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Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$, the second sheet contains pages $3$ and $4$, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the n... | 13 | 1. **Define Variables:**
Let $c$ be the number of consecutive sheets Hiram’s roommate borrows, and let $b$ be the number of sheets preceding the $c$ borrowed sheets. For example, if the roommate borrows sheets $3$, $4$, and $5$, then $c=3$ and $b=2$.
2. **Calculate the Sum of Page Numbers:**
The sum of the page ... |
On a certain math exam, $10\%$ of the students got $70$ points, $25\%$ got $80$ points, $20\%$ got $85$ points, $15\%$ got $90$ points, and the rest got $95$ points. What is the difference between the mean and the median score on this exam? | 1 | 1. **Calculate the percentage of students scoring 95 points**:
Given that $10\%$ scored 70 points, $25\%$ scored 80 points, $20\%$ scored 85 points, and $15\%$ scored 90 points, the remaining percentage of students who scored 95 points is:
\[
100\% - (10\% + 25\% + 20\% + 15\%) = 100\% - 70\% = 30\%
\]
2.... |
The sum of the numerical coefficients in the expansion of the binomial $(a+b)^6$ is: | 64 | To find the sum of the numerical coefficients in the expansion of the binomial $(a+b)^6$, we can substitute $a = 1$ and $b = 1$ into the binomial. This simplifies the expression to just the sum of the coefficients, as each term in the expansion will have the form $\binom{6}{k} a^{6-k} b^k$ and substituting $a = 1$ and ... |
What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594? | 330 | 1. **Find the prime factorizations** of 180 and 594:
- $180 = 2^2 \cdot 3^2 \cdot 5$
- $594 = 2 \cdot 3^3 \cdot 11$
2. **Calculate the greatest common factor (GCF)**:
- The GCF is found by taking the lowest power of all prime factors common to both numbers.
- Common prime factors are $2$ and $3$.
- GCF ... |
There are 24 four-digit whole numbers that use each of the four digits 2, 4, 5 and 7 exactly once. Only one of these four-digit numbers is a multiple of another one. What is it? | 7425 | To solve this problem, we need to determine which of the given four-digit numbers is a multiple of another four-digit number formed using the digits 2, 4, 5, and 7 exactly once.
1. **Identify the Range of Numbers**: The numbers formed by the digits 2, 4, 5, and 7 are between the smallest number 2457 and the largest nu... |
Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$? | 6 | 1. **Given Information and Assumptions**:
- Convex quadrilateral $ABCD$ has $AB = 9$, $CD = 12$, $AC = 14$.
- Diagonals $AC$ and $BD$ intersect at $E$.
- $\triangle AED$ and $\triangle BEC$ have equal areas.
2. **Using Equal Areas to Infer Ratios**:
- Since $\triangle AED$ and $\triangle BEC$ have equal ar... |
Isabella's house has $3$ bedrooms. Each bedroom is $12$ feet long, $10$ feet wide, and $8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $60$ square feet in each bedroom. How many square feet of walls must be painted? | 876 | 1. **Calculate the area of the walls in one bedroom**:
Each bedroom has four walls, two pairs of opposite walls. Each pair consists of:
- Two walls of dimensions $12$ feet (length) by $8$ feet (height).
- Two walls of dimensions $10$ feet (width) by $8$ feet (height).
The area of the walls for each dimens... |
Karl's car uses a gallon of gas every $35$ miles, and his gas tank holds $14$ gallons when it is full. One day, Karl started with a full tank of gas, drove $350$ miles, bought $8$ gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that da... | 525 | 1. **Calculate the gas consumption for the first leg of the trip**: Karl's car uses 1 gallon of gas every 35 miles. Therefore, for the first 350 miles, the amount of gas used is calculated by:
\[
\frac{350 \text{ miles}}{35 \text{ miles per gallon}} = 10 \text{ gallons}
\]
2. **Determine the remaining gas aft... |
Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms? | 64 | 1. **Calculate the total number of students in all classrooms**:
Each classroom has $18$ students. Since there are $4$ classrooms, the total number of students is:
\[
18 \times 4 = 72 \text{ students}
\]
2. **Calculate the total number of rabbits in all classrooms**:
Each classroom has $2$ rabbits. Sin... |
What is the area of the triangle formed by the lines $y=5$, $y=1+x$, and $y=1-x$? | 16 | 1. **Identify the intersection points of the lines**:
- The lines given are $y = 5$, $y = 1 + x$, and $y = 1 - x$.
- To find the intersection of $y = 5$ and $y = 1 + x$, set $5 = 1 + x$. Solving for $x$, we get $x = 4$. Thus, the intersection point is $(4, 5)$.
- To find the intersection of $y = 5$ and $y = 1... |
Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$, $BC = 14$, and $AD = 2\sqrt{65}$. Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$. Find the area of quadrilateral $ABCD$. | 70 | 1. **Identify the Relationship Between Angles and Sides**:
Given that $\angle APB = \angle CPD$ and $\angle APD = \angle BPC$, and using the property that $\sin(\theta) = \sin(180^\circ - \theta)$, we can apply the sine area formula for triangles. This leads to the equation:
\[
BP \cdot AP + CP \cdot DP = BP \... |
In the figure, the length of side $AB$ of square $ABCD$ is $\sqrt{50}$ and $BE=1$. What is the area of the inner square $EFGH$? | 36 | 1. **Identify the Geometry and Given Information**: We are given a square $ABCD$ with side length $\sqrt{50}$, and a smaller square $EFGH$ inside it. The segment $BE$ is given as $1$.
2. **Analyze the Right Triangle Formed**: Since $ABCD$ is a square, each angle is $90^\circ$. The segment $BE$ forms a right triangle $... |
Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of $5$ cups. What percent of the total capacity of the pitcher did each cup receive? | 15 | 1. **Assume the total capacity of the pitcher**: Let's denote the total capacity of the pitcher as $C$ ounces.
