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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 |
For how many nonempty subsets $S \subseteq \{1, 2, \ldots , 10\}$ is the sum of all elements in $S$ even?
[i]Proposed by Andrew Wen[/i] | 1. **Understanding the Problem:**
We need to find the number of nonempty subsets \( S \subseteq \{1, 2, \ldots, 10\} \) such that the sum of all elements in \( S \) is even.
2. **Total Number of Subsets:**
The set \(\{1, 2, \ldots, 10\}\) has \(2^{10} = 1024\) subsets, including the empty set.
3. **Bijection Ar... | 511 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the product of all possible real values for $k$ such that the system of equations
$$x^2+y^2= 80$$
$$x^2+y^2= k+2x-8y$$
has exactly one real solution $(x,y)$.
[i]Proposed by Nathan Xiong[/i] | 1. **Identify the given equations and their geometric interpretations:**
The given system of equations is:
\[
x^2 + y^2 = 80
\]
\[
x^2 + y^2 = k + 2x - 8y
\]
The first equation represents a circle \(\Gamma\) centered at the origin \((0,0)\) with radius \(\sqrt{80}\).
2. **Rearrange and complete... | 960 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Bob has $30$ identical unit cubes. He can join two cubes together by gluing a face on one cube to a face on the other cube. He must join all the cubes together into one connected solid. Over all possible solids that Bob can build, what is the largest possible surface area of the solid?
[i]Proposed by Nathan Xiong[/i] | 1. To maximize the surface area of the solid formed by gluing the 30 unit cubes together, we need to consider the arrangement that exposes the maximum number of faces.
2. Each unit cube has 6 faces. When two cubes are glued together by one face, the number of exposed faces decreases by 2 (one face from each cube is n... | 122 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider the polynomial
\[P(x)=x^3+3x^2+6x+10.\]
Let its three roots be $a$, $b$, $c$. Define $Q(x)$ to be the monic cubic polynomial with roots $ab$, $bc$, $ca$. Compute $|Q(1)|$.
[i]Proposed by Nathan Xiong[/i] | 1. Given the polynomial \( P(x) = x^3 + 3x^2 + 6x + 10 \), we know its roots are \( a, b, c \). By Vieta's formulas, we have:
\[
a + b + c = -3, \quad ab + bc + ca = 6, \quad abc = -10.
\]
2. Define \( Q(x) \) to be the monic cubic polynomial with roots \( ab, bc, ca \). By the Factor Theorem, we can write:
... | 75 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $\triangle ABC$ be a triangle with side length $BC= 4\sqrt{6}$. Denote $\omega$ as the circumcircle of $\triangle{ABC}$. Point $D$ lies on $\omega$ such that $AD$ is the diameter of $\omega$. Let $N$ be the midpoint of arc $BC$ that contains $A$. $H$ is the intersection of the altitudes in $\triangle{ABC}$ and it i... | 1. **Identify Key Elements and Relationships:**
- Given: $\triangle ABC$ with $BC = 4\sqrt{6}$.
- $\omega$ is the circumcircle of $\triangle ABC$.
- $AD$ is the diameter of $\omega$.
- $N$ is the midpoint of arc $BC$ that contains $A$.
- $H$ is the orthocenter of $\triangle ABC$.
- $HN = HD = 6$.
2. ... | 52 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i] | 1. **Identify the total number of outcomes:**
- When rolling two dice, each die has 6 faces, so the total number of outcomes is:
\[
6 \times 6 = 36
\]
2. **Identify non-composite products:**
- A product is non-composite if it is either 1 or a prime number.
- The product 1 can only occur with th... | 65 | 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 |
William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per hour during uphill $20^\circ$ section... | To find the total time William takes to bike to school and back, we need to calculate the time for each segment of the trip and then sum these times. We will consider the distances and speeds for the uphill, flat, and downhill sections separately.
1. **Calculate the time for the uphill sections:**
- Distance for on... | 29 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
We say that an ordered pair $(a,b)$ of positive integers with $a>b$ is square-ish if both $a+b$ and $a-b$ are perfect squares. For example, $(17,8)$ is square-ish because $17+8=25$ and $17-8=9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a+b<100$ are there?
[i]Proposed by Nathan Xiong[/i] | To solve the problem, we need to find the number of ordered pairs \((a, b)\) of positive integers with \(a > b\) such that both \(a + b\) and \(a - b\) are perfect squares, and \(a + b < 100\).
1. **Express \(a\) and \(b\) in terms of perfect squares:**
Let \(a + b = m^2\) and \(a - b = n^2\), where \(m\) and \(n\)... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Andy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registered in the system is $2469$. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) $\underline{a}$ and $\underline{b}$ on the keypad are broken $-$... | To solve this problem, we need to consider all possible scenarios where two different buttons on the keypad are broken and swapped. Let's break down the problem step-by-step:
1. **Identify the possible swaps:**
- The password is \(2469\).
- We need to consider all possible pairs of digits that could be swapped. ... | 540 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $1,7,19,\ldots$ be the sequence of numbers such that for all integers $n\ge 1$, the average of the first $n$ terms is equal to the $n$th perfect square. Compute the last three digits of the $2021$st term in the sequence.
[i]Proposed by Nathan Xiong[/i] | 1. Let \( x_n \) be the \( n \)-th term and \( s_n \) be the sum of the first \( n \) terms for all \( n \in \mathbb{Z}_{+} \). Since the average of the first \( n \) terms is \( n^2 \), we have:
\[
\frac{s_n}{n} = n^2 \implies s_n = n^3
\]
This implies that the sum of the first \( n \) terms is \( n^3 \).
... | 261 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$, while $AD = BC$. It is given that $O$, the circumcenter of $ABCD$, lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$. Given that $OT = 18$, the area of $ABCD$ can be expressed as $a + b\sqrt{c}$ where $a$, $b... | 1. **Identify the given information and set up the problem:**
- Isosceles trapezoid \(ABCD\) with \(AB = 6\), \(CD = 12\), and \(AD = BC\).
- \(O\) is the circumcenter of \(ABCD\) and lies in the interior.
- Extensions of \(AD\) and \(BC\) intersect at \(T\).
- \(OT = 18\).
2. **Determine the height \(h\) ... | 84 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest positive integer $n$ such that the number $(2n)!$ ends with $10$ more zeroes than the number $n!$.
[i]Proposed by Andy Xu[/i] | To solve the problem, we need to find the largest positive integer \( n \) such that the number \( (2n)! \) ends with 10 more zeroes than the number \( n! \).
