problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Compute the value of $x$ such that
$\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\cdots\right)\left(1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots\right)=1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\cdots$. | 4 |
Herbert rolls 6 fair standard dice and computes the product of all of his rolls. If the probability that the product is prime can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$. | 2692 |
Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpicks wide, then the number of toothpicks used is | 430 |
What is the product of the solutions of the equation $45 = -x^2 - 4x?$ | -45 |
In triangle $ABC$, $BC = 23$, $CA = 27$, and $AB = 30$. Points $V$ and $W$ are on $\overline{AC}$ with $V$ on $\overline{AW}$, points $X$ and $Y$ are on $\overline{BC}$ with $X$ on $\overline{CY}$, and points $Z$ and $U$ are on $\overline{AB}$ with $Z$ on $\overline{BU}$. In addition, the points are positioned so that ... | 318 |
Find the maximum of $x^{2} y^{2} z$ under the condition that $x, y, z \geq 0$ and $2 x + 3 x y^{2} + 2 z = 36$. | 144 |
Three fair six-sided dice are rolled (meaning all outcomes are equally likely). What is the probability that the numbers on the top faces form an arithmetic sequence with a common difference of 1? | $\frac{1}{9}$ |
Bella bought stamps at the post office. Some of the stamps had a snowflake design, some had a truck design, and some had a rose design. Bella bought 11 snowflake stamps. She bought 9 more truck stamps than snowflake stamps, and 13 fewer rose stamps than truck stamps. How many stamps did Bella buy in all? | The number of truck stamps is 11 + 9 = <<11+9=20>>20.
The number of rose stamps is 20 − 13 = <<20-13=7>>7.
Bella bought 11 + 20 + 7 = <<11+20+7=38>>38 stamps in all.
#### 38 |
Find \(x\) if
\[2 + 7x + 12x^2 + 17x^3 + \dotsb = 100.\] | 0.6 |
How many integers between 1 and 300 are multiples of both 2 and 5 but not of either 3 or 8? | 14 |
For how many integers \( n \), with \( 2 \leq n \leq 80 \), is \( \frac{(n-1)(n)(n+1)}{8} \) equal to an integer? | 49 |
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
[asy]
size(250);defaultpen(linewidth(0.8));
draw(ellipse(origin, 3, 1)... | 240 |
Chris and Paul each rent a different room of a hotel from rooms $1-60$. However, the hotel manager mistakes them for one person and gives "Chris Paul" a room with Chris's and Paul's room concatenated. For example, if Chris had 15 and Paul had 9, "Chris Paul" has 159. If there are 360 rooms in the hotel, what is the pro... | \frac{153}{1180} |
Evaluate the infinite geometric series: $$\frac{5}{3} - \frac{3}{5} + \frac{9}{25} - \frac{27}{125} + \dots$$ | \frac{125}{102} |
The perimeter of a rectangle is 30 inches. One side of the rectangle is fixed at 7 inches. What is the number of square inches in the maximum possible area for this rectangle? | 56 |
Let $f$ be a quadratic function which satisfies the following condition. Find the value of $\frac{f(8)-f(2)}{f(2)-f(1)}$ .
For two distinct real numbers $a,b$ , if $f(a)=f(b)$ , then $f(a^2-6b-1)=f(b^2+8)$ . | 13 |
Given the equation of an ellipse is $\dfrac {x^{2}}{a^{2}} + \dfrac {y^{2}}{b^{2}} = 1 (a > b > 0)$, a line passing through the right focus of the ellipse and perpendicular to the $x$-axis intersects the ellipse at points $P$ and $Q$. The directrix of the ellipse on the right intersects the $x$-axis at point $M$. If $\... | \dfrac { \sqrt {3}}{3} |
Grayson drives a motorboat for 1 hour at 25 mph and then 0.5 hours for 20 mph. Rudy rows in his rowboat for 3 hours at 10 mph. How much farther, in miles, does Grayson go in his motorboat compared to Rudy? | Grayson first travels 1 * 25 = <<1*25=25>>25 miles
Then Grayson travels 0.5 * 20 = <<0.5*20=10>>10 miles
Grayson travels a total of 25 + 10 = <<25+10=35>>35 miles
Rudy travels 3 * 10 = <<3*10=30>>30 miles
Grayson travels 35 - 30 = <<35-30=5>>5 miles more than Rudy
#### 5 |
A right circular cone has base radius $r$ and height $h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to i... | 14 |
Given that 600 athletes are numbered from 001 to 600 and divided into three color groups (red: 001 to 311, white: 312 to 496, and yellow: 497 to 600), calculate the probability of randomly drawing an athlete wearing white clothing. | \frac{8}{25} |
Ephraim has two machines that make necklaces for his store. On Sunday the first machine made 45 necklaces. The second machine made 2.4 times as many necklaces as the first machine. How many necklaces were made in total on Sunday? | The second machine made 2.4 times as many necklaces as the first machine, so the second machine made 2.4 * 45 = <<2.4*45=108>>108 necklaces on Sunday.