2. **Calculate the amount of juice in the pitcher**: Since the pitcher is filled to three-fourths of its capacity with pineapple juice, the amount of juice in the pitcher is $\frac{3}{4}C$ ounces.
3. **Deter... |
What is the difference between the sum of the first $2003$ even counting numbers and the sum of the first $2003$ odd counting numbers? | 2003 | To find the difference between the sum of the first $2003$ even counting numbers and the sum of the first $2003$ odd counting numbers, we first need to calculate each sum separately.
1. **Sum of the first $2003$ odd counting numbers:**
The sequence of the first $2003$ odd numbers is $1, 3, 5, \ldots, 4005$. This is... |
The expression $\frac{x^2-3x+2}{x^2-5x+6} \div \frac{x^2-5x+4}{x^2-7x+12}$, when simplified is: | 1 | 1. **Factorize each quadratic expression** in the given complex fraction:
- Numerator of the first fraction: $x^2 - 3x + 2 = (x-2)(x-1)$.
- Denominator of the first fraction: $x^2 - 5x + 6 = (x-3)(x-2)$.
- Numerator of the second fraction: $x^2 - 5x + 4 = (x-4)(x-1)$.
- Denominator of the second fraction: $... |
A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars? | 100 | 1. **Calculate the cost price of the candy bars:**
The troop buys candy bars at a rate of 5 bars for $2 dollars. Therefore, the cost per bar is:
\[
\frac{2 \text{ dollars}}{5 \text{ bars}} = 0.4 \text{ dollars per bar}
\]
Since they buy 1000 bars, the total cost is:
\[
1000 \text{ bars} \times 0.4 ... |
For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?
a square
a rectangle that is not a square
a rhombus that is not a square
a parallelogram that is not a rectangle or a rhombus
an isosceles tr... | 3 | To solve this problem, we need to determine which of the given types of quadrilaterals have a point that is equidistant from all four vertices. This point is the circumcenter of the quadrilateral, and the quadrilateral must be cyclic (i.e., it can be inscribed in a circle) for such a point to exist.
1. **Square**:
... |
Assuming $a \neq 3$, $b \neq 4$, and $c \neq 5$, what is the value in simplest form of the following expression?
\[\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}\] | -1 | 1. **Identify the pattern in the expression**: We are given the expression:
\[
\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}
\]
We notice that each fraction has a numerator and a denominator that are differences of variables and constants.
2. **Apply the property of differences**: Recall the ... |
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$ | 50 | 1. **Identify the region and lattice points**:
The region is bounded by the lines $y = \pi x$, $y = -0.1$, and $x = 5.1$. We consider only the lattice points (points with integer coordinates) that lie within this region. The relevant lattice points along the x-axis from $x = 0$ to $x = 5$ are:
- $(0,0)$
- $(1,... |
Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,2,3,....$ More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\frac p... | 55 | To solve this problem, we need to calculate the probability that three balls tossed into bins end up evenly spaced in distinct bins. We start by considering the possible configurations for the balls and their corresponding probabilities.
#### Step 1: Define the problem
We are given that the probability that a ball is ... |
A store owner bought $1500$ pencils at $\$ 0.10$ each. If he sells them for $\$ 0.25$ each, how many of them must he sell to make a profit of exactly $\$ 100.00$? | 1000 | 1. **Calculate the total cost of the pencils**: The store owner bought $1500$ pencils at $0.10$ each. Therefore, the total cost is:
\[
1500 \times 0.10 = 150
\]
So, the total cost is $150$ dollars.
2. **Determine the profit required**: The store owner wants to make a profit of $100$ dollars.
3. **Calculat... |
Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ What is the area of $ABCD?$ | 240 | 1. **Identify the Geometry and Given Values**: We are given a rectangle $ABCD$ inscribed in a semicircle with diameter $\overline{FE}$. The lengths $DA = 16$ and $FD = AE = 9$ are provided.
2. **Complete the Circle**: To utilize symmetry and the properties of a circle, consider the full circle with diameter $\overline... |
Figures $I$, $II$, and $III$ are squares. The perimeter of $I$ is $12$ and the perimeter of $II$ is $24$. The perimeter of $III$ is | 36 | 1. **Identify the side lengths of squares $I$ and $II$:**
- Let the side length of square $I$ be $x$. Since the perimeter of a square is four times its side length, we have:
\[
4x = 12 \implies x = \frac{12}{4} = 3
\]
- Let the side length of square $II$ be $y$. Similarly, since the perimeter of a ... |
Driving at a constant speed, Sharon usually takes $180$ minutes to drive from her house to her mother's house. One day Sharon begins the drive at her usual speed, but after driving $\frac{1}{3}$ of the way, she hits a bad snowstorm and reduces her speed by $20$ miles per hour. This time the trip takes her a total of $2... | 135 | 1. **Define Variables:**
Let the total distance from Sharon's house to her mother's house be $x$ miles.
2. **Calculate Speeds:**
Sharon's usual speed is $\frac{x}{180}$ miles per minute, since she normally takes 180 minutes to cover $x$ miles.
3. **Distance and Speed Adjustments:**
After driving $\frac{1}{3... |
The number of distinct points in the $xy$-plane common to the graphs of $(x+y-5)(2x-3y+5)=0$ and $(x-y+1)(3x+2y-12)=0$ is | 1 | To find the number of distinct points common to the graphs of $(x+y-5)(2x-3y+5)=0$ and $(x-y+1)(3x+2y-12)=0$, we need to solve the systems of equations formed by each factor being zero.
