problem stringlengths 11 5.15k | answer stringlengths 1 42 | solution stringlengths 0 8.75k |
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Doug constructs a square window using $8$ equal-size panes of glass. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window? | 26 | 1. **Identify the dimensions of each pane**: Given that the ratio of the height to the width of each pane is $5:2$, let the height of each pane be $5x$ inches and the width be $2x$ inches.
2. **Calculate the total dimensions of the window**: The window is constructed with $8$ panes arranged in $2$ rows and $4$ columns... |
Let $P(x)$ be a polynomial of degree $3n$ such that
\begin{align*} P(0) = P(3) = \dots = P(3n) &= 2, \\ P(1) = P(4) = \dots = P(3n+1-2) &= 1, \\ P(2) = P(5) = \dots = P(3n+2-2) &= 0. \end{align*}
Also, $P(3n+1) = 730$. Determine $n$. | 1 | To solve for $n$, we start by analyzing the polynomial $P(x)$ given its values at specific points and its degree. We use Lagrange Interpolation Formula to express $P(x)$, and then evaluate it at $x = 3n+1$ to find $n$.
1. **Constructing the Polynomial Using Lagrange Interpolation:**
The polynomial $P(x)$ is defined... |
At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, $8$ students in Mr. Newton's class, and $9$ students in Mrs. Young's class taking the AMC $8$ this year. How many mathematics students at Euclid Middle School are taking the co... | 28 | 1. **Identify the number of students in each class:**
- Mrs. Germain's class: 11 students
- Mr. Newton's class: 8 students
- Mrs. Young's class: 9 students
2. **Assumption of no overlap in students:**
Since the problem does not mention any students being in more than one class, we assume that all students ... |
If $991+993+995+997+999=5000-N$, then $N=$ | 25 | 1. **Identify the problem and express each term in a form that reveals a pattern:**
\[
991+993+995+997+999=5000-N
\]
We can rewrite each term as $1000$ minus a small number:
\[
(1000-9) + (1000-7) + (1000-5) + (1000-3) + (1000-1)
\]
2. **Simplify the expression by factoring out $1000$ and summing ... |
The total in-store price for an appliance is $99.99$. A television commercial advertises the same product for three easy payments of $29.98$ and a one-time shipping and handling charge of $9.98$. How many cents are saved by buying the appliance from the television advertiser? | 7 | 1. **Calculate the total cost of buying the appliance from the television advertiser:**
The advertisement mentions three payments of $\textdollar 29.98$ each and an additional one-time shipping and handling charge of $\textdollar 9.98$. First, calculate the total of the three payments:
\[
3 \times 29.98 = 89.9... |
A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\frac{7}{17}$. What number was erased? | 7 | 1. **Identify the sum of the first $n$ positive integers**: The sum of the first $n$ positive integers is given by the formula for the sum of an arithmetic series:
\[
S = \frac{n(n+1)}{2}
\]
2. **Expression for the average after erasing one number**: If one number $x$ is erased, the sum of the remaining numbe... |
What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are $(\cos 40^\circ,\sin 40^\circ)$, $(\cos 60^\circ,\sin 60^\circ)$, and $(\cos t^\circ,\sin t^\circ)$ is isosceles? | 380 | Let $A = (\cos 40^\circ, \sin 40^\circ)$, $B = (\cos 60^\circ, \sin 60^\circ)$, and $C = (\cos t^\circ, \sin t^\circ)$. We need to find the values of $t$ such that $\triangle ABC$ is isosceles. We consider three cases based on which sides of the triangle are equal.
**Case 1: $AB = AC$**
For $AB = AC$, $C$ must lie on... |
In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB, BC$, and $CD$ are diameters of circle $O, N$, and $P$, respectively. Circles $O, N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points $E$ and $F$, then chord $EF$ has len... | 20 | 1. **Identify the Configuration**: Given that $AB$, $BC$, and $CD$ are diameters of circles $O$, $N$, and $P$ respectively, and all circles have a radius of $15$. Since $B$ and $C$ lie on $AD$, and each segment is a diameter, $AB = BC = CD = 30$. Therefore, $AD = AB + BC + CD = 30 + 30 + 30 = 90$.
2. **Position of $N$... |
The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is: | 0 | 1. **Identify the common difference**:
Given the arithmetic progression (AP) terms are $x - 1$, $x + 1$, and $2x + 3$. In an AP, the difference between consecutive terms is constant. Therefore, the common difference $d$ can be calculated as:
\[
d = (x + 1) - (x - 1)
\]
Simplifying this, we get:
\[
... |
An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, $B$, and $C$ are $12$, $9$, and $10$ meter... | 17 | 1. **Establishing the Coordinate System and Points**:
Let's assume the side length of the hexagon is $6$ meters for simplicity. We place the hexagon in a 3D coordinate system with $A$ at the origin, i.e., $A = (0, 0, 0)$. The coordinates of $B$ and $C$ can be calculated based on the geometry of a regular hexagon:
... |
The points $(2,-3)$, $(4,3)$, and $(5, k/2)$ are on the same straight line. The value(s) of $k$ is (are): | 12 | To determine the value(s) of $k$ such that the points $(2,-3)$, $(4,3)$, and $(5, k/2)$ lie on the same straight line, we need to ensure that the slope between any two pairs of these points is the same.
1. **Calculate the slope between the points $(2,-3)$ and $(4,3)$:**
The slope $m$ of a line through points $(x_1,... |
In a certain year the price of gasoline rose by $20\%$ during January, fell by $20\%$ during February, rose by $25\%$ during March, and fell by $x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x$ | 17 | 1. **Assume the initial price**: Let's assume the initial price of gasoline at the beginning of January is $P_0 = 100$ dollars.
