problem stringlengths 11 5.15k | answer stringlengths 1 42 | solution stringlengths 0 8.75k |
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The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43$ dollars. Some of the $30$ sixth graders each bought a pencil, and they paid a total of $1.95$ dollars. How many more sixth graders than seventh graders bought a pencil? | 4 | 1. **Convert the total payments to cents**:
- Seventh graders paid $1.43$ dollars, which is $143$ cents.
- Sixth graders paid $1.95$ dollars, which is $195$ cents.
2. **Determine the cost of one pencil**:
- The cost of a pencil must be a common factor of both $143$ and $195$ since each group paid a whole num... |
Two walls and the ceiling of a room meet at right angles at point $P.$ A fly is in the air one meter from one wall, eight meters from the other wall, and nine meters from point $P$. How many meters is the fly from the ceiling? | 4 | 1. **Identify the coordinates of the fly and point $P$:**
- Let's assume point $P$ is at the origin $(0,0,0)$ of a three-dimensional coordinate system.
- The fly is 1 meter from one wall, 8 meters from another wall, and 9 meters from point $P$. We can assume the fly is at point $(1, 8, z)$, where $z$ is the dista... |
What is the minimum possible product of three different numbers of the set $\{-8,-6,-4,0,3,5,7\}$? | -280 | To find the minimum possible product of three different numbers from the set $\{-8,-6,-4,0,3,5,7\}$, we need to consider the sign of the product and the magnitude of the numbers involved.
1. **Sign of the Product**:
- A product of three numbers is negative if and only if exactly one or all three of the numbers are ... |
The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216 \pi$. What is the length $AB$? | 20 | 1. **Understanding the Geometry**: The region described in the problem is the set of all points in three-dimensional space that are within $3$ units of the line segment $\overline{AB}$. This region can be visualized as a cylinder with radius $3$ and axis along $\overline{AB}$, capped with two hemispheres of radius $3$ ... |
At the beginning of the school year, Lisa's goal was to earn an $A$ on at least $80\%$ of her $50$ quizzes for the year. She earned an $A$ on $22$ of the first $30$ quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an $A$? | 2 | 1. **Determine the total number of quizzes Lisa needs to score an A on to meet her goal:**
Lisa's goal is to earn an A on at least 80% of her 50 quizzes. Therefore, the number of quizzes she needs to score an A on is:
\[
0.8 \times 50 = 40
\]
quizzes.
2. **Calculate the number of quizzes she has alrea... |
Given the sets of consecutive integers $\{1\}$,$\{2, 3\}$,$\{4,5,6\}$,$\{7,8,9,10\}$,$\; \cdots \;$, where each set contains one more element than the preceding one, and where the first element of each set is one more than the last element of the preceding set. Let $S_n$ be the sum of the elements in the nth set. The... | 4641 | To solve for $S_{21}$, the sum of the elements in the 21st set, we first need to determine the first and last elements of this set.
1. **Determine the first element of the 21st set:**
Each set starts where the previous one ended, plus one. The $n$-th set starts with the element that is one more than the sum of the ... |
A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures $8$ inches high and $10$ inches wide. What is the area of the border, in square inches? | 88 | 1. **Calculate the area of the photograph**:
The photograph measures $8$ inches in height and $10$ inches in width. Therefore, the area of the photograph is calculated as:
\[
\text{Area of photograph} = 8 \times 10 = 80 \text{ square inches}
\]
2. **Determine the dimensions of the entire framed area**:
... |
The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum? | 10 | To solve this problem, we need to find the common sum of the numbers in each row, column, and diagonal of a $5 \times 5$ square matrix using the integers from $-10$ to $14$ inclusive.
1. **Calculate the Total Sum of All Numbers:**
The integers from $-10$ to $14$ form an arithmetic sequence with the first term $a = ... |
One dimension of a cube is increased by $1$, another is decreased by $1$, and the third is left unchanged. The volume of the new rectangular solid is $5$ less than that of the cube. What was the volume of the cube? | 125 | 1. **Define the dimensions and volume of the original cube:**
Let the edge length of the original cube be $a$. Therefore, the volume of the cube is given by:
\[
V_{\text{cube}} = a^3
\]
2. **Define the dimensions and volume of the new rectangular solid:**
After modifying the dimensions of the cube, the ... |
Let N = $69^5 + 5 \cdot 69^4 + 10 \cdot 69^3 + 10 \cdot 69^2 + 5 \cdot 69 + 1$. How many positive integers are factors of $N$? | 216 | 1. **Rewriting the Expression**: Let $a = 69$. Then, the expression for $N$ becomes:
\[
N = a^5 + 5a^4 + 10a^3 + 10a^2 + 5a + 1
\]
2. **Recognizing the Binomial Expansion**: The expression can be recognized as the expansion of $(a+1)^5$ using the binomial theorem:
\[
(a+1)^5 = \sum_{k=0}^{5} \binom{5}{k... |
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions? | 9 | 1. **Identify Special Fractions**: A fraction $\frac{a}{b}$ is special if $a+b=15$ where $a$ and $b$ are positive integers. The possible pairs $(a, b)$ are:
- $(1, 14), (2, 13), (3, 12), (4, 11), (5, 10), (6, 9), (7, 8), (8, 7), (9, 6), (10, 5), (11, 4), (12, 3), (13, 2), (14, 1)$.
2. **List Special Fractions**: Co... |
Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How m... | 23 | 1. **Understanding the problem setup**: Each school sends 3 students, so the total number of students is $3n$, where $n$ is the number of schools. Andrea's score is the median, and she scored the highest in her team. Her teammates Beth and Carla placed 37th and 64th, respectively.