#### System 1: $x+y=5$ and $x-y=-1$
1. Adding these two equations:
\[
(x+y) + (x-y) = 5 + (-1) \implies 2x = 4 \implies x = 2
... |
In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of 21 conference games were played during the 2012 season, how many teams were members of the BIG N conference? | 7 | To determine the number of teams in the BIG N conference, we need to consider that each team plays every other team exactly once. This setup is a classic example of a round-robin tournament, where the number of games played can be calculated using the combination formula for choosing 2 teams out of $n$ total teams. The... |
Find the value of $x$ that satisfies the equation $25^{-2} = \frac{5^{48/x}}{5^{26/x} \cdot 25^{17/x}}.$ | 3 | 1. **Simplify the given equation**: Start by simplifying the right-hand side of the equation:
\[
\frac{5^{48/x}}{5^{26/x} \cdot 25^{17/x}}.
\]
Since $25 = 5^2$, we can rewrite $25^{17/x}$ as $(5^2)^{17/x} = 5^{34/x}$. Thus, the expression simplifies to:
\[
\frac{5^{48/x}}{5^{26/x} \cdot 5^{34/x}} = 5^... |
What is the value of the product
\[\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)?\] | 7 | 1. **Identify the Expression**: We start by simplifying each term in the product:
\[
\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)
\]
Simplifying each term, we get:
\[
... |
The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is: | 50 | 1. **Calculate the circumference of the Earth at the equator:**
The formula for the circumference, $C$, of a circle is given by:
\[
C = 2\pi r
\]
where $r$ is the radius of the circle. Given that the radius of the Earth at the equator is approximately 4000 miles, we substitute $r = 4000$ miles into the... |
The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\sqrt{m} + \sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$ | 23 | 1. **Identify Key Triangles and Midpoint**: Let $M$ be the midpoint of $CD$. Since each side of the pentagon and the internal segments are of length $2$, and given the symmetry and equal lengths, we can infer that triangles $AED$ and $ABC$ are isosceles with a vertex angle of $120^\circ$ formed by extending sides of eq... |
Erin the ant starts at a given corner of a cube and crawls along exactly 7 edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions? | 6 | 1. **Identify the problem**: Erin the ant needs to visit each corner of a cube exactly once using exactly 7 edges, and she cannot return to the starting point along an edge. We need to count the number of such paths.
2. **Visualize the cube and label corners**: Assume Erin starts at corner $(0,0,0)$. The cube's corner... |
Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first meet at the point A again, then the number of times they meet, excluding the start and finish, is | 13 | 1. **Define the problem setup**: Two boys start at the same point A on a circular track and move in opposite directions with speeds of 5 ft/s and 9 ft/s respectively. We need to find the number of times they meet excluding the start and finish.
2. **Calculate relative speed**: Since they are moving in opposite directi... |
A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square? | 4 | 1. **Understanding the Problem:**
- We need to find the smallest integer side length of a square that can be divided into 10 smaller squares, where at least 8 of these squares have an area of 1 (i.e., side length of 1).
2. **Eliminating the Smallest Option:**
- **Option (A) 3:** A square of side length 3 has an ... |
For how many integers $n$ is $\frac n{20-n}$ the square of an integer? | 4 | To solve the problem, we need to find the integers $n$ for which $\frac{n}{20-n}$ is a perfect square.
1. **Examine the range of $n$:**
- If $n < 0$ or $n > 20$, the fraction $\frac{n}{20-n}$ is negative, and thus cannot be a perfect square.
- If $n = 20$, the fraction is undefined.
- If $n \in \{1, 2, \dots,... |
What is the sum of the prime factors of $2010$? | 77 | 1. **Find the prime factorization of $2010$:**
- Start by checking divisibility by the smallest prime, $2$. Since $2010$ is even, it is divisible by $2$.
\[
2010 \div 2 = 1005
\]
- Next, check divisibility by $3$. The sum of the digits of $1005$ is $1 + 0 + 0 + 5 = 6$, which is divisible by $3$. T... |
Ann and Sue bought identical boxes of stationery. Ann used hers to write $1$-sheet letters and Sue used hers to write $3$-sheet letters.
Ann used all the envelopes and had $50$ sheets of paper left, while Sue used all of the sheets of paper and had $50$ envelopes left.
The number of sheets of paper in each box was | 150 | Let $S$ represent the number of sheets of paper in each box, and let $E$ represent the number of envelopes in each box. We can set up the following equations based on the problem statement:
1. Ann used all the envelopes and had 50 sheets of paper left. This implies:
\[ S - E = 50 \]
2. Sue used all the sheets of p... |
What is the value of $\frac{11! - 10!}{9!}$? | 100 | 1. Start by simplifying the expression given in the problem:
\[
\frac{11! - 10!}{9!}
\]
Here, $n!$ denotes the factorial of $n$, which is the product of all positive integers up to $n$.
2. Factor out $10!$ from the numerator:
\[
\frac{11! - 10!}{9!} = \frac{11 \cdot 10! - 10!}{9!} = \frac{10!(11 - 1)... |
The ratio of boys to girls in Mr. Brown's math class is $2:3$. If there are $30$ students in the class, how many more girls than boys are in the class? | 6 | 1. **Understanding the Ratio**: The problem states that the ratio of boys to girls in Mr. Brown's math class is $2:3$. This means for every 2 boys, there are 3 girls.
2. **Setting Up Variables**: Let the number of boys be $2x$ and the number of girls be $3x$. Here, $x$ is a common multiplier that will help us find the... |
Let $a$ and $b$ be relatively prime positive integers with $a>b>0$ and $\frac{a^3-b^3}{(a-b)^3} = \frac{73}{3}$. What is $a-b$? | 3 | 1. **Start by simplifying the given expression:**
\[
\frac{a^3-b^3}{(a-b)^3} = \frac{(a-b)(a^2+ab+b^2)}{(a-b)^3} = \frac{a^2+ab+b^2}{(a-b)^2}
\]
Given that this expression equals $\frac{73}{3}$, we have:
\[
\frac{a^2+ab+b^2}{(a-b)^2} = \frac{73}{3}
\]
2. **Rearrange the equation:**
\[
a^2 + ... |
For any positive integer $n$, define $\boxed{n}$ to be the sum of the positive factors of $n$.