2. **Price after January's increase**: The price increased by 20% in January. Therefore, the new price at the end of January is:
\[
P_1 = P_0 + 0.20 \times P_0 = 100 + 20 = 120 \text{ d... |
Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC=20,$ and $CD=30.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$ | 360 | 1. **Assign Variables and Use Pythagorean Theorem in $\triangle ABC$:**
Let $AB = x$ and $BC = y$. Since $\angle ABC = 90^\circ$, by the Pythagorean theorem, we have:
\[
x^2 + y^2 = AC^2 = 20^2 = 400.
\]
2. **Calculate Area of $\triangle ACD$:**
Since $\angle ACD = 90^\circ$, the area of $\triangle ACD$... |
A set $S$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y=x$. If $(2,3)$ is in $S$, what is the smallest number of points in $S$? | 8 | 1. **Identify Symmetry Requirements**: The problem states that the set $S$ is symmetric about the origin, both coordinate axes, and the line $y=x$. This implies:
- Symmetry about the origin: If $(a, b) \in S$, then $(-a, -b) \in S$.
- Symmetry about the $x$-axis: If $(a, b) \in S$, then $(a, -b) \in S$.
- Symm... |
At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values... | 60 | 1. **Define the variables:**
Let $M$ represent the temperature in Minneapolis at noon, and $L$ represent the temperature in St. Louis at noon. Given that Minneapolis is $N$ degrees warmer than St. Louis at noon, we can express this relationship as:
\[
M = L + N
\]
2. **Temperature changes by 4:00 PM:**
... |
Consider all 1000-element subsets of the set $\{1, 2, 3, \dots , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | 2016 |
To solve this problem, we need to find the arithmetic mean of the least elements of all 1000-element subsets of the set $\{1, 2, 3, \ldots, 2015\}$. We will use combinatorial arguments to derive the solution.
#### Step 1: Counting subsets with a fixed least element
Let $i$ be the least element of a 1000-element subse... |
A man on his way to dinner shortly after $6:00$ p.m. observes that the hands of his watch form an angle of $110^{\circ}$. Returning before $7:00$ p.m. he notices that again the hands of his watch form an angle of $110^{\circ}$. The number of minutes that he has been away is: | 40 | 1. **Define the angles of the hands**: Let $n$ be the number of minutes after 6:00 PM. The angle of the hour hand from the top (12:00 position) can be calculated as follows:
- In one hour, the hour hand moves $30^\circ$ (since $360^\circ / 12 = 30^\circ$).
- In $n$ minutes, the hour hand moves $\frac{30}{60}n = \... |
A 3x3x3 cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube? | 90 | 1. **Understanding the Cube Configuration**:
- A 3x3x3 cube consists of 27 smaller cubes.
- The smaller cubes on the corners have three faces visible.
- The smaller cubes on the edges have two faces visible.
- The smaller cubes in the center of each face have one face visible.
2. **Counting Visible Faces**... |
In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A+B$?
$\begin{array}{cccc} & A & B & A\\ \times & & C & D\\ \hline C & D & C & D\\ \end{array}$ | 1 | 1. **Identify the value of $A$:**
Given the multiplication problem:
\[
\begin{array}{cccc}
& A & B & A\\
\times & & C & D\\
\hline
C & D & C & D\\
\end{array}
\]
We observe that the product of $A$ and $D$ results in a number ending in $D$. This implies that $A \times D$ must be a num... |
Andrea and Lauren are $20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $1$ kilometer per minute. After $5$ minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from t... | 65 | 1. **Define Variables:**
Let $v_A$ be Andrea's speed and $v_L$ be Lauren's speed, both in kilometers per hour.
2. **Set Up Equations:**
Given that Andrea travels at three times the speed of Lauren, we have:
\[ v_A = 3v_L \]
Also, the rate at which the distance between them decreases is $1$ kilometer per mi... |
The sum of two natural numbers is $17402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers? | 14238 | 1. **Identify the relationship between the two numbers**: Given that one number is divisible by $10$ and removing its units digit (which is $0$) gives the other number, we can denote the smaller number as $a$ and the larger number as $10a$.
2. **Set up the equation for their sum**: The sum of the two numbers is given ... |
What is the value of $2^{0^{1^9}} + (2^0)^{1^9}$? | 2 | To solve the expression \[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9,\] we need to evaluate each term separately.
1. **Evaluate the first term: $2^{\left(0^{\left(1^9\right)}\right)}$**
- First, simplify the exponent's base: $1^9 = 1$.
- Then, calculate $0^1 = 0$.
- Finally, compu... |
If $a$ and $b$ are digits for which
$\begin{array}{ccc}& 2 & a\ \times & b & 3\ \hline & 6 & 9\ 9 & 2 & \ \hline 9 & 8 & 9\end{array}$
then $a+b =$ | 7 | 1. **Identify the multiplication setup**: The multiplication setup is given as:
\[
\begin{array}{ccc}
& 2 & a\\
\times & b & 3\\
\hline
& 6 & 9\\
9 & 2\\
\hline
9 & 8 & 9
\end{array}
\]
This represents the multiplication of a two-digit number $2a$ by another two-digit number $b... |
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might be... | 37 | 1. **Understanding the Sequence Property**: The sequence is such that the tens and units digits of each term become the hundreds and tens digits of the next term. This implies that each digit in a position (hundreds, tens, units) of one term will appear in the next position in the next term and eventually cycle back to... |
Chandler wants to buy a $500$ dollar mountain bike. For his birthday, his grandparents send him $50$ dollars, his aunt sends him $35$ dollars and his cousin gives him $15$ dollars. He earns $16$ dollars per week for his paper route. He will use all of his birthday money and all of the money he earns from his paper rout... | 25 | 1. **Identify the total amount of money Chandler needs:** Chandler wants to buy a mountain bike that costs $500.
2. **Calculate the total amount of birthday money Chandler received:**
- From his grandparents: $50$
- From his aunt: $35$
- From his cousin: $15$
- Total birthday money: $50 + 35 + 15 = 100$
3... |
Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, what number of words would be an appropriate length for her speech? | 5650 | 1. **Convert Time to Minutes**:
- One-half hour is $\frac{1}{2} \times 60 = 30$ minutes.