In total, 45 + 108 = <<45+108=153>>153 necklaces were made on Sunday.
#### 153 |
Find the sum of all real numbers $x$ such that $5 x^{4}-10 x^{3}+10 x^{2}-5 x-11=0$. | 1 |
At the beginning of the day, Principal Kumar instructed Harold to raise the flag up the flagpole. The flagpole is 60 feet long, and when fully raised, the flag sits on the very top of the flagpole. Later that morning, Vice-principal Zizi instructed Harold to lower the flag to half-mast. So, Harold lowered the flag h... | Half of the distance up the flagpole is 60/2 = <<60/2=30>>30 feet.
Thus, Harold moved the flag 60 up + 30 down + 30 up + 60 down = <<60+30+30+60=180>>180 feet.
#### 180 |
Consider equations of the form $x^2 + bx + c = 0$. How many such equations have real roots and have coefficients $b$ and $c$ selected from the set of integers $\{1,2,3, 4, 5,6\}$? | 19 |
How many two-digit numbers have digits whose sum is a prime number? | 35 |
Simplify the expression $\dfrac{45}{28} \cdot \dfrac{49}{75} \cdot \dfrac{100}{63}$. | \frac{5}{3} |
Given that there are 20 cards numbered from 1 to 20 on a table, and Xiao Ming picks out 2 cards such that the number on one card is 2 more than twice the number on the other card, find the maximum number of cards Xiao Ming can pick. | 12 |
Triangle $ABC$ lies in the cartesian plane and has an area of $70$. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20),$ respectively, and the coordinates of $A$ are $(p,q).$ The line containing the median to side $BC$ has slope $-5.$ Find the largest possible value of $p+q.$
[asy]defaultpen(fontsize(8)); size(1... | 47 |
What is the largest number, with its digits all different, whose digits add up to 16? | 643210 |
A four-digit number \((xyzt)_B\) is called a stable number in base \(B\) if \((xyzt)_B = (dcba)_B - (abcd)_B\), where \(a \leq b \leq c \leq d\) are the digits \(x, y, z, t\) arranged in ascending order. Determine all the stable numbers in base \(B\).
(Problem from the 26th International Mathematical Olympiad, 1985) | (1001)_2, (3021)_4, (3032)_5, (3B/5, B/5-1, 4B/5-1, 2B/5)_B, 5 | B |
Given the function $f(x)=2\sin(\omega x+\varphi)$, where $(\omega > 0, |\varphi| < \frac{\pi}{2})$, the graph passes through the point $B(0,-1)$, and is monotonically increasing on the interval $\left(\frac{\pi}{18}, \frac{\pi}{3}\right)$. Additionally, the graph of $f(x)$ coincides with its original graph after being ... | -1 |
Billy and George are picking dandelions. At first, Billy picks 36 and then George picks 1/3 as many as Billy. When they see the pile, they each decide to pick 10 more each. How many have they picked on average? | George picked at first 12 because 36 / 3 = <<36/3=12>>12.