2. **Determining Andrea's rank**: Sin... |
At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts? | 32 | 1. **Total number of students and pairs:**
Given that there are $57$ students wearing blue shirts and $75$ students wearing yellow shirts, the total number of students is $57 + 75 = 132$. These students are divided into $66$ pairs.
2. **Pairs with both students wearing blue shirts:**
It is given that in $23$ pai... |
Ara and Shea were once the same height. Since then Shea has grown 20% while Ara has grown half as many inches as Shea. Shea is now 60 inches tall. How tall, in inches, is Ara now? | 55 | 1. **Determine Shea's original height**:
Given that Shea has grown by 20% and is now 60 inches tall, we can set up the equation for her original height $x$:
\[
1.2x = 60
\]
Solving for $x$, we divide both sides by 1.2:
\[
x = \frac{60}{1.2} = 50
\]
Thus, Shea's original height was 50 inches.... |
In the adjoining figure the five circles are tangent to one another consecutively and to the lines $L_1$ and $L_2$. If the radius of the largest circle is $18$ and that of the smallest one is $8$, then the radius of the middle circle is | 12 | 1. **Identify the Configuration**: We are given five circles tangent to each other and to two lines $L_1$ and $L_2$. The radii of the largest and smallest circles are given as $18$ and $8$, respectively.
2. **Understanding the Geometry**: The centers of three consecutive circles are collinear. Let these centers be $P$... |
Paul owes Paula $35$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, and $25$-cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her? | 5 | To solve this problem, we need to determine the minimum and maximum number of coins Paul can use to pay Paula exactly $35$ cents using $5$-cent coins, $10$-cent coins, and $25$-cent coins.
#### Minimum Number of Coins:
To minimize the number of coins, Paul should try to use the coin with the highest value first. The h... |
Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the greatest number of additional cars she must buy in order to be able to arrange all her cars this way? | 1 | 1. **Identify the Requirement**: Danica wants to arrange her model cars in rows with exactly 6 cars in each row. This means that the total number of cars must be a multiple of 6.
2. **Current Number of Cars**: She currently has 23 model cars.
3. **Finding the Next Multiple of 6**: We need to find the smallest multipl... |
Six trees are equally spaced along one side of a straight road. The distance from the first tree to the fourth is 60 feet. What is the distance in feet between the first and last trees? | 100 | 1. **Identify the number of spaces between trees**:
- The problem states that there are six trees equally spaced along one side of a straight road.
- The trees are numbered from 1 to 6.
- The distance from the first tree to the fourth tree is given as 60 feet.
2. **Calculate the distance between consecutiv... |
What is the area of the shaded figure shown below? | 6 | To find the area of the shaded figure with vertices given as $(1,0)$, $(3,2)$, $(5,0)$, and $(3,5)$, we can use the Shoelace Theorem. The Shoelace Theorem states that the area of a polygon with vertices $(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$ is given by:
\[
\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i ... |
The sum of the base-10 logarithms of the divisors of $10^n$ is $792$. What is $n$? | 11 |
To solve the problem, we need to find the value of $n$ such that the sum of the base-$10$ logarithms of the divisors of $10^n$ equals $792$.
#### Step 1: Understanding the divisors of $10^n$
$10^n = 2^n \cdot 5^n$. Any divisor of $10^n$ can be expressed in the form $2^a \cdot 5^b$ where $0 \leq a \leq n$ and $0 \leq ... |
Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly? | 15 | 1. **Understanding the problem**: Cindy was supposed to subtract 3 from a number $x$ and then divide by 9. Instead, she subtracted 9 and divided by 3, resulting in 43.
2. **Setting up the equation for Cindy's incorrect operation**:
\[
\frac{x - 9}{3} = 43
\]
3. **Solving for $x$**:
\[
x - 9 = 43 \time... |
Let $S_n=1-2+3-4+\cdots +(-1)^{n-1}n$, where $n=1,2,\cdots$. Then $S_{17}+S_{33}+S_{50}$ equals: | 1 | To solve for $S_n = 1 - 2 + 3 - 4 + \cdots + (-1)^{n-1}n$, we need to consider the cases when $n$ is even and when $n$ is odd separately.
1. **Case 1: $n$ is even.**
- Pair the terms: $(1-2) + (3-4) + \cdots + (n-1 - n)$.
- Each pair sums to $-1$.
- There are $n/2$ pairs.
- Therefore, $S_n = -1 \times (n/2... |
The perimeter of the polygon shown is | 28 | To solve this problem, we need to determine the perimeter of the polygon. The solution suggests visualizing the polygon as part of a rectangle and then using the properties of the rectangle to find the perimeter of the polygon.
1. **Visualize the Polygon as Part of a Rectangle**:
- Assume the polygon is part of a r... |
In the adjoining figure $TP$ and $T'Q$ are parallel tangents to a circle of radius $r$, with $T$ and $T'$ the points of tangency. $PT''Q$ is a third tangent with $T''$ as a point of tangency. If $TP=4$ and $T'Q=9$ then $r$ is
[asy]
unitsize(45); pair O = (0,0); pair T = dir(90); pair T1 = dir(270); pair T2 = dir(25); p... | 6 | 1. **Identify the Geometry and Given Information:**
- The problem involves a circle with two parallel tangents $TP$ and $T'Q$, and a third tangent $PT''Q$.
- $T$ and $T'$ are points of tangency for the tangents $TP$ and $T'Q$, respectively.
- $T''$ is the point of tangency for the tangent $PT''Q$.