1. **Understanding the number of trailing zeroes in a factorial:**
The number of trailing zeroes in \( k! \) is given by:
\[
\left\lfloor \frac{k}{5} \... | 42 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of triples $(a,b,c)$ of non-negative integers with $a+b+c$ even. The value of the sum
\[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\]
can be expressed as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i] | 1. **Identify the sets and conditions:**
Let \( E \) be the set of even nonnegative integers and \( O \) be the set of odd nonnegative integers. We need to sum over all triples \((a, b, c)\) such that \(a + b + c\) is even. This implies there must be an even number of odd numbers among \(a, b, c\).
2. **Case Analys... | 37 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
King William is located at $(1, 1)$ on the coordinate plane. Every day, he chooses one of the eight lattice points closest to him and moves to one of them with equal probability. When he exits the region bounded by the $x, y$ axes and $x+y = 4$, he stops moving and remains there forever. Given that after an arbitrarily... | 1. **Define the problem and variables:**
- Let \( p_1 \) be the probability that King William ends up on the line \( x + y = 4 \) starting from \((1,1)\).
- Let \( p_2 \) be the probability that King William ends up on the line \( x + y = 4 \) starting from \((2,1)\).
2. **Set up the equations:**
- From \((1,... | 17 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $p$ and $q$ be positive integers such that $p$ is a prime, $p$ divides $q-1$, and $p+q$ divides $p^2+2020q^2$. Find the sum of the possible values of $p$.
[i]Proposed by Andy Xu[/i] | 1. **Given Conditions and Initial Setup:**
- \( p \) is a prime number.
- \( p \) divides \( q-1 \), i.e., \( q \equiv 1 \pmod{p} \).
- \( p+q \) divides \( p^2 + 2020q^2 \).
2. **Simplifying the Divisibility Condition:**
- We start with the condition \( p+q \mid p^2 + 2020q^2 \).
- Rewrite \( p^2 + 202... | 35 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $P$ be a point chosen on the interior of side $\overline{BC}$ of triangle $\triangle ABC$ with side lengths $\overline{AB} = 10, \overline{BC} = 10, \overline{AC} = 12$. If $X$ and $Y$ are the feet of the perpendiculars from $P$ to the sides $AB$ and $AC$, then the minimum possible value of $PX^2 + PY^2$ can be exp... | 1. **Calculate the area of triangle \(\triangle ABC\)**:
- Given side lengths: \(AB = 10\), \(BC = 10\), \(AC = 12\).
- We can split \(\triangle ABC\) into two right triangles by drawing an altitude from \(A\) to \(BC\). Let \(D\) be the foot of this altitude.
- Since \(\triangle ABC\) is isosceles with \(AB =... | 1936 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[u]Set 1[/u]
[b]G1.[/b] The Daily Challenge office has a machine that outputs the number $2.75$ when operated. If it is operated $12$ times, then what is the sum of all $12$ of the machine outputs?
[b]G2.[/b] A car traveling at a constant velocity $v$ takes $30$ minutes to travel a distance of $d$. How long does it ... | 9. A number \(n\) can be represented in base \(6\) as \(\underline{aba}_6\) and base \(15\) as \(\underline{ba}_{15}\), where \(a\) and \(b\) are not necessarily distinct digits. Find \(n\).
We have:
\[
n = 6^2 \cdot a + 6 \cdot b + a = 36a + 6b + a = 37a + 6b
\]
\[
n = 15 \cdot b + a
\]
Equating the two expression... | 61 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[u]Set 4[/u]
[b]G10.[/b] Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The minimum possible area of the resulting shape is $A$. Find the integer closest to $100A$.
[b]G11.[/b] The $10$-digit number $\underline{1A2B3C5D6E}$ is a... | 1. To solve the problem, we need to use the divisibility rules for 9 and 11. A number is divisible by 99 if and only if it is divisible by both 9 and 11.
2. For a number to be divisible by 9, the sum of its digits must be divisible by 9. Let the sum of the digits of the number be \( S \). Then, we have:
\[
1 + A ... | 28 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[u]Set 7[/u]
[b]G19.[/b] How many ordered triples $(x, y, z)$ with $1 \le x, y, z \le 50$ are there such that both $x + y + z$ and $xy + yz + zx$ are divisible by$ 6$?
[b]G20.[/b] Triangle $ABC$ has orthocenter $H$ and circumcenter $O$. If $D$ is the foot of the perpendicular from $A$ to $BC$, then $AH = 8$ and $HD... | To solve the problem, we need to find the probability that Nate flips the coin exactly 12 times and gets two heads in a row immediately followed by a tail (HHT) in the last three flips. We will use the Principle of Inclusion-Exclusion (PIE) to count the number of sequences where HHT does not occur in the first 9 flips.... | 4239 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments $AB = 10$ and $CD = 7$ on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral $ACBD$.
| To determine the maximum possible area of quadrilateral \(ACBD\) formed by two intersecting segments \(AB = 10\) and \(CD = 7\), we need to consider two cases: when the quadrilateral is convex and when it is non-convex.
1. **Convex Case:**
- Let segment \(AB\) intersect segment \(CD\) at point \(O\).
- Draw line... | 35 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$, $BC = 12$, $CA = 13$. The probability that a circle with radius $\frac13$ centered at $P$ does not intersect the perimeter of $ABC$ can be written as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers... | 1. **Identify the type of triangle and its properties:**
- Given triangle \(ABC\) with side lengths \(AB = 5\), \(BC = 12\), and \(CA = 13\).
- Verify that \(ABC\) is a right triangle by checking the Pythagorean theorem:
\[
AB^2 + BC^2 = 5^2 + 12^2 = 25 + 144 = 169 = 13^2 = CA^2
\]
Therefore, ... | 61 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In the coordinate plane, the line passing through points $(2023,0)$ and $(-2021,2024)$ also passes through $(1,c)$ for a constant $c$. Find $c$.
[i]Proposed by Andy Xu[/i] | 1. To find the value of \( c \) for the point \((1, c)\) on the line passing through \((2023, 0)\) and \((-2021, 2024)\), we first need to determine the equation of the line.
2. The slope \( m \) of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x... | 1012 | Geometry | 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 |
Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$. How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$? Note that $\lfloor x\rfloor$ is known as the $\textit{floor}$ function, which returns the greatest int... | To determine how many integers from $1$ to $2023$ inclusive are in the range of the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$, we need to analyze the behavior of the function and find the range of values it can produce.
1. **Analyze the function components:**
-... | 2071 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Pentagon $ANDD'Y$ has $AN \parallel DY$ and $AY \parallel D'N$ with $AN = D'Y$ and $AY = DN$. If the area of $ANDY$ is 20, the area of $AND'Y$ is 24, and the area of $ADD'$ is 26, the area of $ANDD'Y$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m+n$.
[i]Proposed... | Given the pentagon \(ANDD'Y\) with the following properties:
- \(AN \parallel DY\)
- \(AY \parallel D'N\)
- \(AN = D'Y\)
- \(AY = DN\)
We are also given the areas of certain regions:
- The area of \(ANDY\) is 20.