George picks 22 by the end because 12 + 10 = <<12+10=22>>22
Billy picks 66 by the end because 36 + 10 = <<36+10=46>>46
In total they picked 68 because 22 + 46 = <<22+46=68>>68
On average they picked 34 because 68 / 2 = <<68/2=34>>34
#### 34 |
In the arithmetic sequence $\{a\_n\}$, it is known that $a\_1 - a\_4 - a\_8 - a\_{12} + a\_{15} = 2$. Find the value of $S\_{15}$. | -30 |
Given the function $f(x) = \cos x \cdot \sin\left(x + \frac{\pi}{3}\right) - \sqrt{3}\cos^2x + \frac{\sqrt{3}}{4}$, where $x \in \mathbb{R}$.
(1) Find the interval of monotonic increase for $f(x)$.
(2) In an acute triangle $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respec... | \frac{3\sqrt{3}}{4} |
Let \( a \) be a nonzero real number. In the Cartesian coordinate system \( xOy \), the quadratic curve \( x^2 + ay^2 + a^2 = 0 \) has a focal distance of 4. Determine the value of \( a \). | \frac{1 - \sqrt{17}}{2} |
What is the sum of all integer solutions to $|n| < |n-3| < 9$? | -14 |
Suppose that \( ABCDEF \) is a regular hexagon with sides of length 6. Each interior angle of \( ABCDEF \) is equal to \( 120^{\circ} \).
(a) A circular arc with center \( D \) and radius 6 is drawn from \( C \) to \( E \). Determine the area of the shaded sector.
(b) A circular arc with center \( D \) and radius 6 i... | 18\pi - 27\sqrt{3} |
Find the minimum positive integer $k$ such that $f(n+k) \equiv f(n)(\bmod 23)$ for all integers $n$. | 2530 |
How many distinct arrangements of the letters in the word "balloon" are there? | 1260 |
Add $26_7 + 245_7.$ Express your answer in base 7. | 304_7 |
Given the functions $f(x)= \begin{cases} 2^{x-2}-1,x\geqslant 0 \\ x+2,x < 0 \end{cases}$ and $g(x)= \begin{cases} x^{2}-2x,x\geqslant 0 \\ \frac {1}{x},x < 0. \end{cases}$, find the sum of all the zeros of the function $f[g(x)]$. | \frac{1}{2} + \sqrt{3} |
Determine the sum of all single-digit replacements for $z$ such that the number ${36{,}z72}$ is divisible by both 6 and 4. | 18 |
Let \( x_{1}, x_{2}, \cdots, x_{n} \) and \( a_{1}, a_{2}, \cdots, a_{n} \) be two sets of arbitrary real numbers (where \( n \geqslant 2 \)) that satisfy the following conditions:
1. \( x_{1} + x_{2} + \cdots + x_{n} = 0 \)
2. \( \left| x_{1} \right| + \left| x_{2} \right| + \cdots + \left| x_{n} \right| = 1 \)
3. \( ... | 1/2 |
The diagonals of a rhombus measure 18 feet and 12 feet. What is the perimeter of the rhombus? Express your answer in simplest radical form. | 12\sqrt{13}\text{ feet} |
The integer $m$ is the largest positive multiple of $18$ such that every digit of $m$ is either $9$ or $0$. Compute $\frac{m}{18}$. | 555 |
Using the digits 0, 2, 3, 5, 7, how many four-digit numbers divisible by 5 can be formed if:
(1) Digits do not repeat;
(2) Digits can repeat. | 200 |
Calculate the area of the parallelogram formed by the vectors \( a \) and \( b \).
Given:
\[ a = 3p - 4q \]
\[ b = p + 3q \]
\[ |p| = 2 \]
\[ |q| = 3 \]
\[ \text{Angle between } p \text{ and } q \text{ is } \frac{\pi}{4} \] | 39\sqrt{2} |
Alice and Bob are playing the Smallest Positive Integer Game. Alice says, "My number is 24." Bob says, "What kind of silly smallest number is that? Every prime factor of your number is also a prime factor of my number."