- Given l... |
$\frac{2}{1-\frac{2}{3}}=$ | 6 | 1. **Simplify the denominator**: Start by simplifying the expression inside the denominator:
\[
1 - \frac{2}{3} = \frac{3}{3} - \frac{2}{3} = \frac{1}{3}
\]
2. **Substitute back into the fraction**: Replace the simplified denominator back into the original fraction:
\[
\frac{2}{1 - \frac{2}{3}} = \frac{... |
What is the remainder when $3^0 + 3^1 + 3^2 + \cdots + 3^{2009}$ is divided by 8? | 4 |
#### Step-by-step Analysis:
**Solution 1:**
1. **Observation of Pattern:** Notice that $3^0 + 3^1 + 3^2 + 3^3 = 1 + 3 + 9 + 27 = 40$, which is divisible by 8. This implies that any sum of four consecutive powers of 3 starting from a power of 0 modulo 4 is divisible by 8.
2. **Grouping Terms:** The sum $3^0 + 3^1 + 3^... |
In $\triangle PAT,$ $\angle P=36^{\circ},$ $\angle A=56^{\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\overline{TP}$ and $\overline{TA},$ respectively, so that $PU=AG=1.$ Let $M$ and $N$ be the midpoints of segments $\overline{PA}$ and $\overline{UG},$ respectively. What is the degree measure of the acute ang... | 80 | 1. **Identify Given Information and Setup:**
- In $\triangle PAT$, we have $\angle P = 36^\circ$, $\angle A = 56^\circ$, and $PA = 10$.
- Points $U$ and $G$ are on $\overline{TP}$ and $\overline{TA}$ respectively, such that $PU = AG = 1$.
- $M$ and $N$ are midpoints of $\overline{PA}$ and $\overline{UG}$ respe... |
In $\triangle ABC$ with integer side lengths, $\cos A = \frac{11}{16}$, $\cos B = \frac{7}{8}$, and $\cos C = -\frac{1}{4}$. What is the least possible perimeter for $\triangle ABC$? | 9 |
#### Step 1: Verify the values of $\sin A$, $\sin B$, and $\sin C$
Given $\cos A = \frac{11}{16}$, $\cos B = \frac{7}{8}$, and $\cos C = -\frac{1}{4}$, we use the identity $\sin^2 x + \cos^2 x = 1$ to find $\sin A$, $\sin B$, and $\sin C$.
- For $\sin A$:
\[
\sin^2 A = 1 - \cos^2 A = 1 - \left(\frac{11}{16}\right... |
For how many ordered pairs $(b,c)$ of positive integers does neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions? | 6 | To determine the number of ordered pairs $(b,c)$ of positive integers such that neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions, we analyze the discriminants of these quadratic equations.
1. **Analyze the discriminants:**
- For $x^2 + bx + c = 0$, the discriminant must be $\Delta_1 = b^2 - 4c... |
Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League? | 91 | 1. **Calculate the cost of one T-shirt**:
Given that each T-shirt costs $5 more than a pair of socks, and each pair of socks costs $4, the cost of one T-shirt is:
\[
4 + 5 = 9 \text{ dollars}
\]
2. **Calculate the total cost for one member**:
Each member requires 2 pairs of socks and 2 T-shirts. The... |
What is the smallest prime number dividing the sum $3^{11}+5^{13}$? | 2 | 1. **Identify the Parity of Each Term**:
- $3^{11}$ is an odd number because any power of an odd number remains odd.
- $5^{13}$ is also an odd number for the same reason.
2. **Sum of Two Odd Numbers**:
- The sum of two odd numbers is always even. This can be shown by considering that an odd number can be exp... |
The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y| \leq 1000$? | 40 | 1. **Identify the axis of symmetry**: Given the focus of the parabola $P$ at $(0,0)$ and points $(4,3)$ and $(-4,-3)$ on $P$, we observe that the line connecting these points has a slope of $\frac{3 - (-3)}{4 - (-4)} = \frac{6}{8} = \frac{3}{4}$. This suggests that the axis of symmetry of the parabola makes an angle $\... |
Margie's car can go $32$ miles on a gallon of gas, and gas currently costs $4$ per gallon. How many miles can Margie drive on $20$ worth of gas? | 160 | 1. **Calculate the number of gallons Margie can buy with $\textdollar 20$:**
Given that the cost of one gallon of gas is $\textdollar 4$, the number of gallons Margie can buy with $\textdollar 20$ is calculated by dividing the total amount of money by the cost per gallon:
\[
\text{Number of gallons} = \frac{\t... |
Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$? | -10 | 1. **Calculate Harry's Answer ($H$):**
Harry follows the correct order of operations (PEMDAS/BODMAS), which prioritizes parentheses first. Thus, he calculates:
\[
H = 8 - (2 + 5) = 8 - 7 = 1
\]
2. **Calculate Terry's Answer ($T$):**
Terry ignores the parentheses and calculates each operation from left t... |
Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's average speed for her entire ride in miles per hour? | 5 | To find Carmen's average speed for her entire ride, we need to use the formula for average speed, which is given by:
\[
\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}
\]
From the problem statement, we are given:
- Total Distance = 35 miles
- Total Time = 7 hours
Using the formula, we calculate... |
What is $100(100-3)-(100 \cdot 100-3)$? | -297 | We are given the expression \(100(100-3)-(100\cdot100-3)\) and need to simplify it.
1. **Distribute and simplify inside the parentheses:**
\[
100(100-3) = 100 \times 97 = 9700
\]
Here, we calculate \(100 - 3 = 97\) and then multiply by 100.
2. **Simplify the second part of the expression:**
\[
100 \... |
Two lines with slopes $\frac{1}{2}$ and $2$ intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10$ ? | 6 | 1. **Identify the equations of the lines**:
- The line with slope $\frac{1}{2}$ passing through $(2,2)$ has the equation $y - 2 = \frac{1}{2}(x - 2)$, which simplifies to $y = \frac{1}{2}x + 1$.