- The area of \(AND'Y\) is 24.
- The area of \(ADD'\) is 26.
We need to find the area of the pentagon \(A... | 71 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\triangle{ABC}$ be a triangle with $AB = 10$ and $AC = 11$. Let $I$ be the center of the inscribed circle of $\triangle{ABC}$. If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$, then $BC$ can be expressed in the form $\frac{\sqrt{a}-b}{c}$ where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.
... | 1. **Given Information and Setup:**
- We are given a triangle \( \triangle ABC \) with \( AB = 10 \) and \( AC = 11 \).
- \( I \) is the incenter of \( \triangle ABC \).
- \( M \) is the midpoint of \( AI \).
- We are given that \( BM = BC \) and \( CM = 7 \).
2. **Using the Given Conditions:**
- We nee... | 622 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a parallelogram with area 160. Let diagonals $AC$ and $BD$ intersect at $E$. Point $P$ is on $\overline{AE}$ such that $EC = 4EP$. If line $DP$ intersects $AB$ at $F$, find the area of $BFPC$.
[i]Proposed by Andy Xu[/i] | 1. Given that $ABCD$ is a parallelogram with area 160. The diagonals $AC$ and $BD$ intersect at $E$. Point $P$ is on $\overline{AE}$ such that $EC = 4EP$. We need to find the area of $BFPC$ where line $DP$ intersects $AB$ at $F$.
2. Since $E$ is the midpoint of both diagonals $AC$ and $BD$, we have $AE = EC$ and $BE =... | 62 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Will is distributing his money to three friends. Since he likes some friends more than others, the amount of money he gives each is in the ratio of $5 : 3 : 2$. If the person who received neither the least nor greatest amount of money was given $42$ dollars, how many dollars did Will distribute in all?
[b]... | 1. Let the total amount of money Will distributes be \( T \).
2. The ratio of the amounts given to the three friends is \( 5:3:2 \).
3. Let the amounts given to the three friends be \( 5x \), \( 3x \), and \( 2x \) respectively.
4. According to the problem, the person who received neither the least nor the greatest amo... | 140 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide]
[b]C1.[/b] A circle has circumference $6\pi$. Find the area of this circle.
[b]C2 / G2.[/b] Points $A$, $B$, and $C$ are on a line such that $AB = 6$ and $BC = 11$. Find all possible values of $AC$.
[b]C3.[/b] A ... | To solve this problem, we need to find the positive difference between the areas of two triangles \(ABC\) with given side lengths \(AB = 13\), \(BC = 12\sqrt{2}\), and \(\angle C = 45^\circ\).
1. **Identify the possible positions of point \(A\):**
- Draw an arc of a circle centered at \(B\) with radius \(13\). This... | 60 | 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 |
The school store is running out of supplies, but it still has five items: one pencil (costing $\$1$), one pen (costing $\$1$), one folder (costing $\$2$), one pack of paper (costing $\$3$), and one binder (costing $\$4$). If you have $\$10$, in how many ways can you spend your money? (You don't have to spend all of you... | 1. **Identify the total number of items and their costs:**
- Pencil: $1$
- Pen: $1$
- Folder: $2$
- Pack of paper: $3$
- Binder: $4$
2. **Calculate the total number of ways to choose any combination of these items:**
Each item can either be chosen or not chosen, which gives us $2$ choices per item. S... | 31 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In the middle of the school year, $40\%$ of Poolesville magnet students decided to transfer to the Blair magnet, and $5\%$ of the original Blair magnet students transferred to the Poolesville magnet. If the Blair magnet grew from $400$ students to $480$ students, how many students does the Poolesville magnet have after... | 1. **Determine the number of students who transferred from Blair to Poolesville:**
- Blair magnet originally had 400 students.
- 5% of Blair students transferred to Poolesville.
\[
0.05 \times 400 = 20 \text{ students}
\]
2. **Determine the number of students who transferred from Poolesville to Blair:**... | 170 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let G, O, D, I, and T be digits that satisfy the following equation:
\begin{tabular}{ccccc}
&G&O&G&O\\
+&D&I&D&I\\
\hline
G&O&D&O&T
\end{tabular}
(Note that G and D cannot be $0$, and that the five variables are not necessarily different.)
Compute the value of GODOT. | 1. We start with the given equation in the problem:
\[
\begin{array}{cccc}
& G & O & G & O \\
+ & D & I & D & I \\
\hline
G & O & D & O & T \\
\end{array}
\]
2. We need to determine the values of the digits \( G, O, D, I, \) and \( T \). Note that \( G \) and \( D \) cannot be \( 0 \).
3. First, consider the units ... | 10908 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The first triangle number is $1$; the second is $1 + 2 = 3$; the third is $1 + 2 + 3 = 6$; and so on. Find the sum of the first $100$ triangle numbers. | 1. The $n$-th triangular number is given by the formula:
\[
T_n = \frac{n(n+1)}{2}
\]
This formula can be derived from the sum of the first $n$ natural numbers.
2. We need to find the sum of the first 100 triangular numbers:
\[
\sum_{n=1}^{100} T_n = \sum_{n=1}^{100} \frac{n(n+1)}{2}
\]
3. We can... | 171700 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A checkerboard is $91$ squares long and $28$ squares wide. A line connecting two opposite vertices of the checkerboard is drawn. How many squares does the line pass through? | To solve this problem, we can use a well-known result in combinatorial geometry. The number of unit squares a diagonal line passes through in a grid of size \( m \times n \) is given by the formula:
\[
m + n - \gcd(m, n)
\]
where \(\gcd(m, n)\) is the greatest common divisor of \(m\) and \(n\).
1. **Identify the dim... | 112 | Geometry | 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 |
A positive integer is called [i]oneic[/i] if it consists of only $1$'s. For example, the smallest three oneic numbers are $1$, $11$, and $111$. Find the number of $1$'s in the smallest oneic number that is divisible by $63$. | To solve this problem, we need to find the smallest oneic number (a number consisting only of the digit 1) that is divisible by 63. Since 63 can be factored into \(63 = 7 \times 9\), the oneic number must be divisible by both 7 and 9.
1. **Divisibility by 9**:
A number is divisible by 9 if the sum of its digits is ... | 18 | Number Theory | 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 |
[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide]
[u] Set 1[/u]
[b]R1.1 / P1.1[/b] Find $291 + 503 - 91 + 492 - 103 - 392$.
[b]R1.2[/b] Let the operation $a$ & $b$ be defined to be $\frac{a-b}{a+b}$. What is $3$ & $-2$?
[b]R1.3[/b]. Joe can trade $5$ appl... | 1. **R1.1 / P1.1** Find \( 291 + 503 - 91 + 492 - 103 - 392 \).
\[
291 + 503 - 91 + 492 - 103 - 392 = (291 + 503 + 492) - (91 + 103 + 392)
\]
\[
= 1286 - 586 = 700
\]
The final answer is \(\boxed{241}\) | 241 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide]
[u]Set 1[/u]
[b]D1 / Z1.[/b] What is $1 + 2 \cdot 3$?