- Three-quarters of an hour is $\frac{3}{4} \times 60 = 45$ minutes.
2. **Calculate the Range of Words**:
- At the ideal rate of 150 words per minute, the minimum number of words Julie should speak is:
\[
30 \t... |
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \cdot AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \cdot BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3 \cdot CA$. What is the ratio of the... | 16 | 1. **Define the side length of $\triangle ABC$:** Let $AB = BC = CA = x$.
2. **Extend the sides of $\triangle ABC$:**
- Extend $AB$ to $B'$ such that $BB' = 3AB$. Thus, $AB' = AB + BB' = x + 3x = 4x$.
- Extend $BC$ to $C'$ such that $CC' = 3BC$. Thus, $BC' = BC + CC' = x + 3x = 4x$.
- Extend $CA$ to $A'$ suc... |
If the following instructions are carried out by a computer, what value of \(X\) will be printed because of instruction \(5\)?
1. START \(X\) AT \(3\) AND \(S\) AT \(0\).
2. INCREASE THE VALUE OF \(X\) BY \(2\).
3. INCREASE THE VALUE OF \(S\) BY THE VALUE OF \(X\).
4. IF \(S\) IS AT LEAST \(10000\),
TH... | 23 | 1. **Initialization**: The program starts with $X = 3$ and $S = 0$.
2. **Loop Execution**:
- **Instruction 2**: Increase the value of $X$ by $2$.
- **Instruction 3**: Increase the value of $S$ by the current value of $X$.
- **Instruction 4**: Check if $S \geq 10000$. If true, go to instruction 5; otherwise, ... |
Letters $A, B, C,$ and $D$ represent four different digits selected from $0, 1, 2, \ldots ,9.$ If $(A+B)/(C+D)$ is an integer that is as large as possible, what is the value of $A+B$? | 17 | 1. **Objective**: We need to maximize the fraction $\frac{A+B}{C+D}$ where $A, B, C, D$ are distinct digits from $0$ to $9$. This involves maximizing $A+B$ and minimizing $C+D$.
2. **Maximizing $A+B$**:
- The maximum value for any digit is $9$. Thus, to maximize $A+B$, we should choose the largest available distin... |
A shopper plans to purchase an item that has a listed price greater than $\$100$ and can use any one of the three coupons. Coupon A gives $15\%$ off the listed price, Coupon B gives $\$30$ off the listed price, and Coupon C gives $25\%$ off the amount by which the listed price exceeds
$\$100$.
Let $x$ and $y$ be the s... | 50 | Let the listed price be $P$, where $P > 100$. We can express $P$ as $P = 100 + p$ where $p > 0$.
1. **Calculate the savings from each coupon:**
- **Coupon A:** This coupon gives $15\%$ off the listed price. Therefore, the savings from Coupon A are:
\[
0.15P = 0.15(100 + p) = 15 + 0.15p
\]
- **Coup... |
Mary's top book shelf holds five books with the following widths, in centimeters: $6$, $\dfrac{1}{2}$, $1$, $2.5$, and $10$. What is the average book width, in centimeters? | 4 | 1. **List the widths of the books**: The widths of the books are given as $6$, $\dfrac{1}{2}$, $1$, $2.5$, and $10$ centimeters.
2. **Calculate the total sum of the widths**:
\[
6 + \dfrac{1}{2} + 1 + 2.5 + 10 = 6 + 0.5 + 1 + 2.5 + 10
\]
\[
= 6 + 0.5 + 1 + 2.5 + 10 = 20
\]
3. **Count the number of b... |
A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly $10$ ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person plannin... | 10 | 1. **Determine the number of students on the council**:
Let the number of students on the council be $x$. The number of ways to select a two-person welcoming committee is given by the combination formula $\binom{x}{2}$, which represents the number of ways to choose 2 students from $x$ without regard to order. This ... |
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ? | 4 | 1. **Define Variables:**
Let $p$ be Pete's current age and $c$ be Claire's current age.
2. **Translate the Problem into Equations:**
- From the statement "Two years ago Pete was three times as old as his cousin Claire," we have:
\[
p - 2 = 3(c - 2)
\]
- From the statement "Two years before that... |
Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$-axis at the same point. What is the sum of all possible $x$-coordinates of these points of intersection? | -8 | 1. **Identify the points of intersection with the x-axis**:
The equations of the lines are given as $y = ax + 5$ and $y = 3x + b$. The points where these lines intersect the x-axis occur when $y = 0$.
2. **Set up equations for x-intercepts**:
For the line $y = ax + 5$, setting $y = 0$ gives:
\[
0 = ax + 5... |
Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ... | 192 | 1. **Understanding the Problem**: We are given four circles with centers at $A$, $B$, $C$, and $D$. Points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and similarly for circles $C$ and $D$. The distances $AB$ and $CD$ are both 39, and the length of segm... |
On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell? | 49 | 1. **Identify the given ratio**: The problem states that for every 4 sports cars sold, 7 sedans are sold. This sets up a ratio of sports cars to sedans as $4:7$.
2. **Set up the proportion**: Given that the dealership predicts selling 28 sports cars next month, we need to find the number of sedans expected to be sold,... |
Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac34$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\frac{... | 17 | 1. **Define Variables:**
Let the number of faces on the two dice be $a$ and $b$, respectively, with $a \geq b$. Assume each die has distinct integers from $1$ to the number of faces on that die.
2. **Analyze the Probability of Rolling a Sum of $7$:**
Since each die has at least $6$ faces, there are always $6$ wa... |
Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$. It rolls over the smaller tube and continues rolling along the flat surface until it comes to re... | 312 | 1. **Understanding the Problem:**
Two cylindrical tubes, one with radius 72 and the other with radius 24, are lying parallel on a flat surface. The larger tube rolls over the smaller tube and completes one full revolution, returning to the same point on its circumference, without slipping.
2. **Initial Setup:**
... |
The number $2.5252525\ldots$ can be written as a fraction.