What is the smallest possible number that Bob could have? (Remember that Bob's number has to be a p... | 6 |
Tall Tuna has twice as many fish as Jerk Tuna. If Jerk Tuna has one hundred forty-four fish, how many fish do they have together? | If Jerk Tuna has one hundred forty-four fish, then Tall Tuna has 2*144 = 288 fish
Together, they have 288+144= <<288+144=432>>432 fish
#### 432 |
If $(X-2)^8 = a + a_1(x-1) + \ldots + a_8(x-1)^8$, then the value of $\left(a_2 + a_4 + \ldots + a_8\right)^2 - \left(a_1 + a_3 + \ldots + a_7\right)^2$ is (Answer in digits). | -255 |
How many positive integers $n$ less than 100 have a corresponding integer $m$ divisible by 3 such that the roots of $x^2-nx+m=0$ are consecutive positive integers? | 32 |
Let ${ a\uparrow\uparrow b = {{{{{a^{a}}^a}^{\dots}}}^{a}}^{a}} $ , where there are $ b $ a's in total. That is $ a\uparrow\uparrow b $ is given by the recurrence \[ a\uparrow\uparrow b = \begin{cases} a & b=1 a^{a\uparrow\uparrow (b-1)} & b\ge2\end{cases} \] What is the remainder of $ 3\uparrow\uparrow( 3\upa... | 27 |
What is the largest number, with its digits all different and none of them being zero, whose digits add up to 20? | 9821 |
Suppose $p(x)$ is a monic cubic polynomial with real coefficients such that $p(3-2i)=0$ and $p(0)=-52$.
Determine $p(x)$ (in expanded form). | x^3-10x^2+37x-52 |
The ratio of books to pens that Arlo has to buy for his school supplies is 7:3. If he bought a total of 400 stationery from the store, calculate the number of books he bought. | The total ratio representing the stationery that Arlo has to buy for his school supplies is 7+3 = <<7+3=10>>10
The fraction representing the number of books that Arlo bought is 7/10, and since he bought a total of 400 books and pens, he bought 7/10*400 = <<7/10*400=280>>280 books.
#### 280 |
Regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1.$ Point $P$ lies inside the circle so that the region bounded by $\overline{PA_1},\overline{PA_2},$ and the minor arc $\widehat{A_1A_2}$ of the circle has area $\tfrac{1}{7},$ while the region bounded by $\overline{PA_3},\overline{PA_4},$ an... | 504 |
Find the smallest number, written using only ones and zeros, that would be divisible by 225. | 11111111100 |
Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positi... | 85 |
Let the roots of the cubic equation \(27x^3 - 81x^2 + 63x - 14 = 0\) be in geometric progression. Find the difference between the square of the largest root and the square of the smallest root. | \frac{5}{3} |
The sequence $(a_n)$ satisfies $a_1 = 1$ and $5^{(a_{n + 1} - a_n)} - 1 = \frac {1}{n + \frac {2}{3}}$ for $n \geq 1$. Let $k$ be the least integer greater than $1$ for which $a_k$ is an integer. Find $k$. | 41 |
In the diagram below, $\|\overrightarrow{OA}\| = 1,$ $\|\overrightarrow{OB}\| = 1,$ and $\|\overrightarrow{OC}\| = \sqrt{2}.$ Also, $\tan \angle AOC = 7$ and $\angle BOC = 45^\circ.$
[asy]
unitsize(2 cm);
pair A, B, C, O;
A = (1,0);
B = (-0.6,0.8);
C = (0.2,1.4);
O = (0,0);
draw(O--A,Arrow(6));
draw(O--B,Arrow(6))... | \left( \frac{5}{4}, \frac{7}{4} \right) |
If
\[x + \sqrt{x^2 - 1} + \frac{1}{x - \sqrt{x^2 - 1}} = 20,\]then find
\[x^2 + \sqrt{x^4 - 1} + \frac{1}{x^2 + \sqrt{x^4 - 1}}.\] | \frac{10201}{200} |
Determine the volume of the original cube given that one dimension is increased by $3$, another is decreased by $2$, and the third is left unchanged, and the volume of the resulting rectangular solid is $6$ more than that of the original cube. | (3 + \sqrt{15})^3 |
Given the ratio of the distance saved by taking a shortcut along the diagonal of a rectangular field to the longer side of the field is $\frac{1}{2}$, determine the ratio of the shorter side to the longer side of the rectangle. | \frac{3}{4} |