[b]D2.[/b] What is the average of the first $9$ positive integers?
[b]D3 / Z2.[/b] A square of side length $2$ is cut into $4$ congruent squares. What i... | To solve the problem, we need to determine the smallest integer \( n \) such that the product of \( n, n+1, n+2, \ldots, n+9 \) ends in three zeros. This means the product must be divisible by \( 10^3 = 1000 \), which requires at least three factors of 2 and three factors of 5 in the prime factorization of the product.... | 16 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide]
[u]Set 4[/u]
[b]D16.[/b] The cooking club at Blair creates $14$ croissants and $21$ danishes. Daniel chooses pastries randomly, stopping when he gets at least one croissant and at least two danishes. How many ... | To solve the problem, we need to determine the digits \(a\) and \(b\) such that the number \(12ab9876543\) is divisible by 101. Let's break down the steps:
1. **Express the number in terms of \(a\) and \(b\):**
\[
N = \overline{12ab9876543} = 12009876543 + (10a + b) \times 10^7
\]
Since \(N\) is divisible ... | 58 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide]
[b]Z15.[/b] Let $AOB$ be a quarter circle with center $O$ and radius $4$. Let $\omega_1$ and $\omega_2$ be semicircles inside $AOB$ with diameters $OA$ and $OB$, respectively. Find the area of the region within ... | 1. Let the integers \(a\), \(b\), and \(c\) form a geometric sequence with a common ratio \(r\). Therefore, we can write:
\[
b = ar \quad \text{and} \quad c = ar^2
\]
2. Given that \(c = a + 56\), we substitute \(c\) with \(ar^2\):
\[
ar^2 = a + 56
\]
3. Rearrange the equation to isolate \(a\):
\[
... | 21 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide]
[u]Set 1[/u]
[b]B1 / G1[/b] Find $20^3 + 2^2 + 3^1$.
[b]B2[/b] A piece of string of length $10$ is cut $4$ times into strings of equal length. What is the length of each small piece of string?
[b]B3 / G2[/b... | 1. **B1 / G1** Find \( 20^3 + 2^2 + 3^1 \).
\[
20^3 + 2^2 + 3^1 = 8000 + 4 + 3 = 8007
\]
\(\boxed{8007}\)
2. **B2** A piece of string of length \( 10 \) is cut \( 4 \) times into strings of equal length. What is the length of each small piece of string?
Since there are \( 4 \) cuts, there are \( 5 \)... | 9900999 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Victor has $3$ piles of $3$ cards each. He draws all of the cards, but cannot draw a card until all the cards above it have been drawn. (For example, for his first card, Victor must draw the top card from one of the $3$ piles.) In how many orders can Victor draw the cards? | 1. We start by naming the cards based on the pile they are in. Let's denote the cards in the first pile as \(A_1, A_2, A_3\), the cards in the second pile as \(B_1, B_2, B_3\), and the cards in the third pile as \(C_1, C_2, C_3\).
2. Victor must draw the cards in such a way that he can only draw a card if all the car... | 1680 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Mr. Rose, Mr. Stein, and Mr. Schwartz start at the same point around a circular track and run clockwise. Mr. Stein completes each lap in $6$ minutes, Mr. Rose in $10$ minutes, and Mr. Schwartz in $18$ minutes. How many minutes after the start of the race are the runners at identical points around the track (that is, th... | To determine when Mr. Rose, Mr. Stein, and Mr. Schwartz will be at the same point on the track for the first time, we need to find the least common multiple (LCM) of their lap times. The lap times are 6 minutes, 10 minutes, and 18 minutes respectively.
1. **Prime Factorization**:
- Mr. Stein's lap time: \(6 = 2 \ti... | 90 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Adam, Bendeguz, Cathy, and Dennis all see a positive integer $n$. Adam says, "$n$ leaves a remainder of $2$ when divided by $3$." Bendeguz says, "For some $k$, $n$ is the sum of the first $k$ positive integers." Cathy says, "Let $s$ be the largest perfect square that is less than $2n$. Then $2n - s = 20$." Dennis says,... | 1. **Adam's Statement**: Adam says that \( n \equiv 2 \pmod{3} \). This means that when \( n \) is divided by 3, the remainder is 2.
2. **Bendeguz's Statement**: Bendeguz says that \( n \) is a triangular number. A triangular number can be expressed as:
\[
n = \frac{k(k+1)}{2}
\]
for some integer \( k \... | 210 | Logic and Puzzles | 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 |
Jane tells you that she is thinking of a three-digit number that is greater than $500$ that has exactly $20$ positive divisors. If Jane tells you the sum of the positive divisors of her number, you would not be able to figure out her number. If, instead, Jane had told you the sum of the \textit{prime} divisors of her n... | 1. **Identify the form of the number:**
Given that the number has exactly 20 positive divisors, we can use the formula for the number of divisors. If \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \), then the number of divisors is given by:
\[
(e_1 + 1)(e_2 + 1) \cdots (e_k + 1) = 20
\]
The possible factor... | 880 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. If the area of triangle $DOC$ is $4$ and the area of triangle $AOB$ is $36$, compute the minimum possible value of the area of $ABCD$. | 1. Given that diagonals \(AC\) and \(BD\) of quadrilateral \(ABCD\) intersect at \(O\), we know that the area of triangle \(DOC\) is 4 and the area of triangle \(AOB\) is 36.
2. We need to find the minimum possible value of the area of quadrilateral \(ABCD\).
To solve this, we will use the fact that the area of a quad... | 80 | Geometry | 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 |
[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide]
[b]D1.[/b] The product of two positive integers is $5$. What is their sum?
[b]D2.[/b] Gavin is $4$ feet tall. He walks $5$ feet before falling forward onto a cushion. How many feet is the top of Gavin’s head fro... | To solve the problem, we need to find the number of possible values of \( m \) such that the quadratic equation \( x^2 + 202200x + 2022m \) has integer roots. This requires the discriminant of the quadratic equation to be a perfect square.
1. **Discriminant Condition**:
The discriminant of the quadratic equation \(... | 50 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide]
[b]B1[/b] What is the sum of the first $5$ positive integers?
[b]B2[/b] Bread picks a number $n$. He finds out that if he multiplies $n$ by $23$ and then subtracts $20$, he gets $46279$. What is $n$?
[b]B3[/b... | 1. **B1** What is the sum of the first $5$ positive integers?