For example, $\boxed{6} = 1 + 2 + 3 + 6 = 12$. Find $\boxed{\boxed{11}}$ . | 28 | To solve for $\boxed{\boxed{11}}$, we first need to determine $\boxed{11}$, which is the sum of the positive factors of $11$.
1. **Finding $\boxed{11}$**:
Since $11$ is a prime number, its only positive factors are $1$ and $11$ itself. Therefore, the sum of the factors of $11$ is:
\[
\boxed{11} = 1 + 11 = 12.... |
The numbers $3, 5, 7, a,$ and $b$ have an average (arithmetic mean) of $15$. What is the average of $a$ and $b$? | 30 | 1. **Calculate the total sum of the numbers given their average**:
The average (arithmetic mean) of the numbers $3, 5, 7, a,$ and $b$ is given as $15$. The formula for the average of a set of numbers is:
\[
\text{Average} = \frac{\text{Sum of all numbers}}{\text{Number of numbers}}
\]
Here, the sum of t... |
Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order)... | 80 |
To solve this problem, we need to ensure that the average after each score is entered is an integer. We are given the scores in ascending order: $71$, $76$, $80$, $82$, and $91$. The total sum of these scores is $400$. We need to determine the last score entered by Mrs. Walter.
#### Step-by-step Analysis:
1. **Divis... |
Josh writes the numbers $1,2,3,\dots,99,100$. He marks out $1$, skips the next number $(2)$, marks out $3$, and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number $(2)$, skips the next number $(4)$, marks out $6$, s... | 64 | Josh's method of marking out numbers can be understood as a process of elimination where in each round, every second number from the remaining list is marked out. Let's analyze the pattern step by step:
1. **First Pass:**
- Josh marks out every second number starting from $1$. This means all odd numbers are marked ... |
The number of distinct ordered pairs $(x,y)$ where $x$ and $y$ have positive integral values satisfying the equation $x^4y^4-10x^2y^2+9=0$ is: | 3 | 1. **Rewrite the given equation**: Start by rewriting the equation $x^4y^4 - 10x^2y^2 + 9 = 0$ in terms of a new variable $a = (xy)^2$. This gives us:
\[
a^2 - 10a + 9 = 0
\]
2. **Factorize the quadratic equation**: The quadratic equation in $a$ can be factorized as:
\[
a^2 - 10a + 9 = (a - 9)(a - 1) = ... |
What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$? | 1 | 1. **Expand the given expression**:
\[
(xy-1)^2 + (x+y)^2 = (x^2y^2 - 2xy + 1) + (x^2 + 2xy + y^2)
\]
Combine like terms:
\[
x^2y^2 + x^2 + 2xy + y^2 - 2xy + 1 = x^2y^2 + x^2 + y^2 + 1
\]
2. **Factor the expression**:
Notice that the expression can be grouped:
\[
x^2y^2 + x^2 + y^2 + 1 = ... |
A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is heavy-tailed if $a_1 + a_2 < a_4 + a_5$. What is the number of heavy-tailed permutations? | 48 | We analyze the problem by considering the position of $a_3$ and how it affects the possible values of $a_1 + a_2$ and $a_4 + a_5$. We need to ensure $a_1 + a_2 < a_4 + a_5$ for the permutation to be heavy-tailed.
1. **Case 1: $a_3 = 1$.**
- The remaining numbers are $2, 3, 4, 5$. We need $a_1 + a_2 < a_4 + a_5$.
... |
A $4\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums? | 4 | 1. **Identify the original diagonals**:
- The main diagonal (from top left to bottom right) consists of the numbers: $1, 9, 17, 25$.
- The secondary diagonal (from top right to bottom left) consists of the numbers: $4, 10, 16, 22$.
2. **Calculate the original diagonal sums**:
- Main diagonal sum: $1 + 9 + 17... |
For a science project, Sammy observed a chipmunk and squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide... | 48 | 1. **Define Variables:**
Let $x$ be the number of holes the chipmunk dug. Each hole contains 3 acorns, so the total number of acorns the chipmunk hid is $3x$.
Let $y$ be the number of holes the squirrel dug. Each hole contains 4 acorns, so the total number of acorns the squirrel hid is $4y$.
2. **Set Up the Equa... |
Suppose that $f(x+3)=3x^2 + 7x + 4$ and $f(x)=ax^2 + bx + c$. What is $a+b+c$? | 2 | 1. **Given Equations**: We are given that $f(x+3) = 3x^2 + 7x + 4$ and $f(x) = ax^2 + bx + c$.
2. **Expression for $f(x+3)$**: Using the expression for $f(x)$, we can write:
\[
f(x+3) = a(x+3)^2 + b(x+3) + c
\]
Expanding $(x+3)^2$ and $(x+3)$, we get:
\[
(x+3)^2 = x^2 + 6x + 9 \quad \text{and} \quad ... |
How many ways are there to write $2016$ as the sum of twos and threes, ignoring order? (For example, $1008 \cdot 2 + 0 \cdot 3$ and $402 \cdot 2 + 404 \cdot 3$ are two such ways.) | 337 | 1. **Express $2016$ in terms of $6$:**
We start by noting that $2016$ can be divided by $6$:
\[
2016 = 6 \times 336
\]
This means we can write $2016$ as the sum of $336$ sixes.
2. **Decompose $6$ into sums of $2$s and $3$s:**
Each $6$ can be expressed in two ways using $2$s and $3$s:
- As th... |
The number $695$ is to be written with a factorial base of numeration, that is, $695=a_1+a_2\times2!+a_3\times3!+ \ldots a_n \times n!$ where $a_1, a_2, a_3 ... a_n$ are integers such that $0 \le a_k \le k,$ and $n!$ means $n(n-1)(n-2)...2 \times 1$. Find $a_4$ | 3 | To solve this problem, we need to express the number 695 in a factorial base. This means we need to find coefficients $a_1, a_2, a_3, \ldots, a_n$ such that:
\[ 695 = a_1 + a_2 \times 2! + a_3 \times 3! + \ldots + a_n \times n! \]
where $0 \leq a_k \leq k$ for each $k$.