When reduced to lowest terms the sum of the numerator and denominator of this fraction is: | 349 |
#### Step 1: Represent the repeating decimal as a fraction
Let $x = 2.5252525\ldots$. We can express $x$ in terms of itself shifted by two decimal places:
\[ 100x = 252.5252525\ldots \]
#### Step 2: Set up the equation
Subtract the original $x$ from $100x$:
\[ 100x - x = 252.5252525\ldots - 2.5252525\ldots \]
\[ 99x ... |
How many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive? | 58 | 1. **Calculate the lower bound**: We start by calculating $2^8 + 1$:
\[
2^8 = 256 \quad \text{so} \quad 2^8 + 1 = 257.
\]
We need to find the smallest perfect cube greater than 257. We know $6^3 = 216$ and $7^3 = 343$. Since $343 > 257$, the smallest cube greater than 257 is $7^3 = 343$.
2. **Calculate the... |
Alicia earns 20 dollars per hour, of which $1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes? | 29 | 1. **Convert Alicia's hourly wage to cents**: Alicia earns $20$ dollars per hour. Since there are $100$ cents in a dollar, her hourly wage in cents is:
\[
20 \text{ dollars} \times 100 \text{ cents/dollar} = 2000 \text{ cents}
\]
2. **Calculate the tax deduction in cents**: The local tax rate is $1.45\%$. To ... |
How many positive factors of 36 are also multiples of 4? | 3 | To solve this problem, we first need to find the prime factorization of 36. Then, we will determine which factors of 36 are also multiples of 4.
1. **Prime Factorization of 36**:
\[
36 = 2^2 \cdot 3^2
\]
This factorization shows that 36 is composed of the square of 2 and the square of 3.
2. **Factors of 3... |
The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$? | 90 | 1. **Understanding the Circle and Arcs**: The circle is divided into 12 equal arcs, and each arc corresponds to a central angle at the center $O$. Since the circle's total degrees is $360^\circ$, each central angle measures:
\[
\frac{360^\circ}{12} = 30^\circ
\]
2. **Central Angles for $x$ and $y$**:
- If... |
Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number? | 12 | 1. **Define the problem in terms of algebra**: Let the two-digit number Mary thought of be $x$. According to the problem, she performs the following operations on $x$:
- Multiplies by $3$: $3x$
- Adds $11$: $3x + 11$
- Switches the digits of the result.
2. **Analyze the range of the final number**: The switch... |
Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}, BC=CD=43$, and $\overline{AD} \perp \overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$... | 194 | 1. **Identify the properties of the trapezoid**: Given that $ABCD$ is a trapezoid with $\overline{AB}\parallel\overline{CD}$ and $BC=CD=43$. Also, $\overline{AD}\perp\overline{BD}$, which implies that $\triangle ABD$ is a right triangle.
2. **Examine the diagonals and intersection**: The diagonals $\overline{AC}$ and ... |
Carrie has a rectangular garden that measures $6$ feet by $8$ feet. She plants the entire garden with strawberry plants. Carrie is able to plant $4$ strawberry plants per square foot, and she harvests an average of $10$ strawberries per plant. How many strawberries can she expect to harvest? | 1920 | 1. **Calculate the area of the garden**:
The garden is a rectangle with dimensions $6$ feet by $8$ feet. The area \( A \) of a rectangle is given by the formula:
\[
A = \text{length} \times \text{width}
\]
Substituting the given dimensions:
\[
A = 6 \text{ ft} \times 8 \text{ ft} = 48 \text{ ft}^... |
The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number. What is the smallest possible value of $s$? | 4 | To find the smallest possible value of $s$ such that the sides $6.5$, $10$, and $s$ can form a triangle, we must apply the triangle inequality theorem. The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. We need to check th... |
Quadrilateral $ABCD$ has $AB = BC = CD$, $m\angle ABC = 70^\circ$ and $m\angle BCD = 170^\circ$. What is the degree measure of $\angle BAD$? | 85 | 1. **Assign the unknown and draw necessary diagonals**: Let $\angle BAD = x$. Draw diagonals $BD$ and $AC$. Let $I$ be the intersection of diagonals $BD$ and $AC$.
2. **Analyze the isosceles triangles**: Since $AB = BC = CD$, triangles $\triangle ABC$ and $\triangle BCD$ are isosceles. Therefore, $\angle DBC = \angle ... |
An $11 \times 11 \times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point? | 331 | To find the greatest number of unit cubes that can be seen from a single point, we consider viewing the cube from one of its corners. From this vantage point, three faces of the cube are visible: the top face, the front face, and the side face.
1. **Counting the visible unit cubes on each face:**
- Each face of the... |
The Fort Worth Zoo has a number of two-legged birds and a number of four-legged mammals. On one visit to the zoo, Margie counted 200 heads and 522 legs. How many of the animals that Margie counted were two-legged birds? | 139 | 1. **Define Variables:**
Let $x$ be the number of two-legged birds and $y$ be the number of four-legged mammals.
2. **Set Up Equations:**
- Each animal has one head, so the total number of heads gives us the equation:
\[ x + y = 200 \]
- Each bird has 2 legs and each mammal has 4 legs, so the total numbe... |
A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies hal... | 2 | 1. **Define the areas**: Let $A_{\text{outer}}$ be the area of the entire floor and $A_{\text{inner}}$ be the area of the painted rectangle. Given that the unpainted border is 1 foot wide, the dimensions of the painted rectangle are $(a-2)$ and $(b-2)$.