To find the sum of the first $5$ positive integers, we use the formula for the sum of the first $n$ positive integers:
\[
S = \frac{n(n+1)}{2}
\]
For \( n = 5 \):
\[
S = \frac{5(5+1)}{2} = \frac{5 \cdot 6}{2} = 15
\]
The final answer is \(\boxed{27}\) | 27 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Kevin the Koala eats $1$ leaf on the first day of its life, $3$ leaves on the second, $5$ on the third, and in general eats $2n-1$ leaves on the $n$th day. What is the smallest positive integer $n>1$ such that the total number of leaves Kevin has eaten his entire $n$-day life is a perfect sixth power?
[i]2015 CCA Math... | 1. First, we need to determine the total number of leaves Kevin has eaten by the \(n\)-th day. The number of leaves eaten on the \(n\)-th day is given by \(2n - 1\). Therefore, the total number of leaves eaten by the \(n\)-th day is the sum of the first \(n\) odd numbers:
\[
1 + 3 + 5 + \cdots + (2n-1)
\]
I... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many positive integers less than or equal to $1000$ are divisible by $2$ and $3$ but not by $5$?
[i]2015 CCA Math Bonanza Individual Round #6[/i] | 1. **Determine the number of integers less than or equal to 1000 that are divisible by 6:**
- Since a number divisible by both 2 and 3 is divisible by their least common multiple (LCM), which is 6, we first find the number of integers less than or equal to 1000 that are divisible by 6.
- This can be calculated as... | 133 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Harry Potter would like to purchase a new owl which cost him 2 Galleons, a Sickle, and 5 Knuts. There are 23 Knuts in a Sickle and 17 Sickles in a Galleon. He currently has no money, but has many potions, each of which are worth 9 Knuts. How many potions does he have to exhange to buy this new owl?
[i]2015 CCA Math Bo... | 1. **Convert the cost of the owl into Knuts:**
- Given: 1 Galleon = 17 Sickles and 1 Sickle = 23 Knuts.
- Therefore, 1 Galleon = \( 17 \times 23 = 391 \) Knuts.
- The cost of the owl is 2 Galleons, 1 Sickle, and 5 Knuts.
- Convert 2 Galleons to Knuts: \( 2 \times 391 = 782 \) Knuts.
- Convert 1 Sickle to... | 90 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A rectangle has an area of $16$ and a perimeter of $18$; determine the length of the diagonal of the rectangle.
[i]2015 CCA Math Bonanza Individual Round #8[/i] | 1. Let the lengths of the sides of the rectangle be \( a \) and \( b \). We are given two pieces of information:
- The area of the rectangle is \( ab = 16 \).
- The perimeter of the rectangle is \( 2(a + b) = 18 \), which simplifies to \( a + b = 9 \).
2. To find the length of the diagonal, we use the Pythagorea... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The fourth-degree equation $x^4-x-504=0$ has $4$ roots $r_1$, $r_2$, $r_3$, $r_4$. If $S_x$ denotes the value of ${r_1}^4+{r_2}^4+{r_3}^4+{r_4}^4$, compute $S_4$.
[i]2015 CCA Math Bonanza Individual Round #10[/i] | 1. Given the polynomial equation \(x^4 - x - 504 = 0\), we know it has four roots \(r_1, r_2, r_3, r_4\).
2. For each root \(r_i\), we have \(r_i^4 = r_i + 504\). This follows directly from substituting \(r_i\) into the polynomial equation.
3. We need to find the value of \(S_4 = r_1^4 + r_2^4 + r_3^4 + r_4^4\).
4. Usi... | 2016 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
How many ways are there to rearrange the letters of the word RAVEN such that no two vowels are consecutive?
[i]2015 CCA Math Bonanza Team Round #2[/i] | 1. **Calculate the total number of arrangements of the letters in "RAVEN":**
The word "RAVEN" consists of 5 distinct letters. The total number of ways to arrange these letters is given by the factorial of the number of letters:
\[
5! = 120
\]
2. **Calculate the number of arrangements where the vowels (A an... | 72 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Emily Thorne is throwing a Memorial Day Party to kick off the Summer in the Hamptons, and she is trying to figure out the seating arrangment for all of her guests. Emily saw that if she seated $4$ guests to a table, there would be $1$ guest left over (how sad); if she seated $5$ to a table, there would be $3$ guests le... | To solve this problem, we need to find a number \( n \) that satisfies the following conditions:
1. \( n \equiv 1 \pmod{4} \)
2. \( n \equiv 3 \pmod{5} \)
3. \( n \equiv 1 \pmod{6} \)
4. \( 100 \leq n \leq 200 \)
We will use the Chinese Remainder Theorem (CRT) to solve this system of congruences.
1. **Combine the con... | 193 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many digits are in the base $10$ representation of $3^{30}$ given $\log 3 = 0.47712$?
[i]2015 CCA Math Bonanza Lightning Round #1.4[/i] | 1. To determine the number of digits in \(3^{30}\), we use the formula for the number of digits of a number \(n\) in base 10, which is given by:
\[
\text{Number of digits} = \lfloor \log_{10} n \rfloor + 1
\]
2. We need to find \(\log_{10} (3^{30})\). Using the properties of logarithms, we have:
\[
\log_... | 15 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The polynomial $x^3-kx^2+20x-15$ has $3$ roots, one of which is known to be $3$. Compute the greatest possible sum of the other two roots.
[i]2015 CCA Math Bonanza Lightning Round #2.4[/i] | 1. **Identify the roots and use Vieta's formulas:**
Given the polynomial \( P(x) = x^3 - kx^2 + 20x - 15 \) and one of the roots is \( 3 \), we can denote the roots as \( 3, x_1, x_2 \).
2. **Apply Vieta's formulas:**
By Vieta's formulas for a cubic polynomial \( ax^3 + bx^2 + cx + d = 0 \), the sum of the roots... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Alice the ant starts at vertex $A$ of regular hexagon $ABCDEF$ and moves either right or left each move with equal probability. After $35$ moves, what is the probability that she is on either vertex $A$ or $C$?
[i]2015 CCA Math Bonanza Lightning Round #5.3[/i] | 1. **Understanding the problem**: Alice starts at vertex \( A \) of a regular hexagon \( ABCDEF \) and moves either right or left with equal probability. We need to find the probability that after 35 moves, Alice is at either vertex \( A \) or \( C \).
2. **Analyzing the movement**: In a regular hexagon, moving right ... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If $a,b,c$ are the roots of $x^3+20x^2+1x+5$, compute $(a^2+1)(b^2+1)(c^2+1)$.
[i]2015 CCA Math Bonanza Tiebreaker Round #2[/i] | 1. Given that \(a, b, c\) are the roots of the polynomial \(x^3 + 20x^2 + x + 5\), we need to compute \((a^2 + 1)(b^2 + 1)(c^2 + 1)\).