1. **Find the largest factorial smaller than 695... |
Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved tw... | 2 | To solve this problem, we need to analyze the movements of each friend and determine the original seating arrangement based on the given movements.
1. **Initial Setup**: There are 5 seats, and each friend occupies one seat. Ada leaves, creating one empty seat.
2. **Movements**:
- **Bea** moves two seats to the rig... |
The graph shows the distribution of the number of children in the families of the students in Ms. Jordan's English class. The median number of children in the family for this distribution is | 4 | To find the median number of children in the families of the students in Ms. Jordan's English class, we follow these steps:
1. **Identify the Total Number of Data Points**: The total number of families (data points) is given as thirteen.
2. **Determine the Position of the Median**: The median is the middle value in a... |
Let $C_1$ and $C_2$ be circles of radius 1 that are in the same plane and tangent to each other. How many circles of radius 3 are in this plane and tangent to both $C_1$ and $C_2$? | 6 | 1. **Identify the configuration of $C_1$ and $C_2$:**
- $C_1$ and $C_2$ are circles of radius 1, tangent to each other. Let's denote the point of tangency as $P$. The distance between the centers of $C_1$ and $C_2$ is thus 2.
2. **Consider circles of radius 3 tangent to both $C_1$ and $C_2$:**
- **Externally tan... |
The coefficient of $x^7$ in the polynomial expansion of $(1+2x-x^2)^4$ is | -8 | 1. **Identify the Polynomial and the Target Term**: We are given the polynomial \((1+2x-x^2)^4\) and need to find the coefficient of \(x^7\).
2. **Expand the Polynomial**: The polynomial can be expanded using the multinomial theorem, which states that:
\[
(a+b+c)^n = \sum_{k_1+k_2+k_3=n} \binom{n}{k_1, k_2, k_3}... |
Seven cubes, whose volumes are $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface ... | 749 | 1. **Identify the side lengths of the cubes**: The volumes of the cubes are given as $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units. The side lengths of these cubes are the cube roots of these volumes, which are $1$, $2$, $3$, $4$, $5$, $6$, and $7$ units respectively.
2. **Determine the surface area of eac... |
In a geometric sequence of real numbers, the sum of the first $2$ terms is $7$, and the sum of the first $6$ terms is $91$. The sum of the first $4$ terms is | 32 | Let the first term of the geometric sequence be $a$ and the common ratio be $r$. Then the terms of the sequence are $a, ar, ar^2, ar^3, \ldots$.
1. **Sum of the first 2 terms:**
\[
a + ar = 7
\]
2. **Sum of the first 6 terms:**
\[
a + ar + ar^2 + ar^3 + ar^4 + ar^5 = 91
\]
We can express the sum... |
The symbol $R_k$ stands for an integer whose base-ten representation is a sequence of $k$ ones. For example, $R_3=111, R_5=11111$, etc. When $R_{24}$ is divided by $R_4$, the quotient $Q=R_{24}/R_4$ is an integer whose base-ten representation is a sequence containing only ones and zeroes. The number of zeros in $Q$ is: | 15 | 1. **Expression for $R_k$:**
The number $R_k$ is defined as an integer consisting of $k$ ones in its decimal representation. We can express $R_k$ as:
\[
R_k = 111\ldots1 \quad (\text{$k$ times})
\]
This can be written as a sum of powers of 10:
\[
R_k = 1 + 10 + 10^2 + \ldots + 10^{k-1}
\]
U... |
If $R_n=\frac{1}{2}(a^n+b^n)$ where $a=3+2\sqrt{2}$ and $b=3-2\sqrt{2}$, and $n=0,1,2,\cdots,$ then $R_{12345}$ is an integer. Its units digit is | 9 | 1. **Identify the recurrence relation**: Given $R_n = \frac{1}{2}(a^n + b^n)$ where $a = 3 + 2\sqrt{2}$ and $b = 3 - 2\sqrt{2}$, we start by expressing $R_{n+1}$ and $R_{n-1}$ in terms of $R_n$:
\[
(a+b)R_n = \frac{1}{2}(a+b)(a^n + b^n) = \frac{1}{2}(a^{n+1} + ab^n + ba^n + b^{n+1})
\]
\[
= \frac{1}{2}(a... |
The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n$? | 12 | 1. **Count the number of factors of 10 in $90!$:**
The number of factors of 10 in $90!$ is determined by the number of factors of 5, as there are more factors of 2 than 5. We calculate this using the formula for the number of factors of a prime $p$ in $n!$:
\[
\left\lfloor \frac{90}{5} \right\rfloor + \left\lf... |
At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage ... | 30 | 1. **Calculate the total cost without any discounts:**
The regular price for one pair of sandals is $50. Therefore, the cost for three pairs of sandals at the regular price is:
\[
3 \times 50 = 150 \text{ dollars}
\]
2. **Calculate the total cost with the "fair special" discounts:**
- The first pair... |
In this addition problem, each letter stands for a different digit.
$\begin{array}{cccc}&T & W & O\\ +&T & W & O\\ \hline F& O & U & R\end{array}$
If T = 7 and the letter O represents an even number, what is the only possible value for W? | 3 | 1. **Identify the value of T**: Given that $T = 7$, we substitute $T$ in the addition problem:
\[
\begin{array}{cccc}
& 7 & W & O \\
+ & 7 & W & O \\
\hline
F & O & U & R \\
\end{array}
\]
2. **Determine the value of O**: Since $7 + 7 = 14$, the sum of $O + O$ must result in a number ending i... |
A dealer bought $n$ radios for $d$ dollars, $d$ a positive integer. He contributed two radios to a community bazaar at half their cost. The rest he sold at a profit of $8 on each radio sold. If the overall profit was $72, then the least possible value of $n$ for the given information is: | 12 | 1. **Calculate the cost and selling price of the radios:**
- The dealer bought $n$ radios for $d$ dollars, so the cost per radio is $\frac{d}{n}$.