Therefore, we have:
\[
A_{\text{outer}} = ab
\]
... |
Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had $56$ dollars. The absolute difference between the amounts Ashley and Betty had to spend was $19$ dollars. The absolute difference between the amounts Betty and Carlos had was $7$ dollars, between Car... | 10 | Let's denote the amount of money each person has as follows:
- Ashley: $A$
- Betty: $B$
- Carlos: $C$
- Dick: $D$
- Elgin: $E$
From the problem, we have the following equations based on the absolute differences:
1. $|A - B| = 19$
2. $|B - C| = 7$
3. $|C - D| = 5$
4. $|D - E| = 4$
5. $|E - A| = 11$
Additionally, we kn... |
What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is not a divisor of the product of the first $n$ positive integers? | 996 |
1. **Understanding the Problem:**
We need to find the largest three-digit integer $n$ such that the sum of the first $n$ positive integers is not a divisor of the product of the first $n$ positive integers.
2. **Sum and Product Formulas:**
- The sum of the first $n$ positive integers is given by the formula:
... |
Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle? | 143 | 1. **Understanding the problem**: Claire calculated the sum of the interior angles of a convex polygon as $2017^\circ$. However, she missed one angle. We need to find the measure of this forgotten angle.
2. **Using the formula for the sum of interior angles**: The sum of the interior angles of an $n$-sided polygon is ... |
Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this? | 12 | 1. **Identify the total number of unrestricted arrangements:**
Let's denote the marbles as $A$ (Aggie), $B$ (Bumblebee), $S$ (Steelie), and $T$ (Tiger). Without any restrictions, the total number of ways to arrange these four marbles is given by the factorial of the number of marbles, which is $4!$.
\[
4! = 4... |
Triangle $ABC$ has vertices $A = (3,0)$, $B = (0,3)$, and $C$, where $C$ is on the line $x + y = 7$. What is the area of $\triangle ABC$? | 6 |
#### Step 1: Understanding the problem
We are given a triangle $ABC$ with vertices $A = (3,0)$, $B = (0,3)$, and $C$ on the line $x + y = 7$. We need to find the area of $\triangle ABC$.
#### Step 2: Analyzing the line $x + y = 7$
The line $x + y = 7$ is parallel to the line connecting $A$ and $B$, because the slope ... |
Everyday at school, Jo climbs a flight of $6$ stairs. Jo can take the stairs $1$, $2$, or $3$ at a time. For example, Jo could climb $3$, then $1$, then $2$. In how many ways can Jo climb the stairs? | 24 | 1. **Define the recursive function**: Let $f(n)$ represent the number of ways Jo can climb to the $n$-th step. We start by defining the base cases:
- $f(0) = 1$: There is only one way to be on the ground (by starting there).
- $f(1) = 1$: There is only one way to reach the first step, which is by taking a single ... |
A team won $40$ of its first $50$ games. How many of the remaining $40$ games must this team win so it will have won exactly $70 \%$ of its games for the season? | 23 | 1. **Identify the total number of games and the desired win percentage**: The team plays a total of $50 + 40 = 90$ games in the season and aims to win 70% of these games.
2. **Convert the percentage to a fraction**: The fraction corresponding to 70% is $\frac{70}{100} = \frac{7}{10}$.
3. **Set up the equation for th... |
In this diagram, not drawn to scale, Figures $I$ and $III$ are equilateral triangular regions with respective areas of $32\sqrt{3}$ and $8\sqrt{3}$ square inches. Figure $II$ is a square region with area $32$ square inches. Let the length of segment $AD$ be decreased by $12\frac{1}{2}$% of itself, while the lengths of ... | 25 | To solve this problem, we first need to determine the side lengths of the equilateral triangles and the square based on the given areas.
1. **Calculate the side length of the equilateral triangles:**
- For an equilateral triangle, the area $A$ is given by $A = \frac{\sqrt{3}}{4} s^2$, where $s$ is the side length.
... |
Positive integers $a$ and $b$ are each less than $6$. What is the smallest possible value for $2 \cdot a - a \cdot b$? | -15 | 1. **Expression Simplification**:
Start by simplifying the given expression:
\[
2a - ab = a(2 - b)
\]
This shows that the expression depends on the values of $a$ and $b$ through the product $a(2 - b)$.
2. **Determine the Range of Values**:
Given that $a$ and $b$ are positive integers each less than $... |
In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR = 6$ and $PR = 7$. What is the area of the square? | 117 | 1. **Identify Similar Triangles**: Notice that $\triangle CRB \sim \triangle BAP$ by AA similarity (both have a right angle and share angle $BRP$).
2. **Set Up Ratio of Sides**: From the similarity, we have the ratio of corresponding sides:
\[
\frac{CB}{CR} = \frac{PB}{AB}
\]
Since $CB = AB = s$ (side leng... |
The equations of $L_1$ and $L_2$ are $y=mx$ and $y=nx$, respectively. Suppose $L_1$ makes twice as large of an angle with the horizontal (measured counterclockwise from the positive x-axis ) as does $L_2$, and that $L_1$ has 4 times the slope of $L_2$. If $L_1$ is not horizontal, then $mn$ is | 2 | 1. **Given Information and Equations:**
- The equations of lines $L_1$ and $L_2$ are $y = mx$ and $y = nx$ respectively.
- $L_1$ makes twice as large of an angle with the horizontal as does $L_2$.
- $L_1$ has 4 times the slope of $L_2$.
- $L_1$ is not horizontal.
2. **Relating Slopes to Angles:**
- Let ... |
An amusement park has a collection of scale models, with a ratio of $1: 20$, of buildings and other sights from around the country. The height of the United States Capitol is $289$ feet. What is the height in feet of its duplicate to the nearest whole number? | 14 | 1. **Identify the scale ratio and the actual height of the building**: The scale model is built at a ratio of $1:20$, and the actual height of the United States Capitol is $289$ feet.
2. **Calculate the height of the scale model**: To find the height of the scale model, we divide the actual height by the scale ratio:
... |
Given $0 \le x_0 < 1$, let
\[x_n = \begin{cases} 2x_{n-1} & \text{ if } 2x_{n-1} < 1 \\ 2x_{n-1} - 1 & \text{ if } 2x_{n-1} \ge 1 \end{cases}\]for all integers $n > 0$. For how many $x_0$ is it true that $x_0 = x_5$? | 31 | 1. **Understanding the Sequence**: The sequence defined by $x_n$ is a binary sequence where each term is generated by doubling the previous term and subtracting 1 if the result is at least 1. This can be interpreted as a shift and truncate operation in binary representation.