Given a sequence $\{a_{n}\}$ where $a_{1}=1$, and ${a}_{n}+(-1)^{n}{a}_{n+1}=1-\frac{n}{2022}$, let $S_{n}$ denote the sum of the first $n$ terms of the sequence $\{a_{n}\}$. Find $S_{2023}$. | 506 |
Circle $\Gamma$ has diameter $\overline{AB}$ with $AB = 6$ . Point $C$ is constructed on line $AB$ so that $AB = BC$ and $A \neq C$ . Let $D$ be on $\Gamma$ so that $\overleftrightarrow{CD}$ is tangent to $\Gamma$ . Compute the distance from line $\overleftrightarrow{AD}$ to the circumcenter of $... | 4\sqrt{3} |
Find the area of quadrilateral $EFGH$, given that $m\angle F = m \angle G = 135^{\circ}$, $EF=4$, $FG=6$, and $GH=8$. | 18\sqrt{2} |
A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are possible? | 351 |
Your video streaming subscription costs $14 a month. If you're evenly splitting the cost with your friend, how much do you pay in total after the first year for this service? | An even split is 50%. If the monthly subscription is $14, but I pay only 50% that means I pay every month $14*50%=$<<14*50*.01=7>>7
We know that in 1 year there are 12 months. Therefore if each month I pay $7, in 12 months I'd pay $7*12=$<<7*12=84>>84
#### 84 |
Kyle knows that $4 = 2^{5r+1}$. What is the value of $r$? Express your answer as a common fraction. | \frac{1}{5} |
Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest number possible of black edges is | 3 |
The line $y = 2x + 7$ is to be parameterized using vectors. Which of the following options are valid parameterizations?
(A) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 7 \end{pmatrix} + t \begin{pmatrix} 2 \\ 1 \end{pmatrix}$
(B) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -7/2 \\ 0 \end... | \text{B,E} |
Given condition p: $|5x - 1| > a$ and condition q: $x^2 - \frac{3}{2}x + \frac{1}{2} > 0$, please choose an appropriate real number value for $a$, and use the given two conditions as A and B to construct the proposition: If A, then B. Make sure the constructed original proposition is true, while its converse is false, ... | a = 4 |
In triangle \(ABC\), point \(N\) lies on side \(AB\) such that \(AN = 3NB\); the median \(AM\) intersects \(CN\) at point \(O\). Find \(AB\) if \(AM = CN = 7\) cm and \(\angle NOM = 60^\circ\). | 4\sqrt{7} |
The perimeter of triangle $BQN$ is $180$, and the angle $QBN$ is a right angle. A circle of radius $15$ with center $O$ on $\overline{BQ}$ is drawn such that it is tangent to $\overline{BN}$ and $\overline{QN}$. Given that $OQ=p/q$ where $p$ and $q$ are relatively prime positive integers, find $p+q$. | 79 |
How many positive three-digit integers with a $7$ in the units place are divisible by $21$? | 39 |
Jude bought three chairs for his house, all at the same price. He also bought a table that costs $50 and two sets of plates at $20 for each set. After giving the cashier $130, Jude got a $4 change. How much did each of the chairs cost? | The two sets of plates amount to $20 x 2 = $<<20*2=40>>40.
So the table and the two sets of plates amount to $50 + $40 = $<<50+40=90>>90 in all.
Since Jude received a $4 change, then he paid a total of $130 - $4 = $<<130-4=126>>126 for the chairs, tables, and plates.
Thus, the three chairs cost $126 - $90 = $<<126-90=3... |
Find the range of the function
\[f(x) = \left( \arccos \frac{x}{2} \right)^2 + \pi \arcsin \frac{x}{2} - \left( \arcsin \frac{x}{2} \right)^2 + \frac{\pi^2}{12} (x^2 + 6x + 8).\] | \left[ \frac{\pi^2}{4}, \frac{9 \pi^2}{4} \right] |
In the Cartesian coordinate system $xoy$, the parametric equation of line $l$ is $\begin{cases} x=1- \frac { \sqrt {3}}{2}t \\ y= \frac {1}{2}t\end{cases}$ (where $t$ is the parameter), and in the polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the equation o... | 2 \sqrt {3} |
Let \[f(x) =
\begin{cases}
2x^2 - 3&\text{if } x\le 2, \\
ax + 4 &\text{if } x>2.