- The line with slope $2$ passing through $(2,2)$ has the equation $y - 2 = 2(x - 2)$, which simplifies to $y = 2x - 2... |
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? | 40 | Let's denote the total number of flowers as $F$. We are given several pieces of information about the distribution of these flowers:
1. Six tenths of the flowers are pink, so the number of pink flowers is $\frac{6}{10}F$.
2. One third of the pink flowers are roses, so the number of pink roses is $\frac{1}{3} \times \f... |
Points $B$, $D$, and $J$ are midpoints of the sides of right triangle $ACG$. Points $K$, $E$, $I$ are midpoints of the sides of triangle $JDG$, etc. If the dividing and shading process is done 100 times (the first three are shown) and $AC=CG=6$, then the total area of the shaded triangles is nearest | 6 |
We are given a right triangle $ACG$ with $AC = CG = 6$ and a process of dividing and shading triangles that continues 100 times. We need to find the total area of the shaded triangles.
#### Step 1: Calculate the area of the initial triangle $ACG$.
Since $ACG$ is a right triangle with legs of length 6, its area is:
\[... |
What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50? | 3127 | 1. **Identify the smallest primes greater than 50**: The problem specifies that the number must not have any prime factors less than 50. The smallest primes greater than 50 are 53 and 59.
2. **Calculate the product of the smallest primes**: To find the smallest number that meets the criteria, we calculate the product ... |
The year 2002 is a palindrome (a number that reads the same from left to right as it does from right to left). What is the product of the digits of the next year after 2002 that is a palindrome? | 4 | To find the next palindromic year after 2002, we need to identify a year that reads the same forwards and backwards. We start by checking the years sequentially from 2003 onwards.
1. **Check year 2003**: This is not a palindrome.
2. **Check year 2004**: This is not a palindrome.
3. **Check year 2005**: This is not a p... |
Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters? | 400 | 1. **Define Variables:**
Let the length of the track be $x$ meters.
2. **Analyze First Meeting:**
Since Brenda and Sally start at diametrically opposite points, they start $x/2$ meters apart. When they first meet, Brenda has run 100 meters. Therefore, Sally must have run the remaining distance to complete half t... |
If $S = i^n + i^{-n}$, where $i = \sqrt{-1}$ and $n$ is an integer, then the total number of possible distinct values for $S$ is: | 3 | 1. **Understanding the Powers of $i$:**
Recall that $i = \sqrt{-1}$, and the powers of $i$ cycle every four terms:
- $i^1 = i$
- $i^2 = -1$
- $i^3 = -i$
- $i^4 = 1$
- $i^5 = i$, and so on.
2. **Expression for $i^{-n}$:**
We know that $i^{-n} = \frac{1}{i^n}$. Using the property of $i$, $\frac{... |
All $20$ diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect? | 70 | To solve this problem, we need to count the number of distinct intersection points formed by the diagonals of a regular octagon. We will use combinatorial geometry to determine this count.
#### Step 1: Counting the Total Number of Diagonals
In an $n$-sided polygon, the number of diagonals can be calculated using the f... |
Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$. The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of this star. What is the difference between the maximum and the minimum possible values of $s$? | 0 | 1. **Identify the Geometry of the Pentagon**: Given that $ABCDE$ is an equiangular convex pentagon, each interior angle is $\frac{180^\circ \times (5-2)}{5} = 108^\circ$.
2. **Extend the Sides to Form a Star**: Extending the sides of the pentagon forms a five-pointed star. The vertices of this star are the intersectio... |
Find the units digit of the decimal expansion of $\left(15 + \sqrt{220}\right)^{19} + \left(15 + \sqrt{220}\right)^{82}$. | 9 | 1. **Define the expressions**:
Let $A = 15 + \sqrt{220}$ and $B = 15 - \sqrt{220}$.
2. **Use the properties of conjugates**:
Note that $A$ and $B$ are conjugates, and any even power of $\sqrt{220}$ in their binomial expansions will result in rational numbers, while odd powers will result in irrational numbers t... |
If $m$ and $n$ are the roots of $x^2+mx+n=0$, $m \ne 0$, $n \ne 0$, then the sum of the roots is: | -1 | 1. **Identify the coefficients and roots of the polynomial:**
Given the quadratic equation $x^2 + mx + n = 0$, where $m$ and $n$ are the roots of the equation.
2. **Apply Vieta's formulas:**
Vieta's formulas state that for a quadratic equation $ax^2 + bx + c = 0$ with roots $r$ and $s$, the sum of the roots ... |
Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last ... | 7 |
We need to find the smallest initial number $N$ such that Bernardo wins the game. The game involves Bernardo doubling the number and Silvia adding 50 to it. The sequence of operations will continue until the number exceeds 999.
#### Step-by-step Analysis:
1. **Sequence of Operations**:
- Bernardo's operation: $B... |
The number $a=\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying
\[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\]is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\}=x- \lfloor x \rfloor... | 929 | 1. **Define Variables**:
Let $w = \lfloor x \rfloor$ and $f = \{x\}$ denote the whole part and the fractional part of $x$, respectively. Thus, $x = w + f$ where $0 \leq f < 1$.
2. **Rewrite the Equation**:
The given equation is:
\[
\lfloor x \rfloor \cdot \{x\} = a \cdot x^2
\]
Substituting $w$ and $... |
An equilateral triangle of side length $10$ is completely filled in by non-overlapping equilateral triangles of side length $1$. How many small triangles are required? | 100 | To solve this problem, we need to determine how many small equilateral triangles of side length $1$ can fit into a larger equilateral triangle of side length $10$.