2. **Binary Representation**: Let's represe... |
In $\triangle PQR$, $PR=15$, $QR=20$, and $PQ=25$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, and points $E$ and $F$ lie on $\overline{PR}$, with $PA=QB=QC=RD=RE=PF=5$. Find the area of hexagon $ABCDEF$. | 150 | 1. **Identify the type of triangle $PQR$**:
Given $PR=15$, $QR=20$, and $PQ=25$, we check if $\triangle PQR$ is a right triangle using the Pythagorean theorem:
\[
PR^2 + QR^2 = 15^2 + 20^2 = 225 + 400 = 625 = 25^2 = PQ^2.
\]
Since $PR^2 + QR^2 = PQ^2$, $\triangle PQR$ is a right triangle with $PQ$ as the... |
What is the tens digit of $7^{2011}$? | 4 | 1. **Identify the goal:** We need to find the tens digit of $7^{2011}$. This is equivalent to finding $7^{2011} \mod 100$ because the tens digit of a number is the second last digit from the right.
2. **Use modular arithmetic to simplify the problem:** We start by calculating the powers of $7$ modulo $100$ to identify... |
For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? | 19 | To solve this problem, we need to consider the possible number of intersection points formed by four distinct lines in a plane. Each pair of lines can intersect at most once, and the maximum number of intersection points is determined by the number of ways to choose 2 lines from 4, which is given by the binomial coeffi... |
Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for $a$, $b$, $c$, and $d$ wer... | 3 | 1. **Substitute the given values into the expression ignoring parentheses:**
Larry ignored the parentheses, so he calculated the expression as:
\[
a - b - c - d + e = 1 - 2 - 3 - 4 + e
\]
Simplifying this, we get:
\[
-8 + e
\]
2. **Substitute the given values into the expression with correct ... |
The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$. | 289 | 1. **Express the given equation in terms of $a$ and $b$:**
Let $z = a + bi$. Then, the given equation $z + |z| = 2 + 8i$ can be rewritten as:
\[
a + bi + \sqrt{a^2 + b^2} = 2 + 8i
\]
Here, $a$ and $b$ are real numbers, and $\sqrt{a^2 + b^2}$ is the modulus of $z$, which is a real number.
2. **Separate... |
There are $120$ seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone? | 40 | To solve this problem, we need to find the minimum number of people that must be seated such that any additional person must sit next to someone already seated. We aim to maximize the number of empty seats between seated people under this constraint.
1. **Understanding the Pattern**:
- If we place a person in a se... |
Jose is $4$ years younger than Zack. Zack is $3$ years older than Inez. Inez is $15$ years old. How old is Jose? | 14 | 1. **Determine Inez's Age:**
Given that Inez is $15$ years old.
2. **Calculate Zack's Age:**
Zack is $3$ years older than Inez. Therefore, we calculate Zack's age as follows:
\[
\text{Zack's age} = \text{Inez's age} + 3 = 15 + 3 = 18 \text{ years}
\]
3. **Calculate Jose's Age:**
Jose is $4$ years yo... |
The sum of all the roots of $4x^3-8x^2-63x-9=0$ is: | 2 | 1. **Identify the polynomial and its degree**:
Given the polynomial equation:
\[
4x^3 - 8x^2 - 63x - 9 = 0
\]
This is a cubic polynomial of degree 3.
2. **Simplify the polynomial**:
Factor out the common coefficient of the highest degree term:
\[
4(x^3 - 2x^2 - \frac{63}{4}x - \frac{9}{4}) = 0
... |
Nonzero real numbers $x$, $y$, $a$, and $b$ satisfy $x < a$ and $y < b$. How many of the following inequalities must be true?
$\textbf{(I)}\ x+y < a+b$
$\textbf{(II)}\ x-y < a-b$
$\textbf{(III)}\ xy < ab$
$\textbf{(IV)}\ \frac{x}{y} < \frac{a}{b}$ | 1 | We are given that $x < a$ and $y < b$ for nonzero real numbers $x$, $y$, $a$, and $b$. We need to determine how many of the given inequalities must be true under these conditions.
1. **Analyzing $\textbf{(I)}\ x+y < a+b$:**
- Since $x < a$ and $y < b$, by adding these inequalities, we get:
\[
x + y < a + ... |
What is the value of the expression $\sqrt{16\sqrt{8\sqrt{4}}}$? | 8 | To solve the expression $\sqrt{16\sqrt{8\sqrt{4}}}$, we will simplify the expression step-by-step:
1. **Simplify the innermost square root:**
\[
\sqrt{4} = 2
\]
2. **Substitute and simplify the next square root:**
\[
\sqrt{8\sqrt{4}} = \sqrt{8 \cdot 2} = \sqrt{16}
\]
3. **Simplify the result obtain... |
Penniless Pete's piggy bank has no pennies in it, but it has 100 coins, all nickels, dimes, and quarters, whose total value is $8.35. It does not necessarily contain coins of all three types. What is the difference between the largest and smallest number of dimes that could be in the bank? | 64 | Let $a$, $b$, and $c$ represent the number of nickels, dimes, and quarters, respectively. We are given two conditions:
1. The total number of coins is 100.
2. The total value of the coins is $8.35, which is equivalent to 835 cents.
From these conditions, we can set up the following equations:
\[
\begin{align*}
a + b +... |
Six distinct positive integers are randomly chosen between $1$ and $2006$, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of $5$? | 1 | 1. **Understanding the Problem:**
We need to find the probability that among six distinct positive integers chosen from the set $\{1, 2, \ldots, 2006\}$, there exists at least one pair whose difference is a multiple of $5$.