2. Notice that \((a^2 + 1)(b^2 + 1)(c^2 + 1)\) can be rewritten using complex numbers. Specifically, we can use the fact that \(a^2 + 1 = (a - i)(a + i)\), \(b^2 + 1 = (b - i)(b + i)\)... | 229 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Positive integers (not necessarily unique) are written, one on each face, on two cubes such that when the two cubes are rolled, each integer $2\leq k\leq12$ appears as the sum of the upper faces with probability $\frac{6-|7-k|}{36}$. Compute the greatest possible sum of all the faces on one cube.
[i]2015 CCA Math Bona... | 1. **Understanding the Problem:**
We need to find the greatest possible sum of all the faces on one cube such that the sum of the numbers on the faces of two cubes follows the same probability distribution as the sum of two standard six-sided dice. The probability distribution for the sum \( k \) of two standard dic... | 33 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Sristan Thin is walking around the Cartesian plane. From any point $\left(x,y\right)$, Sristan can move to $\left(x+1,y\right)$ or $\left(x+1,y+3\right)$. How many paths can Sristan take from $\left(0,0\right)$ to $\left(9,9\right)$?
[i]2019 CCA Math Bonanza Individual Round #3[/i] | 1. **Understanding the movement constraints**:
- Sristan can move from \((x, y)\) to either \((x+1, y)\) or \((x+1, y+3)\).
- To reach from \((0,0)\) to \((9,9)\), the x-coordinate must increase by 1 in each step, requiring exactly 9 steps.
2. **Analyzing the y-coordinate changes**:
- We need to increase the ... | 84 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many ways are there to rearrange the letters of CCAMB such that at least one C comes before the A?
[i]2019 CCA Math Bonanza Individual Round #5[/i] | 1. **Calculate the total number of arrangements of the letters in "CCAMB":**
- The word "CCAMB" consists of 5 letters where 'C' appears twice.
- The total number of distinct permutations of these letters is given by:
\[
\frac{5!}{2!} = \frac{120}{2} = 60
\]
2. **Determine the number of arrangement... | 40 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If distinct digits $D,E,L,M,Q$ (between $0$ and $9$ inclusive) satisfy
\begin{tabular}{c@{\,}c@{\,}c@{\,}c}
& & $E$ & $L$ \\
+ & $M$ & $E$ & $M$ \\\hline
& $Q$ & $E$ & $D$ \\
\end{tabular}
what is the maximum possible value of the three digit integer $QED$?
[i]2019 CCA Math Bonanza Individual... | 1. We start with the given equation in columnar form:
\[
\begin{array}{c@{\,}c@{\,}c@{\,}c}
& & E & L \\
+ & M & E & M \\
\hline
& Q & E & D \\
\end{array}
\]
This translates to the equation:
\[
100E + 10L + E + 100M + 10E + M = 100Q + 10E + D
\]
Simplifyi... | 893 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Call an odd prime $p$ [i]adjective[/i] if there exists an infinite sequence $a_0,a_1,a_2,\ldots$ of positive integers such that \[a_0\equiv1+\frac{1}{a_1}\equiv1+\frac{1}{1+\frac{1}{a_2}}\equiv1+\frac{1}{1+\frac{1}{1+\frac{1}{a_3}}}\equiv\ldots\pmod p.\] What is the sum of the first three odd primes that are [i]not[/i]... | 1. **Define the sequence and the problem conditions:**
We need to determine the sum of the first three odd primes that are not "adjective." A prime \( p \) is called "adjective" if there exists an infinite sequence \( a_0, a_1, a_2, \ldots \) of positive integers such that:
\[
a_0 \equiv 1 + \frac{1}{a_1} \equ... | 33 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A triangle has side lengths of $x,75,100$ where $x<75$ and altitudes of lengths $y,28,60$ where $y<28$. What is the value of $x+y$?
[i]2019 CCA Math Bonanza Team Round #2[/i] | 1. **Understanding the problem**: We are given a triangle with side lengths \(x, 75, 100\) and altitudes \(y, 28, 60\) corresponding to these sides. We need to find the value of \(x + y\) given the constraints \(x < 75\) and \(y < 28\).
2. **Using the area relationship**: The area of a triangle can be expressed using ... | 56 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
What is the sum of all possible values of $\cos\left(2\theta\right)$ if $\cos\left(2\theta\right)=2\cos\left(\theta\right)$ for a real number $\theta$?
[i]2019 CCA Math Bonanza Team Round #3[/i] | 1. We start with the given equation:
\[
\cos(2\theta) = 2\cos(\theta)
\]
2. Using the double-angle identity for cosine, we know:
\[
\cos(2\theta) = 2\cos^2(\theta) - 1
\]
3. Substituting this identity into the given equation, we get:
\[
2\cos^2(\theta) - 1 = 2\cos(\theta)
\]
4. Rearrange t... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ordered tuples $\left(C,A,M,B\right)$ of non-negative integers such that \[C!+C!+A!+M!=B!\]
[i]2019 CCA Math Bonanza Team Round #4[/i] | To find the number of ordered tuples \((C, A, M, B)\) of non-negative integers such that
\[ C! + C! + A! + M! = B!, \]
we will analyze the possible values for \(B\) by bounding and listing out the solutions.
1. **Bounding \(B\):**
Since factorials grow very quickly, we can start by considering small values for \(B... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the smallest positive integer $n$ such that there exists a choice of signs for which \[1^2\pm2^2\pm3^2\ldots\pm n^2=0\] is true?
[i]2019 CCA Math Bonanza Team Round #5[/i] | To find the smallest positive integer \( n \) such that there exists a choice of signs for which
\[ 1^2 \pm 2^2 \pm 3^2 \pm \ldots \pm n^2 = 0 \]
is true, we need to consider the sum of the squares of the first \( n \) integers.
1. **Sum of Squares Formula**:
The sum of the squares of the first \( n \) integers is... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
fantasticbobob is proctoring a room for the SiSiEyMB with $841$ seats arranged in $29$ rows and $29$ columns. The contestants sit down, take part $1$ of the contest, go outside for a break, and come back to take part $2$ of the contest. fantasticbobob sits among the contestants during part $1$, also goes outside during... | 1. **Understanding the Problem:**
- We have a $29 \times 29$ grid of seats, totaling $841$ seats.
- There are $840$ contestants, each sitting in one seat.
- After a break, each contestant moves to a seat that is horizontally or vertically adjacent to their original seat.
- We need to determine how many poss... | 421 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many integers divide either $2018$ or $2019$? Note: $673$ and $1009$ are both prime.
[i]2019 CCA Math Bonanza Lightning Round #1.1[/i] | To determine how many integers divide either \(2018\) or \(2019\), we need to find the number of divisors of each number and then account for any overlap (common divisors).
1. **Find the prime factorizations:**
- \(2018 = 2 \times 1009\)
- \(2019 = 3 \times 673\)
2. **Determine the number of divisors:**
- Fo... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest positive integer $n$ for which there are no [i]positive[/i] integers $a,b$ such that $8a+11b=n$?