#### Method 1: Summing the Number of Triangles in Each Row
1. **Visualize the Triangle**: Imagine the large triangle divided into rows of smaller equilater... |
A large urn contains $100$ balls, of which $36 \%$ are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be $72 \%$? (No red balls are to be removed.) | 36 | 1. **Calculate the initial number of red and blue balls:**
Given that $36\%$ of the balls are red, and there are $100$ balls in total, the number of red balls is:
\[
0.36 \times 100 = 36 \text{ red balls}
\]
The rest of the balls are blue, so the number of blue balls is:
\[
100 - 36 = 64 \text{ blu... |
Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. How many monotonous positive integers are there? | 1524 | 1. **Understanding Monotonous Numbers**:
A monotonous number is defined as a one-digit number or a number whose digits form a strictly increasing or strictly decreasing sequence.
2. **Counting One-Digit Monotonous Numbers**:
There are 9 one-digit monotonous numbers (1 through 9).
3. **Counting Multi-Digit Incre... |
Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d(12)=6$. (This function is known as the divisor function.) Let\[f(n)=\frac{d(n)}{\sqrt [3]n}.\]There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\ne N$. Wh... | 9 | 1. **Understanding the function $f(n)$**: We define $f(n) = \frac{d(n)}{\sqrt[3]{n}}$, where $d(n)$ is the number of divisors of $n$. We need to find the integer $N$ such that $f(N) > f(n)$ for all $n \neq N$.
2. **Prime factorization and divisor function**: For any integer $n$ with prime factorization $n = \prod_{i=1... |
A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$? | 9 | 1. **Identify the formula for the sum of the first $N$ natural numbers**: The sum of the first $N$ natural numbers is given by the formula:
\[
S = 1 + 2 + 3 + \cdots + N = \frac{N(N+1)}{2}
\]
We need to find $N$ such that this sum equals $2016$.
2. **Set up the equation**:
\[
\frac{N(N+1)}{2} = 2016
... |
Lines $L_1, L_2, \dots, L_{100}$ are distinct. All lines $L_{4n}$, where $n$ is a positive integer, are parallel to each other. All lines $L_{4n-3}$, where $n$ is a positive integer, pass through a given point $A$. The maximum number of points of intersection of pairs of lines from the complete set $\{L_1, L_2, \dots, ... | 4351 | We begin by partitioning the set of lines $\{L_1, L_2, \dots, L_{100}\}$ into three subsets based on their properties:
1. **Set $X$**: Contains lines $L_n$ where $n \equiv 0 \pmod{4}$. These lines are all parallel to each other.
2. **Set $Y$**: Contains lines $L_n$ where $n \equiv 1 \pmod{4}$. These lines all pass thr... |
A particle is placed on the parabola $y = x^2- x -6$ at a point $P$ whose $y$-coordinate is $6$. It is allowed to roll along the parabola until it reaches the nearest point $Q$ whose $y$-coordinate is $-6$. The horizontal distance traveled by the particle (the numerical value of the difference in the $x$-coordinates of... | 4 | To solve this problem, we need to find the $x$-coordinates of points $P$ and $Q$ on the parabola $y = x^2 - x - 6$ where $y = 6$ and $y = -6$, respectively, and then calculate the horizontal distance between these points.
1. **Finding $x$-coordinates for $P$ where $y = 6$:**
\[
x^2 - x - 6 = 6
\]
Simplifyi... |
A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list? | 225 | 1. **Understanding the Problem:**
- We have a list of $2018$ positive integers.
- The mode (most frequently occurring number) appears exactly $10$ times.
- We need to find the least number of distinct values in the list.
2. **Setting Up the Equation:**
- To minimize the number of distinct values, we should... |
How many ordered triples of integers $(a,b,c)$, with $a \ge 2$, $b\ge 1$, and $c \ge 0$, satisfy both $\log_a b = c^{2005}$ and $a + b + c = 2005$? | 2 | We are given two conditions:
1. $\log_a b = c^{2005}$
2. $a + b + c = 2005$
From the first condition, we can rewrite it using the definition of logarithms:
\[ a^{c^{2005}} = b \]
We will analyze the possible values of $c$ and their implications on $a$ and $b$.
#### Case 1: $c = 0$
From the equation $a^{c^{2005}} = b... |
What is the value of \(2^{1+2+3}-(2^1+2^2+2^3)?\) | 50 | 1. **Simplify the Exponent:**
The expression given is $2^{1+2+3} - (2^1 + 2^2 + 2^3)$. First, simplify the exponent in the term $2^{1+2+3}$:
\[
1+2+3 = 6
\]
Thus, the expression becomes:
\[
2^6 - (2^1 + 2^2 + 2^3)
\]
2. **Evaluate Powers of 2:**
Calculate each power of 2:
\[
2^6 = ... |
Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of $10$ hours per week helping around the house for $6$ weeks. For the first $5$ weeks she helps around the house for $8$, $11$, $7$, $12$ and $10$ hours. How many hours must she work for the final week to earn the ticket... | 12 | 1. **Define the variable**: Let $x$ be the number of hours Theresa needs to work in the sixth week to achieve an average of 10 hours per week over 6 weeks.
2. **Set up the equation for the average**: The average number of hours worked per week is given by the total hours worked divided by the number of weeks. Therefor... |
Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of $50$ units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles? | 128 | 1. **Understanding the Problem:**
- We are given a rectangle with a perimeter of 50 units.
- The side lengths of the rectangle are integers.
- We need to find the difference between the largest and smallest possible areas of such rectangles.
2. **Expressing the Perimeter in Terms of Side Lengths:**
- Let t... |
After school, Maya and Naomi headed to the beach, $6$ miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds? | 24 | 1. **Convert Naomi's travel time to hours**: Naomi travels $6$ miles in $10$ minutes. To convert minutes to hours, we divide by $60$:
\[
10 \text{ minutes} = \frac{10}{60} \text{ hours} = \frac{1}{6} \text{ hours}
\]
2. **Calculate Naomi's average speed**: Using the formula for speed, which is $\text{speed} =... |
Nine chairs in a row are to be occupied by six students and Professors Alpha, Beta and Gamma. These three professors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can Professors Alpha, Beta and Gamma choose their chairs? | 60 | 1. **Identify the constraints for the professors' seating**: Professors Alpha, Beta, and Gamma must each be seated between two students. This means they cannot occupy the first or last chair in the row, as these positions do not allow a student to be seated on both sides.