2. **Using Modular Arithmetic:**
For two numbers to have a difference that is a multiple ... |
The ratio of the radii of two concentric circles is $1:3$. If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB=12$, then the radius of the larger circle is | 18 | 1. **Identify the Geometry and Setup the Problem:**
- We have two concentric circles with radii ratio $1:3$. Let the radius of the smaller circle be $r$ and the radius of the larger circle be $3r$.
- $\overline{AC}$ is a diameter of the larger circle, so $AC = 2 \times 3r = 6r$.
- $\overline{BC}$ is a chord of... |
What is the hundreds digit of $(20! - 15!)?$ | 0 | To find the hundreds digit of $(20! - 15!)$, we need to analyze the factorials and their properties modulo $1000$.
1. **Factorial Properties**:
- $n!$ (where $n \geq 5$) contains at least one factor of $5$ and at least one factor of $2$, making it divisible by $10$.
- $n!$ (where $n \geq 10$) contains at least t... |
How many positive integer factors of $2020$ have more than $3$ factors? | 7 | To solve this problem, we first need to find the prime factorization of $2020$. We have:
\[ 2020 = 2^2 \times 5 \times 101. \]
Next, we use the formula for the number of divisors of a number given its prime factorization. If $n = p^a \times q^b \times r^c \times \ldots$, then the number of divisors of $n$, denoted as ... |
Before the district play, the Unicorns had won $45$% of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all? | 48 | 1. Let $y$ be the total number of games the Unicorns played before the district play, and $x$ be the number of games they won. According to the problem, they won 45% of these games, so we have the equation:
\[
\frac{x}{y} = 0.45
\]
Multiplying both sides by $y$ gives:
\[
x = 0.45y
\]
2. During the... |
Oscar buys $13$ pencils and $3$ erasers for $1.00$. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser? | 10 | 1. **Define Variables:**
Let $p$ represent the cost of one pencil and $e$ represent the cost of one eraser in cents. We are given the equation:
\[
13p + 3e = 100
\]
where $p > e > 0$ and both $p$ and $e$ are whole numbers.
2. **Constraints on $p$ and $e$:**
Since $p$ and $e$ are positive integers, we... |
What is the greatest number of consecutive integers whose sum is $45?$ | 90 | To find the greatest number of consecutive integers whose sum is $45$, we need to consider sequences of integers, both positive and negative.
1. **Understanding the sum of consecutive integers**:
The sum of $N$ consecutive integers starting from $a$ can be expressed as:
\[
a + (a+1) + (a+2) + \cdots + (a+N-1)... |
What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$? | 22 | To find the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$, we first need to determine the maximum number of digits that a base-seven number can have without exceeding $2019$ in decimal.
1. **Convert $2019$ to base-seven:**
- The largest power of $7$ less... |
The measures of the interior angles of a convex polygon of $n$ sides are in arithmetic progression. If the common difference is $5^{\circ}$ and the largest angle is $160^{\circ}$, then $n$ equals: | 16 | 1. **Identify the formula for the sum of interior angles of a polygon**: The sum of the interior angles of a polygon with $n$ sides is given by the formula:
\[
S = 180^\circ (n-2)
\]
2. **Set up the arithmetic sequence**: Given that the angles are in arithmetic progression with a common difference of $5^\circ... |
What is the value of $\frac{11! - 10!}{9!}$? | 100 | 1. Start by rewriting the expression $\dfrac{11!-10!}{9!}$ using the definition of factorial:
\[
\frac{11! - 10!}{9!} = \frac{11 \times 10! - 10!}{9!}
\]
2. Factor out $10!$ from the numerator:
\[
\frac{11 \times 10! - 10!}{9!} = \frac{10!(11 - 1)}{9!}
\]
3. Simplify the expression inside the parent... |
Let $A, M$, and $C$ be digits with $(100A+10M+C)(A+M+C) = 2005$. What is $A$? | 4 | 1. **Identify the Equation**: We are given the equation \[(100A+10M+C)(A+M+C) = 2005.\] Here, $A$, $M$, and $C$ are digits, meaning each of them can be any integer from $0$ to $9$.
2. **Prime Factorization of 2005**: To find possible values for $100A+10M+C$ and $A+M+C$, we first factorize 2005. The prime factorization... |
If $f(a)=a-2$ and $F(a,b)=b^2+a$, then $F(3,f(4))$ is: | 7 | 1. **Calculate $f(4)$**: Given the function $f(a) = a - 2$, substitute $a = 4$:
\[
f(4) = 4 - 2 = 2
\]
2. **Evaluate $F(3, f(4))$**: With $f(4) = 2$, we need to find $F(3, 2)$. The function $F(a, b) = b^2 + a$ is given, so substitute $a = 3$ and $b = 2$:
\[
F(3, 2) = 2^2 + 3 = 4 + 3 = 7
\]
3. **Conc... |
The number of digits in $4^{16}5^{25}$ (when written in the usual base $10$ form) is | 28 | 1. **Rewrite the expression using properties of exponents:**
The given expression is $4^{16}5^{25}$. We can express $4$ as $2^2$, so:
\[
4^{16} = (2^2)^{16} = 2^{32}
\]
Therefore, the expression becomes:
\[
4^{16}5^{25} = 2^{32}5^{25}
\]
2. **Combine powers of 2 and 5 to form powers of 10:** ... |
How many integers $n \geq 2$ are there such that whenever $z_1, z_2, \dots, z_n$ are complex numbers such that
\[|z_1| = |z_2| = \dots = |z_n| = 1 \text{ and } z_1 + z_2 + \dots + z_n = 0,\]
then the numbers $z_1, z_2, \dots, z_n$ are equally spaced on the unit circle in the complex plane? | 2 | 1. **Case $n=2$:**
Consider $z_1$ and $z_2$ on the unit circle such that $|z_1| = |z_2| = 1$ and $z_1 + z_2 = 0$. This implies $z_2 = -z_1$. Since both $z_1$ and $-z_1$ lie on the unit circle and are diametrically opposite, they are equally spaced on the unit circle.