[i]2019 CCA Math Bonanza Lightning Round #2.2[/i] | To solve this problem, we will use the Chicken McNugget Theorem (also known as the Frobenius Coin Problem). The theorem states that for two coprime integers \( m \) and \( n \), the largest integer that cannot be expressed as \( am + bn \) for nonnegative integers \( a \) and \( b \) is \( mn - m - n \).
1. **Identify... | 88 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $N$ is a three digit number divisible by $7$ such that upon removing its middle digit, the remaining two digit number is also divisible by $7$. What is the minimum possible value of $N$?
[i]2019 CCA Math Bonanza Lightning Round #3.1[/i] | 1. Let \( N = 100a + 10b + c \), where \( a, b, c \) are digits and \( a \neq 0 \) since \( N \) is a three-digit number.
2. Given that \( N \) is divisible by \( 7 \), we have:
\[
100a + 10b + c \equiv 0 \pmod{7}
\]
3. Removing the middle digit \( b \) from \( N \) results in the two-digit number \( 10a + c \... | 154 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What is the area of a triangle with side lengths $17$, $25$, and $26$?
[i]2019 CCA Math Bonanza Lightning Round #3.2[/i] | 1. **Identify the side lengths and calculate the semi-perimeter:**
The side lengths of the triangle are \(a = 17\), \(b = 25\), and \(c = 26\).
The semi-perimeter \(s\) is given by:
\[
s = \frac{a + b + c}{2} = \frac{17 + 25 + 26}{2} = \frac{68}{2} = 34
\]
2. **Apply Heron's Formula:**
Heron's Formul... | 204 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine the maximum possible value of \[\frac{\left(x^2+5x+12\right)\left(x^2+5x-12\right)\left(x^2-5x+12\right)\left(-x^2+5x+12\right)}{x^4}\] over all non-zero real numbers $x$.
[i]2019 CCA Math Bonanza Lightning Round #3.4[/i] | To determine the maximum possible value of the given expression, we start by simplifying and analyzing the expression:
\[
\frac{\left(x^2+5x+12\right)\left(x^2+5x-12\right)\left(x^2-5x+12\right)\left(-x^2+5x+12\right)}{x^4}
\]
1. **Simplify the expression:**
Let us denote the terms in the numerator as follows:
... | 576 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The Garfield Super Winners play $100$ games of foosball, in which teams score a non-negative integer number of points and the team with more points after ten minutes wins (if both teams have the same number of points, it is a draw). Suppose that the Garfield Super Winners score an average of $7$ points per game but all... | 1. Let \( W \) be the number of games the Garfield Super Winners (GSW) win, and \( L \) be the number of games they lose. Since they play 100 games in total, we have:
\[
W + L = 100
\]
2. The average points scored by GSW per game is 7, and the average points allowed per game is 8. Therefore, the average point... | 81 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
GM Bisain's IQ is so high that he can move around in $10$ dimensional space. He starts at the origin and moves in a straight line away from the origin, stopping after $3$ units. How many lattice points can he land on? A lattice point is one with all integer coordinates.
[i]2019 CCA Math Bonanza Lightning Round #4.2[/i... | To determine the number of lattice points GM Bisain can land on in a 10-dimensional space after moving 3 units from the origin, we need to find the number of integer solutions to the equation:
\[ a^2 + b^2 + c^2 + \cdots + j^2 = 3^2 \]
This equation can be written as:
\[ a^2 + b^2 + c^2 + \cdots + j^2 = 9 \]
We nee... | 1720 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that a planet contains $\left(CCAMATHBONANZA_{71}\right)^{100}$ people ($100$ in decimal), where in base $71$ the digits $A,B,C,\ldots,Z$ represent the decimal numbers $10,11,12,\ldots,35$, respectively. Suppose that one person on this planet is snapping, and each time they snap, at least half of the current po... | 1. First, we need to understand the problem. We are given a population of $\left(CCAMATHBONANZA_{71}\right)^{100}$ people, where $CCAMATHBONANZA_{71}$ is a number in base 71. The digits $A, B, C, \ldots, Z$ represent the decimal numbers $10, 11, 12, \ldots, 35$ respectively.
2. We need to convert $CCAMATHBONANZA_{71}$... | 8326 | Combinatorics | other | Yes | Yes | aops_forum | false |
Isosceles triangle $\triangle{ABC}$ has $\angle{ABC}=\angle{ACB}=72^\circ$ and $BC=1$. If the angle bisector of $\angle{ABC}$ meets $AC$ at $D$, what is the positive difference between the perimeters of $\triangle{ABD}$ and $\triangle{BCD}$?
[i]2019 CCA Math Bonanza Tiebreaker Round #2[/i] | 1. Given that $\triangle{ABC}$ is an isosceles triangle with $\angle{ABC} = \angle{ACB} = 72^\circ$ and $BC = 1$. We need to find the positive difference between the perimeters of $\triangle{ABD}$ and $\triangle{BCD}$, where $D$ is the point where the angle bisector of $\angle{ABC}$ meets $AC$.
2. Let $AB = AC = x$. S... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
An ant is crawling along the coordinate plane. Each move, it moves one unit up, down, left, or right with equal probability. If it starts at $(0,0)$, what is the probability that it will be at either $(2,1)$ or $(1,2)$ after $6$ moves?
[i]2020 CCA Math Bonanza Individual Round #1[/i] | 1. **Understanding the problem**: The ant starts at $(0,0)$ and makes 6 moves, each move being one unit up, down, left, or right with equal probability. We need to determine the probability that the ant will be at either $(2,1)$ or $(1,2)$ after 6 moves.
2. **Sum of coordinates**: Notice that each move changes the sum... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)=x^2-kx+(k-1)^2$ for some constant $k$. What is the largest possible real value of $k$ such that $f$ has at least one real root?
[i]2020 CCA Math Bonanza Individual Round #5[/i] | 1. **Identify the discriminant of the quadratic function:**
The given quadratic function is \( f(x) = x^2 - kx + (k-1)^2 \). For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is given by:
\[
\Delta = b^2 - 4ac
\]
Here, \( a = 1 \), \( b = -k \), and \( c = (k-1)^2 \). Substi... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Compute the remainder when the largest integer below $\frac{3^{123}}{5}$ is divided by $16$.
[i]2020 CCA Math Bonanza Individual Round #8[/i] | 1. We need to compute the remainder when the largest integer below $\frac{3^{123}}{5}$ is divided by $16$.