- He sold two radios at half their cost, which is $\frac{1}{2} \times \frac{d}{n} = \frac{d}{2n}$ each. Therefore, the total income from these two radios is $2 \times ... |
A $\text{palindrome}$, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$? | 24 |
1. **Identify the range of $x$ and $x+32$:**
- Since $x$ is a three-digit palindrome, the maximum value of $x$ is $999$.
- Consequently, the maximum value of $x+32$ is $999 + 32 = 1031$.
- The minimum value of $x+32$ is $1000$ (since it is a four-digit palindrome).
2. **Determine the possible values for $x+3... |
A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle ABC of side length $3$. The perimeter of the remaining quadrilateral is | 8 | 1. **Identify the Similar Triangles**:
- $\triangle DBE$ is cut from $\triangle ABC$. Since $\triangle ABC$ is equilateral, all its angles are $60^\circ$.
- $\triangle DBE$ is also equilateral because $DB = EB$ and each angle in $\triangle DBE$ is $60^\circ$.
2. **Determine the Side Lengths**:
- Since $\tri... |
All three vertices of $\triangle ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$? | 8 | 1. **Identify the Coordinates of Points**:
- Since $A$ is at the origin, $A = (0,0)$.
- The line $\overline{BC}$ is parallel to the $x$-axis, and since all vertices lie on the parabola $y = x^2$, the coordinates of $B$ and $C$ must be symmetric about the $y$-axis. Let's denote the $x$-coordinates of $B$ and $C$ ... |
Each face of a regular tetrahedron is painted either red, white, or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible? | 15 | 1. **Identify the Problem Context**: We are given a regular tetrahedron with each face painted either red, white, or blue. We need to find the number of distinguishable colorings, considering that colorings are indistinguishable if one tetrahedron can be rotated to look like another.
2. **Count the Faces**: A tetrahed... |
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible? | 132 |
We are given that Amy, Beth, and Jo listen to four different songs and discuss which ones they like, with the conditions that no song is liked by all three and for each pair of girls, there is at least one song liked by those two but disliked by the third. We need to find the number of ways this can happen.
#### Step... |
Sides $AB, BC, CD$ and $DA$ of convex polygon $ABCD$ have lengths 3, 4, 12, and 13, respectively, and $\angle CBA$ is a right angle. The area of the quadrilateral is | 36 | 1. **Identify the triangles**: Given that $\angle CBA$ is a right angle and the sides $AB = 3$ and $BC = 4$, triangle $ABC$ is a right triangle. By the Pythagorean theorem, the hypotenuse $AC$ can be calculated as follows:
\[
AC = \sqrt{AB^2 + BC^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.
\]
Thu... |
The state income tax where Kristin lives is levied at the rate of $p\%$ of the first $\$28000$ of annual income plus $(p + 2)\%$ of any amount above $\$28000$. Kristin noticed that the state income tax she paid amounted to $(p + 0.25)\%$ of her annual income. What was her annual income? | 32000 | Let $A$ represent Kristin's annual income and $T$ represent the total income tax she paid. The tax rate is $p\%$ for the first $\$28000$ and $(p+2)\%$ for any amount above $\$28000$. We are given that the total tax $T$ is also $(p+0.25)\%$ of her annual income $A$.
1. **Express the total tax $T$ in terms of $p$ and $A... |
A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is: | 540 | Let's consider a quadrilateral $ABCD$ inscribed in a circle. We need to find the sum of the angles inscribed in the four segments outside the quadrilateral.
1. **Identify the Segments and Angles**:
- The four segments outside the quadrilateral are the regions outside $ABCD$ but inside the circle.
- Let $\alpha$,... |
A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have? | 84 | To solve this problem, we need to understand the structure of the solid after the smaller cubes are removed from each corner of the larger cube.
1. **Original Cube Characteristics**:
- The original cube has a side length of $3$.
- It has $8$ vertices (corners).
- It has $12$ edges.
- It has $6$ faces.
2. ... |
Ralph went to the store and bought 12 pairs of socks for a total of $24. Some of the socks he bought cost $1 a pair, some of the socks he bought cost $3 a pair, and some of the socks he bought cost $4 a pair. If he bought at least one pair of each type, how many pairs of $1 socks did Ralph buy? | 7 |
Let $x$ be the number of $1$ pairs of socks, $y$ be the number of $3$ pairs of socks, and $z$ be the number of $4$ pairs of socks that Ralph bought. We are given the following system of equations based on the problem statement:
1. **Total number of pairs**: $x + y + z = 12$
2. **Total cost of socks**: $x + 3y + 4z = ... |
The area of a pizza with radius $4$ is $N$ percent larger than the area of a pizza with radius $3$ inches. What is the integer closest to $N$? | 78 | 1. **Calculate the area of both pizzas**:
- The area of a circle is given by the formula $A = \pi r^2$.
- For the pizza with radius $4$ inches, the area is:
\[
A_1 = \pi \times 4^2 = 16\pi \text{ square inches}
\]
- For the pizza with radius $3$ inches, the area is:
\[
A_2 = \pi \times... |
The number of solutions in positive integers of $2x+3y=763$ is: | 127 | 1. **Formulate the Equation**: We start with the Diophantine equation given:
\[
2x + 3y = 763
\]
2. **Find a Particular Solution**: We need to find one particular solution $(x_0, y_0)$ in integers. By trial and error or using the extended Euclidean algorithm, we find:
\[
x_0 = 380, \quad y_0 = 1
\]
... |
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a mu... | 11 | 1. **Identify the ages of Joey, Chloe, and Zoe**: Let's denote Chloe's age today as $C$, Joey's age as $J$, and Zoe's age as $Z$. Given that Zoe is exactly 1 year old today, we have $Z = 1$. Joey is 1 year older than Chloe, so $J = C + 1$.