2. **Case $n=3$:**
Without loss of genera... |
The set $\{3,6,9,10\}$ is augmented by a fifth element $n$, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of $n$? | 26 | We are given the set $\{3,6,9,10\}$ and an additional element $n$, which is distinct from the other elements. We need to find the sum of all possible values of $n$ such that the median and the mean of the augmented set are equal.
#### Case 1: Median is $6$
For $6$ to be the median, $n$ must be less than or equal to $6... |
Odell and Kershaw run for $30$ minutes on a circular track. Odell runs clockwise at $250 m/min$ and uses the inner lane with a radius of $50$ meters. Kershaw runs counterclockwise at $300 m/min$ and uses the outer lane with a radius of $60$ meters, starting on the same radial line as Odell. How many times after the sta... | 47 | 1. **Determine the Circumference of Each Track**:
- Odell's track radius = $50$ meters, so the circumference is $C_O = 2\pi \times 50 = 100\pi$ meters.
- Kershaw's track radius = $60$ meters, so the circumference is $C_K = 2\pi \times 60 = 120\pi$ meters.
2. **Calculate the Speed in Terms of Radians per Minute**... |
The 64 whole numbers from 1 through 64 are written, one per square, on a checkerboard (an 8 by 8 array of 64 squares). The first 8 numbers are written in order across the first row, the next 8 across the second row, and so on. After all 64 numbers are written, the sum of the numbers in the four corners will be | 130 | 1. **Identify the positions of the corner numbers on the checkerboard:**
- The top left corner is the first square, so the number is $1$.
- The top right corner is the last square of the first row, which is the 8th square, so the number is $8$.
- The bottom right corner is the last square of the last row, whic... |
Six rectangles each with a common base width of $2$ have lengths of $1, 4, 9, 16, 25$, and $36$. What is the sum of the areas of the six rectangles? | 182 | 1. **Identify the dimensions of each rectangle**: Each rectangle has a common base width of $2$. The lengths of the rectangles are given as $1, 4, 9, 16, 25$, and $36$.
2. **Calculate the area of each rectangle**: The area of a rectangle is calculated by multiplying its length by its width. Therefore, the area of each... |
In counting $n$ colored balls, some red and some black, it was found that $49$ of the first $50$ counted were red.
Thereafter, $7$ out of every $8$ counted were red. If, in all, $90$ % or more of the balls counted were red, the maximum value of $n$ is: | 210 | 1. **Understanding the Problem:**
We are given that $49$ of the first $50$ balls are red, and thereafter, $7$ out of every $8$ balls counted are red. We need to find the maximum number of balls, $n$, such that at least $90\%$ of them are red.
2. **Setting Up the Equation:**
Let $x$ be the number of batches of $8... |
The figure may be folded along the lines shown to form a number cube. Three number faces come together at each corner of the cube. What is the largest sum of three numbers whose faces come together at a corner? | 14 | To solve this problem, we need to understand how a cube is formed and how the numbers on the faces are arranged. In a standard die, opposite faces sum up to 7. This means:
- If one face shows 1, the opposite face shows 6.
- If one face shows 2, the opposite face shows 5.
- If one face shows 3, the opposite face shows 4... |
Many calculators have a reciprocal key $\boxed{\frac{1}{x}}$ that replaces the current number displayed with its reciprocal. For example, if the display is $\boxed{00004}$ and the $\boxed{\frac{1}{x}}$ key is depressed, then the display becomes $\boxed{000.25}$. If $\boxed{00032}$ is currently displayed, what is the ... | 2 | 1. **Define the function**: Let $f(x) = \frac{1}{x}$ represent the operation of the reciprocal key on the calculator.
2. **Apply the function**: We need to determine how many times we must apply $f(x)$ to return to the original number. Start by applying $f(x)$ to the number 32:
\[
f(32) = \frac{1}{32}
\]
3. ... |
The digits 1, 2, 3, 4 and 9 are each used once to form the smallest possible even five-digit number. The digit in the tens place is | 9 | 1. **Identify the requirement for the number to be even**: The number must end in an even digit. The available even digits are 2 and 4.
2. **Determine the smallest possible even digit for the units place**: To form the smallest number, we prefer the smallest digits in the higher place values. Since the number must be ... |
In a magic triangle, each of the six whole numbers $10-15$ is placed in one of the circles so that the sum, $S$, of the three numbers on each side of the triangle is the same. The largest possible value for $S$ is
[asy] draw(circle((0,0),1)); draw(dir(60)--6*dir(60)); draw(circle(7*dir(60),1)); draw(8*dir(60)--13*dir(... | 39 | 1. **Assign Variables to Circles**: Let the numbers in the circles be $a$, $b$, $c$, $d$, $e$, and $f$ starting from the top circle and moving clockwise.
2. **Set Up Equations for Each Side of the Triangle**:
- The sum of the numbers on the first side is $S = a + b + c$.
- The sum of the numbers on the second si... |
Supposed that $x$ and $y$ are nonzero real numbers such that $\frac{3x+y}{x-3y}=-2$. What is the value of $\frac{x+3y}{3x-y}$? | 2 | 1. Start with the given equation:
\[
\frac{3x+y}{x-3y} = -2
\]
2. Cross-multiply to eliminate the fraction:
\[
3x + y = -2(x - 3y)
\]
3. Distribute the -2 on the right-hand side:
\[
3x + y = -2x + 6y
\]
4. Rearrange the equation to isolate terms involving \(x\) and \(y\) on opposite sides:... |
Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
[asy]
unitsize(6mm);
defaultpen(linewidth(.8pt));
path p=(1,1)--(-2,1)--(-2,2)--(1,2);
draw(p);
draw... | 3 | 1. **Identify the dimensions of the squares and rectangles**:
Let the side length of the inner square be $s$. Assume the shorter side of each rectangle is $y$ and the longer side is $x$. The rectangles are congruent and placed around the inner square such that their longer sides and shorter sides together form the ... |
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