be the same. Therefore, we can write the relationship as:
\[
b - a = ab - b
\]
Simplifying this equation, we get:
\[
b - a = ab - b \implies 2b = a + ab
\]
2. Rearrange the equation to isolate... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If $x$, $y$, $z$ are nonnegative integers satisfying the equation below, then compute $x+y+z$.
\[\left(\frac{16}{3}\right)^x\times \left(\frac{27}{25}\right)^y\times \left(\frac{5}{4}\right)^z=256.\]
[i]Proposed by Jeffrey Shi[/i] | 1. Start with the given equation:
\[
\left(\frac{16}{3}\right)^x \times \left(\frac{27}{25}\right)^y \times \left(\frac{5}{4}\right)^z = 256
\]
2. Rewrite each fraction in terms of prime factors:
\[
\frac{16}{3} = \frac{2^4}{3}, \quad \frac{27}{25} = \frac{3^3}{5^2}, \quad \frac{5}{4} = \frac{5}{2^2}
... | 6 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Four students Alice, Bob, Charlie, and Diana want to arrange themselves in a line such that Alice is at either end of the line, i.e., she is not in between two students. In how many ways can the students do this?
[i]Proposed by Nathan Xiong[/i] | 1. **Determine the possible positions for Alice:**
- Alice can either be at the first position or the last position in the line. Therefore, there are 2 choices for Alice's position.
2. **Arrange the remaining students:**
- Once Alice's position is fixed, we need to arrange the remaining three students (Bob, Char... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Andover has a special weather forecast this week. On Monday, there is a $\frac{1}{2}$ chance of rain. On Tuesday, there is a $\frac{1}{3}$ chance of rain. This pattern continues all the way to Sunday, when there is a $\frac{1}{8}$ chance of rain. The probability that it doesn't rain in Andover all week can be expressed... | 1. The probability that it doesn't rain on a given day is the complement of the probability that it rains. For Monday, the probability that it doesn't rain is:
\[
1 - \frac{1}{2} = \frac{1}{2}
\]
2. For Tuesday, the probability that it doesn't rain is:
\[
1 - \frac{1}{3} = \frac{2}{3}
\]
3. This pat... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Andy and Harry are trying to make an O for the MOAA logo. Andy starts with a circular piece of leather with radius 3 feet and cuts out a circle with radius 2 feet from the middle. Harry starts with a square piece of leather with side length 3 feet and cuts out a square with side length 2 feet from the middle. In square... | 1. **Calculate the area of Andy's final product:**
- Andy starts with a circular piece of leather with radius 3 feet.
- The area of the initial circle is given by:
\[
A_{\text{initial}} = \pi \times (3)^2 = 9\pi \text{ square feet}
\]
- Andy cuts out a smaller circle with radius 2 feet from the ... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Mr. Stein is ordering a two-course dessert at a restaurant. For each course, he can choose to eat pie, cake, rødgrød, and crème brûlée, but he doesn't want to have the same dessert twice. In how many ways can Mr. Stein order his meal? (Order matters.) | 1. Mr. Stein has 4 options for the first course: pie, cake, rødgrød, and crème brûlée.
2. Since he doesn't want to have the same dessert twice, he will have 3 remaining options for the second course after choosing the first dessert.
3. Therefore, the number of ways to choose and order the two desserts is calculated by ... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many lattice points are exactly twice as close to $(0,0)$ as they are to $(15,0)$? (A lattice point is a point $(a,b)$ such that both $a$ and $b$ are integers.) | 1. **Set up the distance equations:**
- The distance from a point \((x, y)\) to \((0, 0)\) is given by:
\[
d_1 = \sqrt{x^2 + y^2}
\]
- The distance from the point \((x, y)\) to \((15, 0)\) is given by:
\[
d_2 = \sqrt{(x - 15)^2 + y^2}
\]
- According to the problem, the point \((x,... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Three real numbers $a$, $b$, and $c$ between $0$ and $1$ are chosen independently and at random. What is the probability that $a + 2b + 3c > 5$? | 1. We start by considering the unit cube in the 3-dimensional space where each of the coordinates \(a\), \(b\), and \(c\) ranges from 0 to 1. This cube represents all possible combinations of \(a\), \(b\), and \(c\).
2. We need to find the region within this cube where the inequality \(a + 2b + 3c > 5\) holds. This in... | 0 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
[u]Set 1[/u]
[b]p1.[/b] Arnold is currently stationed at $(0, 0)$. He wants to buy some milk at $(3, 0)$, and also some cookies at $(0, 4)$, and then return back home at $(0, 0)$. If Arnold is very lazy and wants to minimize his walking, what is the length of the shortest path he can take?
[b]p2.[/b] Dilhan selects... | 1. To find the shortest path Arnold can take, we need to consider the distances between the points and the possible paths. The points are $(0,0)$ (home), $(3,0)$ (milk), and $(0,4)$ (cookies).
- Distance from $(0,0)$ to $(3,0)$:
\[
\sqrt{(3-0)^2 + (0-0)^2} = \sqrt{9} = 3
\]
- Distance from $(3,0)$... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[u]Set 4[/u]
[b]p16.[/b] Albert, Beatrice, Corey, and Dora are playing a card game with two decks of cards numbered $1-50$ each. Albert, Beatrice, and Corey draw cards from the same deck without replacement, but Dora draws from the other deck. What is the probability that the value of Corey’s card is the highest valu... | To find the last two digits of \(3^{361}\), we can use Euler's theorem. Euler's theorem states that if \(a\) and \(n\) are coprime, then \(a^{\phi(n)} \equiv 1 \pmod{n}\), where \(\phi(n)\) is the Euler's totient function.
1. **Calculate \(\phi(100)\):**
\[
\phi(100) = \phi(2^2 \cdot 5^2) = 100 \left(1 - \frac{1... | 03 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A blind ant is walking on the coordinate plane. It is trying to reach an anthill, placed at all points where both the $x$-coordinate and $y$-coordinate are odd. The ant starts at the origin, and each minute it moves one unit either up, down, to the right, or to the left, each with probability $\frac{1}{4}$. The ant mov... | 1. **Understanding the Problem:**
- The ant starts at the origin \((0,0)\).
- It moves one unit in one of the four directions (up, down, left, right) with equal probability \(\frac{1}{4}\).
- The ant needs to reach a point where both coordinates are odd.
- We need to find the expected number of additional m... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_n =\sum_{d|n} \frac{1}{2^{d+ \frac{n}{d}}}$. In other words, $a_n$ is the sum of $\frac{1}{2^{d+ \frac{n}{d}}}$ over all divisors $d$ of $n$.
Find
$$\frac{\sum_{k=1} ^{\infty}ka_k}{\sum_{k=1}^{\infty} a_k} =\frac{a_1 + 2a_2 + 3a_3 + ....}{a_1 + a_2 + a_3 +....}$$ | 1. We start by analyzing the given sequence \( a_n = \sum_{d|n} \frac{1}{2^{d + \frac{n}{d}}} \), where the sum is taken over all divisors \( d \) of \( n \).
2. We need to find the value of the expression:
\[
\frac{\sum_{k=1}^\infty k a_k}{\sum_{k=1}^\infty a_k}
\]
3. First, we compute the numerator \(\sum_... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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