2. **Understand the condition for Chloe's age**: It is given that today is the ... |
Mary is $20\%$ older than Sally, and Sally is $40\%$ younger than Danielle. The sum of their ages is $23.2$ years. How old will Mary be on her next birthday? | 8 | 1. **Define the variables:**
Let $m$ be Mary's age, $s$ be Sally's age, and $d$ be Danielle's age.
2. **Set up the relationships:**
- Sally is $40\%$ younger than Danielle, which means Sally's age is $60\%$ of Danielle's age:
\[
s = 0.6d
\]
- Mary is $20\%$ older than Sally, so Mary's age is $1... |
Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea's next $n$ shots are bullseyes she will be guaranteed victory. What is the minimu... | 42 | 1. **Identify the current situation**: Chelsea is leading by 50 points after 50 shots. Let $k$ be the number of points Chelsea has scored so far.
2. **Determine the maximum possible score for the opponent**: Since the opponent is 50 points behind Chelsea, if the opponent scores bullseyes (10 points each) for the remai... |
Maria buys computer disks at a price of $4$ for $\$5$ and sells them at a price of $3$ for $\$5$. How many computer disks must she sell in order to make a profit of $\$100$? | 240 | 1. **Calculate the cost and selling price per disk:**
- Maria buys disks at a rate of 4 disks for $5 dollars, so the cost per disk is:
\[
\frac{5}{4} = 1.25 \text{ dollars per disk}
\]
- She sells disks at a rate of 3 disks for $5 dollars, so the selling price per disk is:
\[
\frac{5}{3} ... |
If the Highest Common Divisor of $6432$ and $132$ is diminished by $8$, it will equal: | 4 | To solve the problem, we need to find the highest common divisor (HCD) or greatest common divisor (GCD) of $6432$ and $132$, and then subtract $8$ from it.
1. **Prime Factorization**:
- For $6432$, we start by finding its prime factors. We see that $6432$ is divisible by $2$ (since it's even). Dividing repeatedly b... |
How many real numbers $x$ satisfy the equation $3^{2x+2}-3^{x+3}-3^x+3=0$? | 2 | 1. **Substitute and Simplify**: Let $a = 3^x$. Then, we can rewrite the given equation $3^{2x+2} - 3^{x+3} - 3^x + 3 = 0$ using the substitution:
\[
3^{2x+2} = (3^x)^2 \cdot 3^2 = 9a^2, \quad 3^{x+3} = 3^x \cdot 3^3 = 27a, \quad 3^x = a
\]
Substituting these into the equation, we get:
\[
9a^2 - 27a - ... |
If $200 \leq a \leq 400$ and $600 \leq b \leq 1200$, then the largest value of the quotient $\frac{b}{a}$ is | 6 | 1. **Identify the range of values for $a$ and $b$:**
Given $200 \leq a \leq 400$ and $600 \leq b \leq 1200$.
2. **Determine the expression to maximize:**
We need to find the maximum value of the quotient $\frac{b}{a}$.
3. **Analyze the behavior of the quotient:**
The quotient $\frac{b}{a}$ increases as $b$ i... |
Suppose that $a * b$ means $3a-b.$ What is the value of $x$ if $2 * (5 * x)=1$ | 10 | 1. **Interpret the operation $*$**: Given that $a * b = 3a - b$, we need to find $x$ such that $2 * (5 * x) = 1$.
2. **Evaluate $5 * x$**: According to the operation definition, $5 * x = 3 \cdot 5 - x = 15 - x$.
3. **Substitute $5 * x$ into the equation**: Now substitute $15 - x$ for $(5 * x)$ in the equation $2 * (5... |
What is the smallest positive integer $n$ such that $\sqrt{n}-\sqrt{n-1}<.01$? | 2501 | 1. **Starting from the inequality:** We are given that $\sqrt{n} - \sqrt{n-1} < 0.01$. To isolate $\sqrt{n}$, we add $\sqrt{n-1}$ to both sides:
\[
\sqrt{n} < \sqrt{n-1} + 0.01
\]
2. **Squaring both sides:** To eliminate the square roots, we square both sides of the inequality:
\[
(\sqrt{n})^2 < (\sqrt{... |
How many of the twelve pentominoes pictured below have at least one line of reflectional symmetry?
[asy] unitsize(5mm); defaultpen(linewidth(1pt)); draw(shift(2,0)*unitsquare); draw(shift(2,1)*unitsquare); draw(shift(2,2)*unitsquare); draw(shift(1,2)*unitsquare); draw(shift(0,2)*unitsquare); draw(shift(2,4)*unitsquare)... | 6 | To solve this problem, we need to identify which of the twelve pentominoes have at least one line of reflectional symmetry. Reflectional symmetry in a shape means that there is at least one line (axis of symmetry) along which the shape can be folded or reflected onto itself perfectly.
1. **Identify each pentomino**: W... |
Let $P$ be the product of any three consecutive positive odd integers. The largest integer dividing all such $P$ is: | 3 | 1. **Express the product $P$:** Let the three consecutive odd integers be $2n-1$, $2n+1$, and $2n+3$. Then the product $P$ is:
\[
P = (2n-1)(2n+1)(2n+3)
\]
2. **Simplify the expression for $P$:** We can expand this product:
\[
P = (2n-1)(2n+1)(2n+3) = ((2n-1)(2n+1))(2n+3)
\]
\[
= (4n^2 - 1)(2n+... |
Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes? | 25 | 1. **Determine individual rates**:
- Cagney's rate of frosting cupcakes is 1 cupcake per 20 seconds.
- Lacey's rate of frosting cupcakes is 1 cupcake per 30 seconds.