2. **Determine the possible seats for the prof... |
Consider those functions $f$ that satisfy $f(x+4)+f(x-4) = f(x)$ for all real $x$. Any such function is periodic, and there is a least common positive period $p$ for all of them. Find $p$. | 24 | 1. **Identify the functional equation and its implications**: We are given that $f(x+4) + f(x-4) = f(x)$ for all real $x$. This equation suggests a relationship between function values at intervals of 4 units.
2. **Derive additional relationships**: Substitute $x+4$ for $x$ in the original equation:
\[
f(x+8) + ... |
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? | 20 | 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 combined rate**:
- The combined rate of frosting cupcakes when Cagney and Lacey work together can be calculated using the formul... |
What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock? | 60 | 1. **Understanding the Clock's Degree System**: A clock face is a full circle, which contains 360 degrees. Since there are 12 hours marked on the clock, the angle between each hour is calculated by dividing the total degrees by the number of hours:
\[
\frac{360^\circ}{12} = 30^\circ
\]
This means each hour ... |
There are $10$ people standing equally spaced around a circle. Each person knows exactly $3$ of the other $9$ people: the $2$ people standing next to her or him, as well as the person directly across the circle. How many ways are there for the $10$ people to split up into $5$ pairs so that the members of each pair know... | 12 | To solve this problem, we need to consider the relationships each person has with others in the circle. Each person knows exactly three others: the two adjacent people and the one directly across the circle. We need to find how many ways we can pair up these 10 people such that each pair consists of people who know eac... |
A man born in the first half of the nineteenth century was $x$ years old in the year $x^2$. He was born in: | 1806 | 1. **Formulate the equation**: Given that a man was $x$ years old in the year $x^2$, the year of birth can be expressed as $x^2 - x$.
2. **Determine the range of $x$**: Since he was born in the first half of the 19th century, we have $x^2 - x < 1850$.
3. **Estimate the possible values of $x$**:
- Calculate for $x ... |
On a $50$-question multiple choice math contest, students receive $4$ points for a correct answer, $0$ points for an answer left blank, and $-1$ point for an incorrect answer. Jesse’s total score on the contest was $99$. What is the maximum number of questions that Jesse could have answered correctly? | 29 | Let's define the variables:
- Let $a$ be the number of questions Jesse answered correctly.
- Let $b$ be the number of questions Jesse left blank.
- Let $c$ be the number of questions Jesse answered incorrectly.
Given the total number of questions is $50$, we have the equation:
\[ a + b + c = 50 \]
Jesse's scoring rul... |
In how many ways can $10001$ be written as the sum of two primes? | 0 | 1. **Identify the Parity of the Sum**: The number $10001$ is odd. To express an odd number as the sum of two primes, one of the primes must be even and the other must be odd. This is because the sum of two odd numbers is even, and the sum of two even numbers is also even. Thus, the only way to get an odd sum is to add ... |
In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200$.
\[1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots, 200, 200, \ldots , 200\]What is the median of the numbers in this list? | 142 | 1. **Understanding the Problem**: We are given a list where each integer $n$ from 1 to 200 appears exactly $n$ times. We need to find the median of this list.
2. **Calculating the Total Number of Elements**: The total number of elements in the list is the sum of the first 200 natural numbers, since $n$ appears $n$ tim... |
Mr. Green measures his rectangular garden by walking two of the sides and finds that it is $15$ steps by $20$ steps. Each of Mr. Green's steps is $2$ feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden? | 600 | 1. **Convert steps to feet**:
Mr. Green's garden measures $15$ steps by $20$ steps. Given that each step is $2$ feet long, we convert the dimensions from steps to feet:
\[
15 \text{ steps} \times 2 \text{ feet/step} = 30 \text{ feet}
\]
\[
20 \text{ steps} \times 2 \text{ feet/step} = 40 \text{ feet}... |
A circular table has 60 chairs around it. There are $N$ people seated at this table in such a way that the next person seated must sit next to someone. What is the smallest possible value for $N$? | 20 | To find the smallest possible value for $N$, we need to ensure that any new person seated must sit next to someone already seated. We will analyze the seating arrangement to determine the minimum $N$ that satisfies this condition.
1. **Understanding the seating arrangement**: The table has 60 chairs arranged in a circ... |
A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{\text{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate t... | 1925 | To solve this problem, we need to determine the smallest number of coins in the chest initially, such that each pirate receives a whole number of coins when taking their share according to the given rule. The rule states that the $k^{\text{th}}$ pirate takes $\frac{k}{12}$ of the remaining coins.
Let $x$ be the initia... |
What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$? | 1 | 1. **Expand and simplify the expression**: Start by expanding the given expression:
\[
(xy-1)^2 + (x+y)^2 = (xy-1)^2 + x^2 + 2xy + y^2.
\]
Expanding $(xy-1)^2$ gives:
\[
(xy-1)^2 = x^2y^2 - 2xy + 1.
\]
Substituting this into the expression, we get:
\[
x^2y^2 - 2xy + 1 + x^2 + 2xy + y^2 = x... |
In the expression $(\underline{\qquad}\times\underline{\qquad})+(\underline{\qquad}\times\underline{\qquad})$ each blank is to be filled in with one of the digits $1,2,3,$ or $4,$ with each digit being used once. How many different values can be obtained? | 4 | To solve this problem, we need to determine how many distinct values can be obtained by filling in the blanks in the expression $\left(a \times b\right) + \left(c \times d\right)$, where $a, b, c, d$ are the digits $1, 2, 3, 4$, each used exactly once.