2. First, let's find the remainder of $3^{123}$ when divided by $5$. We use Fermat's Little Theorem, which states that if $p$ is a prime number and $a$ is an integer not divisible by $p$, then $a^{p-1} \equiv 1 \p... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A sequence $a_n$ of real numbers satisfies $a_1=1$, $a_2=0$, and $a_n=(S_{n-1}+1)S_{n-2}$ for all integers $n\geq3$, where $S_k=a_1+a_2+\dots+a_k$ for positive integers $k$. What is the smallest integer $m>2$ such that $127$ divides $a_m$?
[i]2020 CCA Math Bonanza Individual Round #9[/i] | 1. Given the sequence \(a_n\) with initial conditions \(a_1 = 1\) and \(a_2 = 0\), and the recurrence relation \(a_n = (S_{n-1} + 1) S_{n-2}\) for \(n \geq 3\), where \(S_k = a_1 + a_2 + \dots + a_k\).
2. We start by calculating the first few terms of the sequence and their corresponding sums:
\[
\begin{aligned}... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Annie takes a $6$ question test, with each question having two parts each worth $1$ point. On each [b]part[/b], she receives one of nine letter grades $\{\text{A,B,C,D,E,F,G,H,I}\}$ that correspond to a unique numerical score. For each [b]question[/b], she receives the sum of her numerical scores on both parts. She kno... | 1. **Determine the placement of A, E, and I:**
- Annie receives two of each of the grades A, E, and I. These grades must be distributed among the 6 questions such that each question has one part with one of these grades.
- The number of ways to distribute 2 A's, 2 E's, and 2 I's among 6 questions is given by the ... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The base $4$ repeating decimal $0.\overline{12}_4$ can be expressed in the form $\frac{a}{b}$ in base 10, where $a$ and $b$ are relatively prime positive integers. Compute the sum of $a$ and $b$.
[i]2020 CCA Math Bonanza Team Round #2[/i] | 1. First, let's express the repeating decimal \(0.\overline{12}_4\) as a geometric series. In base 4, the repeating decimal \(0.\overline{12}_4\) can be written as:
\[
0.\overline{12}_4 = 0.121212\ldots_4
\]
2. We can express this as an infinite series:
\[
0.\overline{12}_4 = \frac{1}{4} + \frac{2}{4^2}... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute
\[
\left(\frac{4-\log_{36} 4 - \log_6 {18}}{\log_4 3} \right) \cdot \left( \log_8 {27} + \log_2 9 \right).
\]
[i]2020 CCA Math Bonanza Team Round #4[/i] | 1. Start by simplifying the expression inside the first fraction:
\[
\left(\frac{4 - \log_{36} 4 - \log_6 18}{\log_4 3}\right)
\]
2. Use the change of base formula for logarithms:
\[
\log_{36} 4 = \frac{\log 4}{\log 36} = \frac{\log 4}{2 \log 6} = \frac{\log 2^2}{2 \log 6} = \frac{2 \log 2}{2 \log ... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Compute the remainder when $99989796\ldots 121110090807 \ldots 01$ is divided by $010203 \ldots 091011 \ldots 9798$ (note that the first one starts at $99$, and the second one ends at $98$).
[i]2020 CCA Math Bonanza Team Round #7[/i] | 1. **Define the numbers:**
Let \( N \) be the second number, which is the concatenation of numbers from 01 to 98. We need to compute the remainder when the first number, which is the concatenation of numbers from 99 to 01, is divided by \( N \).
2. **Express \( N \) in a mathematical form:**
\[
N = 010203 \ld... | 9801 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ such that $201+k$ and $9+k$ give the same remainder when divided by $24$?
[i]2020 CCA Math Bonanza Lightning Round #1.1[/i] | 1. We start with the given congruence:
\[
201 \equiv 9 \pmod{24}
\]
This means that when 201 and 9 are divided by 24, they leave the same remainder.
2. We need to find the smallest positive integer \( k \) such that:
\[
201 + k \equiv 9 + k \pmod{24}
\]
This implies:
\[
(201 + k) \mod 24 ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ such that $201+k$ and $9+k$ give the same remainder when divided by $24+k$?
[i]2020 CCA Math Bonanza Lightning Round #2.1[/i] | 1. We start with the given condition that \(201\) and \(9\) give the same remainder when divided by \(24\). This can be written as:
\[
201 \equiv 9 \pmod{24}
\]
Simplifying this, we get:
\[
201 - 9 = 192 \equiv 0 \pmod{24}
\]
This confirms that \(192\) is divisible by \(24\).
2. We need to find... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A rectangular box with side lengths $1$, $2$, and $16$ is cut into two congruent smaller boxes with integer side lengths. Compute the square of the largest possible length of the space diagonal of one of the smaller boxes.
[i]2020 CCA Math Bonanza Lightning Round #2.2[/i] | 1. We start with a rectangular box with side lengths \(1\), \(2\), and \(16\). We need to cut this box into two congruent smaller boxes with integer side lengths.
2. To achieve this, we can cut the original box along one of its dimensions. We have three possible ways to cut the box:
- Cut along the side of length \(... | 258 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$3$ uncoordinated aliens launch a $3$-day attack on $4$ galaxies. Each day, each of the three aliens chooses a galaxy uniformly at random from the remaining galaxies and destroys it. They make their choice simultaneously and independently, so two aliens could destroy the same galaxy. If the probability that every galax... | 1. **Understanding the problem**: We need to find the probability that every galaxy is destroyed at least once over the course of 3 days by 3 aliens, each choosing a galaxy uniformly at random from 4 galaxies.
2. **Complementary counting**: Instead of directly calculating the probability that every galaxy is destroyed... | 111389 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If
\[
\sum_{k=1}^{1000}\left( \frac{k+1}{k}+\frac{k}{k+1}\right)=\frac{m}{n}
\]
for relatively prime positive integers $m,n$, compute $m+n$.
[i]2020 CCA Math Bonanza Lightning Round #2.4[/i] | 1. We start with the given sum:
\[
\sum_{k=1}^{1000}\left( \frac{k+1}{k}+\frac{k}{k+1}\right)
\]
2. We can split the sum into two separate sums:
\[
\sum_{k=1}^{1000} \frac{k+1}{k} + \sum_{k=1}^{1000} \frac{k}{k+1}
\]
3. Simplify each term in the first sum:
\[
\frac{k+1}{k} = 1 + \frac{... | 2004000 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For some positive integer $n$, the sum of all odd positive integers between $n^2-n$ and $n^2+n$ is a number between $9000$ and $10000$, inclusive. Compute $n$.
[i]2020 CCA Math Bonanza Lightning Round #3.1[/i] | 1. First, observe that \( n^2 - n \) and \( n^2 + n \) are both even numbers. We need to find the sum of all odd integers between these two even numbers.
2. The sequence of odd integers between \( n^2 - n \) and \( n^2 + n \) is:
\[
n^2 - n + 1, \, n^2 - n + 3, \, \ldots, \, n^2 + n - 1
\]
3. This sequence i... | 21 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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