2. **Calculate the combined rate**:
- The combined rate \( R \) when two people are working together can be calculated using the formula for th... |
Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M,32,33,34,35,36,37,38,39,$ and $40.$ What is the value of $\frac{N}{M}$? | 74 | 1. **Calculate $M$:**
$M$ is the least common multiple (LCM) of all integers from $10$ to $30$. To find $M$, we need to consider the highest powers of all prime numbers that appear in the factorizations of these numbers:
- $2^4$ comes from $16 = 2^4$.
- $3^3$ comes from $27 = 3^3$.
- $5^2$ comes from $25 ... |
When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. The smallest possible value of $S$ is | 337 | 1. **Define the transformation**: Let $d_i$ be the number on the $i$-th die. We define a transformation where each die's number is replaced by $d_i' = 7 - d_i$. This transformation is an involution, meaning applying it twice returns the original value: $7 - (7 - d_i) = d_i$.
2. **Effect of the transformation on the su... |
In a mathematics contest with ten problems, a student gains 5 points for a correct answer and loses 2 points for an incorrect answer. If Olivia answered every problem and her score was 29, how many correct answers did she have? | 7 | Let $c$ be the number of correct answers Olivia had, and $w$ be the number of incorrect answers. Since there are 10 problems in total, we have:
\[ c + w = 10 \]
For each correct answer, Olivia gains 5 points, and for each incorrect answer, she loses 2 points. Therefore, her total score can be expressed as:
\[ 5c - 2w ... |
A parabolic arch has a height of $16$ inches and a span of $40$ inches. The height, in inches, of the arch at the point $5$ inches from the center $M$ is: | 15 | 1. **Identify the shape and equation of the arch**: Given that the arch is parabolic, we can model it using a quadratic equation of the form:
\[
y = ax^2 + k
\]
where \(y\) is the height of the arch at a horizontal distance \(x\) from the center, \(a\) is a constant that determines the curvature of the para... |
The value of $x$ at the intersection of $y=\frac{8}{x^2+4}$ and $x+y=2$ is: | 0 | 1. **Substitute for $y$ in the second equation:**
Given the equations $y = \frac{8}{x^2 + 4}$ and $x + y = 2$, we can substitute $y$ from the second equation into the first:
\[
x + y = 2 \implies y = 2 - x.
\]
Substituting $y = 2 - x$ into $y = \frac{8}{x^2 + 4}$, we get:
\[
2 - x = \frac{8}{x^2 ... |
On a sheet of paper, Isabella draws a circle of radius $2$, a circle of radius $3$, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly $k \ge 0$ lines. How many different values of $k$ are possible? | 5 | To determine the number of different values of $k$, the number of lines simultaneously tangent to both circles, we analyze the possible configurations of the two circles:
1. **Concentric Circles**: If the circles are concentric (i.e., they have the same center), no tangent lines can be drawn that are tangent to both c... |
If two factors of $2x^3-hx+k$ are $x+2$ and $x-1$, the value of $|2h-3k|$ is | 0 | 1. **Using the Remainder Theorem**: Given that $x+2$ and $x-1$ are factors of $2x^3 - hx + k$, we can use the Remainder Theorem to set up equations. The Remainder Theorem states that if $x - c$ is a factor of a polynomial $p(x)$, then $p(c) = 0$.
2. **Setting up equations**:
- For $x + 2$, substitute $x = -2$ into ... |
Gilda has a bag of marbles. She gives $20\%$ of them to her friend Pedro. Then Gilda gives $10\%$ of what is left to another friend, Ebony. Finally, Gilda gives $25\%$ of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself? | 54 | 1. **Initial Amount of Marbles**: Let's denote the initial number of marbles Gilda has as $M$.
2. **Marbles Given to Pedro**: Gilda gives $20\%$ of her marbles to Pedro. Therefore, the number of marbles she gives to Pedro is $0.20M$. The number of marbles left with Gilda after giving to Pedro is:
\[
M - 0.20M = ... |
Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perim... | 60 | 1. **Define the problem in terms of variables:**
Let the dimensions of the rectangular pan be $m$ (length) and $n$ (width). The total number of brownies is $mn$.
2. **Determine the number of interior and perimeter pieces:**
- The interior pieces are formed by excluding the outermost row and column, thus having d... |
Three congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circle centered at $Q$ has diameter $4$ and passes through points $P$ and $R$. The area of the rectangle is | 32 | 1. **Identify the diameters of the circles**: Given that the circle centered at $Q$ has a diameter of $4$, and since the circles are congruent, the circles centered at $P$ and $R$ also have diameters of $4$.
2. **Determine the height of the rectangle**: Consider a diameter of circle $P$ that is parallel to side $AD$ o... |
A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted $7$ children and $19$ wheels. How many tricycles were there? | 5 |
We are given that there are 7 children and 19 wheels. Each child is riding either a bicycle (which has 2 wheels) or a tricycle (which has 3 wheels). We need to determine the number of tricycles.
#### Solution 1: Using a substitution method
1. Let $b$ be the number of bicycles and $t$ be the number of tricycles.
2. We... |
A store prices an item in dollars and cents so that when 4% sales tax is added, no rounding is necessary because the result is exactly $n$ dollars where $n$ is a positive integer. The smallest value of $n$ is | 13 | 1. **Define the variables**: Let the price of the item in cents be $x$. Since the sales tax is 4%, the total cost including tax is $x + 0.04x = 1.04x$ cents.
2. **Convert the total cost to dollars**: Since $1$ dollar = $100$ cents, the total cost in dollars is $\frac{1.04x}{100} = 0.0104x$ dollars.
3. **Set up the eq... |
Some boys and girls are having a car wash to raise money for a class trip to China. Initially $40\%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $30\%$ of the group are girls. How many girls were initially in the group? | 8 | 1. **Define Variables:**
Let $p$ be the total number of people initially in the group. Since $40\%$ of the group are girls, the number of girls initially is $0.4p$.
2. **Change in Group Composition:**
After two girls leave and two boys join, the total number of people remains $p$. The new number of girls is $0.4... |
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