1. **Identify Possible Products**: First, we calculate all possibl... |
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$? | 20 | 1. **Calculate the total amount spent and the individual shares:**
- Total amount spent by Tom, Dorothy, and Sammy:
\[
105 + 125 + 175 = 405 \text{ dollars}
\]
- Since they agreed to split the costs evenly, each person's share is:
\[
\frac{405}{3} = 135 \text{ dollars}
\]
2. **Deter... |
What is the smallest result that can be obtained from the following process?
Choose three different numbers from the set $\{3,5,7,11,13,17\}$.
Add two of these numbers.
Multiply their sum by the third number. | 36 | To find the smallest result from the given process, we need to consider the operations involved: addition and multiplication. Both operations are increasing functions when dealing with positive integers. Therefore, to minimize the result, we should ideally start with the smallest numbers in the set.
1. **Select the sm... |
A computer can do $10,000$ additions per second. How many additions can it do in one hour? | 36000000 | 1. **Identify the rate of additions per second**: The computer can perform $10,000$ additions per second.
2. **Convert the rate to additions per hour**:
- There are $3600$ seconds in one hour.
- To find the total number of additions in one hour, multiply the number of additions per second by the number of secon... |
In $\triangle ABC$, $AB=6$, $AC=8$, $BC=10$, and $D$ is the midpoint of $\overline{BC}$. What is the sum of the radii of the circles inscribed in $\triangle ADB$ and $\triangle ADC$? | 3 | 1. **Identify the triangles and their properties**:
Given $\triangle ABC$ with $AB = 6$, $AC = 8$, and $BC = 10$. Point $D$ is the midpoint of $\overline{BC}$, so $BD = DC = \frac{10}{2} = 5$.
2. **Use the triangle side lengths to find the length of $AD$**:
Since $D$ is the midpoint of $BC$, we can use Apolloni... |
The $8 \times 18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y$? | 6 | 1. **Understanding the problem**: We are given an $8 \times 18$ rectangle that is cut into two congruent hexagons. These hexagons are rearranged to form a square. We need to find the value of $y$, which is a dimension in the hexagon.
2. **Area of the rectangle**: The area of the rectangle is calculated as:
\[
\t... |
A stone is dropped into a well and the report of the stone striking the bottom is heard $7.7$ seconds after it is dropped. Assume that the stone falls $16t^2$ feet in t seconds and that the velocity of sound is $1120$ feet per second. The depth of the well is: | 784 | 1. **Define Variables:**
Let $d$ be the depth of the well in feet, $t_1$ be the time in seconds it takes for the stone to fall to the bottom of the well, and $t_2$ be the time in seconds it takes for the sound to travel back up to the top of the well. We know that the total time for the stone to fall and the sound t... |
Josanna's test scores to date are $90, 80, 70, 60,$ and $85$. Her goal is to raise her test average at least $3$ points with her next test. What is the minimum test score she would need to accomplish this goal? | 95 | 1. **Calculate the current average score**:
Josanna's current test scores are $90, 80, 70, 60,$ and $85$. The average of these scores is calculated as follows:
\[
\text{Average} = \frac{90 + 80 + 70 + 60 + 85}{5} = \frac{385}{5} = 77
\]
2. **Determine the desired average score**:
Josanna wants to raise ... |
A positive integer $N$ is a palindrome if the integer obtained by reversing the sequence of digits of $N$ is equal to $N$. The year 1991 is the only year in the current century with the following 2 properties:
(a) It is a palindrome
(b) It factors as a product of a 2-digit prime palindrome and a 3-digit prime palindrom... | 4 | 1. **Identify the properties of a four-digit palindrome**: A four-digit palindrome has the form $abba$, where $a$ and $b$ are digits and $a \neq 0$. For example, $1991$ is a palindrome because reversing the digits gives the same number.
2. **Check divisibility by 11**: A number $abba$ is divisible by 11 if and only if... |
How many solutions does the equation $\tan(2x)=\cos(\frac{x}{2})$ have on the interval $[0,2\pi]?$ | 5 | To solve the equation $\tan(2x) = \cos\left(\frac{x}{2}\right)$ on the interval $[0, 2\pi]$, we analyze the behavior and intersections of the functions $y = \tan(2x)$ and $y = \cos\left(\frac{x}{2}\right)$.
1. **Properties of $y = \tan(2x)$**:
- The function $\tan(2x)$ has a period of $\frac{\pi}{2}$.
- Vertical... |
[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair A=(0,0), B=(16,0), C=(16,16), D=(0,16), E=(32,0), F=(48,0), G=(48,16), H=(32,16), I=(0,8), J=(10,8), K=(10,16), L=(32,6), M=(40,6), N=(40,16);
draw(A--B--C--D--A^^E--F--G--H--E^^I--J--K^^L--M--N);
label("S", (18,8));
label("S", (50,8));
label("Figure 1", (A+B)/2, S);
... | 13 | To determine the smallest integer value of $S$ for which the table can be moved as desired without tilting it or taking it apart, we need to consider the diagonal of the table, as this is the longest dimension when the table is rotated.
1. **Calculate the diagonal of the table**: The table has dimensions $8'$ by $10'$... |
Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it ... | 4 | 1. **Identify the movement pattern**: The bug starts at point 5. According to the rules, since 5 is odd, it moves one point to point 1.
2. **Continue the pattern**:
- From point 1 (odd), it moves one point to point 2.
- From point 2 (even), it moves two points to point 4.
- From point 4 (even), it moves two p... |
A pyramid has a square base $ABCD$ and vertex $E$. The area of square $ABCD$ is $196$, and the areas of $\triangle ABE$ and $\triangle CDE$ are $105$ and $91$, respectively. What is the volume of the pyramid? | 784 | 1. **Identify the side length of the square base**: Given the area of square $ABCD$ is $196$, we find the side length by taking the square root:
\[
s = \sqrt{196} = 14
\]
2. **Calculate the heights of triangles $ABE$ and $CDE$**: The area of a triangle is given by $\frac{1}{2} \times \text{base} \times \text{... |
The dimensions of a rectangular box in inches are all positive integers and the volume of the box is $2002$ in$^3$. Find the minimum possible sum of the three dimensions. | 38 | 1. **Understanding the Problem**: We need to find the minimum possible sum of the three dimensions of a rectangular box with a volume of $2002$ cubic inches, where all dimensions are positive integers.
2. **Prime Factorization of the Volume**: The volume of the box is given as $2002$ cubic inches. We start by finding ... |
Let $z$ be a complex number satisfying $12|z|^2=2|z+2|^2+|z^2+1|^2+31.$ What is the value of $z+\frac 6z?$ | -2 | 1. **Rewrite the given equation using $w \overline{w} = |w|^2$:**
\[
12|z|^2 = 2|z+2|^2 + |z^2+1|^2 + 31
\]
Expressing in terms of $z$ and $\overline{z}$:
\[
12z\overline{z} = 2(z+2)(\overline{z}+2) + (z^2+1)(\overline{z}^2+1) + 31
\]
Expanding the terms:
\[
12z\overline{z} = 2z\overline{z... |
A child's wading pool contains 200 gallons of water. If water evaporates at the rate of 0.5 gallons per day and no other water is added or removed, how many gallons of water will be in the pool after 30 days? | 185 | 1. **Identify the initial amount of water**: The pool initially contains 200 gallons of water.
2. **Determine the rate of evaporation**: Water evaporates at a rate of 0.5 gallons per day.
3. **Calculate the total evaporation over 30 days**: Multiply the daily evaporation rate by the number of days:
\[
30 \text{... |
How many 4-digit numbers greater than 1000 are there that use the four digits of 2012? | 9 | To solve this problem, we need to determine how many distinct 4-digit numbers can be formed using the digits of 2012, ensuring that each number is greater than 1000. We will use casework based on the leading digit of the number.
#### Case 1: Leading digit is 1
The digits available for the remaining three positions are... |
A number of linked rings, each $1$ cm thick, are hanging on a peg. The top ring has an outside diameter of $20$ cm. The outside diameter of each of the outer rings is $1$ cm less than that of the ring above it. The bottom ring has an outside diameter of $3$ cm. What is the distance, in cm, from the top of the top ring ... | 173 | 1. **Identify the number of rings**:
The top ring has an outside diameter of $20$ cm, and the bottom ring has an outside diameter of $3$ cm. Each subsequent ring's outside diameter decreases by $1$ cm. Therefore, the number of rings is $20 - 3 + 1 = 18$ rings.
2. **Calculate the inside diameters**:
The inside d... |
A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base $5$.
Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base $5$ and the elements of the set
$\{V, W, X, Y, Z\}$. Using this correspondence, the cryptog... | 108 | 1. **Identify the pattern and the base-5 system**: The problem states that three consecutive integers are coded as $VYZ, VYX, VVW$. We know that in base-5, each digit represents powers of 5, starting from the rightmost digit (units place) to the leftmost digit.
2. **Analyze the change from $VYX$ to $VVW$**: Since $VYX... |
Let $R$ be a set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of possible values of the median of $R$? | 7 | 1. **Understanding the Problem**: We need to find the number of possible values for the median of a set $R$ containing nine distinct integers, where six of these integers are already given as $2$, $3$, $4$, $6$, $9$, and $14$.
2. **Properties of the Median**: Since $R$ has nine elements, the median will be the fifth e... |
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$? | 385 | 1. **Determine the total number of teams**: Given that each team won $10$ games and lost $10$ games, and there were no ties, each team played $20$ games. Since each game involves two teams, the total number of games played is $\frac{21 \times 20}{2} = 210$. This confirms there are $21$ teams in the tournament.
2. **To... |
Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars? | 80 | 1. **Identify the total sales**: The problem provides the sales for each month from January through April as $100$, $60$, $40$, and $120$ dollars respectively.
2. **Calculate the total sales**: Add the sales for each month to find the total sales over the four months.
\[
100 + 60 + 40 + 120 = 320 \text{ dollars}... |
The twelve-sided figure shown has been drawn on $1 \text{ cm}\times 1 \text{ cm}$ graph paper. What is the area of the figure in $\text{cm}^2$?
[asy] unitsize(8mm); for (int i=0; i<7; ++i) { draw((i,0)--(i,7),gray); draw((0,i+1)--(7,i+1),gray); } draw((1,3)--(2,4)--(2,5)--(3,6)--(4,5)--(5,5)--(6,4)--(5,3)--(5,2)--... | 13 | To find the area of the twelve-sided figure, we can break it down into simpler shapes whose areas we can easily calculate. The figure is composed of several unit squares and triangles on a $1 \text{ cm} \times 1 \text{ cm}$ grid.
1. **Count the full unit squares**:
- There are 9 full unit squares completely inside... |
Which number in the array below is both the largest in its column and the smallest in its row? (Columns go up and down, rows go right and left.)
\[\begin{tabular}{ccccc} 10 & 6 & 4 & 3 & 2 \\ 11 & 7 & 14 & 10 & 8 \\ 8 & 3 & 4 & 5 & 9 \\ 13 & 4 & 15 & 12 & 1 \\ 8 & 2 & 5 & 9 & 3 \end{tabular}\] | 7 | To solve this problem, we need to identify the number that is both the largest in its column and the smallest in its row from the given array. We will proceed by first finding the largest number in each column and then checking if any of these numbers are also the smallest in their respective rows.
The array is:
\[
\b... |
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... |
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