\end{cases}
\]Find $a$ if the graph of $y=f(x)$ is continuous (which means the graph can be drawn without lifting your pencil from the paper). | \frac{1}{2} |
Given a prism \(ABC-A'B'C'\) with a base that is an equilateral triangle with side length 2, the lateral edge \(AA'\) forms a 45-degree angle with the edges \(AB\) and \(AC\) of the base. Point \(A'\) is equidistant from the planes \(ABC\) and \(BB'C'C\). Find \(A'A = \_\_\_\_\_ \). | \sqrt{6} |
What is the period of $y = \cos \frac{x}{2}$? | 4 \pi |
A right circular cone has a volume of $12\pi$ cubic centimeters. The height of the cone is 4 cm. How many centimeters is the circumference of the base of the cone, in terms of $\pi$? | 6\pi |
Solve the following equation and provide its root. If the equation has multiple roots, provide their product.
\[ \sqrt{2 x^{2} + 8 x + 1} - x = 3 \] | -8 |
Summer performs 5 sun salutation yoga poses as soon as she gets out of bed, on the weekdays. How many sun salutations will she perform throughout an entire year? | She performs 5 sun salutations on the weekdays so that’s 5 days so that’s 5*5 = <<5*5=25>>25 sun salutations a week
There are 52 weeks in a year and she performs 25 sun salutations every week for a total of 52*25 = <<52*25=1300>>1,300 sun salutations a year
#### 1300 |
The length of the segment between the points $(2a, a-4)$ and $(4, -1)$ is $2\sqrt{10}$ units. What is the product of all possible values for $a$? | -3 |
A regular dodecagon ($12$ sides) is inscribed in a circle with radius $r$ inches. The area of the dodecagon, in square inches, is: | 3r^2 |
Factorize the expression $27x^6 - 512y^6$ and find the sum of all integer coefficients in its factorized form. | 92 |
A larger square is constructed, and another square is formed inside it by connecting the midpoints of each side of the larger square. If the area of the larger square is 100, what is the area of the smaller square formed inside? | 25 |
For what real values of $a$ is the expression $\frac{a+3}{a^2-4}$ undefined? List your answers in increasing order separated by commas. | -2, 2 |
Simplify $16^{\frac{1}{2}}-625^{\frac{1}{2}}$. | -21 |
What is the measure of an orthogonal trihedral angle? What is the sum of the measures of polyhedral angles that share a common vertex, have no common interior points, and together cover the entire space? | 4\pi |
Moving only south and east along the line segments, how many paths are there from $A$ to $B$? [asy]
import olympiad; size(250); defaultpen(linewidth(0.8)); dotfactor=4;
for(int i = 0; i <= 9; ++i)
if (i!=4 && i !=5)
draw((2i,0)--(2i,3));
for(int j = 0; j <= 3; ++j)
draw((0,j)--(18,j));
draw((2*4,0)--(2*4,1));
draw(... | 160 |
What is the sum of the two solutions to the equation $54-15x-x^2=0$? | -15 |
Kate wants to buy a special pen as a gift for her friend. The pen costs $30, and Kate only has money for a third of that amount. How much more money does she need to buy the pen? | Currently, Kate has only 1/3 * 30 = $<<1/3*30=10>>10.
That means Kate needs 30 - 10 = $<<30-10=20>>20 more to buy the pen.
#### 20 |
Max attended 40 college courses in 2 years. Sid attended four times as many college courses as Max in the same period. What's the total of college courses they both attended? | If Sid attended four times as many college courses as Max, he attended 4*40=<<4*40=160>>160 college courses.
The total of college courses they both attended is 160+40 = <<160+40=200>>200
#### 200 |
The product of the base seven numbers $24_7$ and $30_7$ is expressed in base seven. What is the base seven sum of the digits of this product? | 6 |
In order to obtain steel for a specific purpose, the golden section method was used to determine the optimal addition amount of a specific chemical element. After several experiments, a good point on the optimal range $[1000, m]$ is in the ratio of 1618, find $m$. | 2000 |
Let \( z \) be a complex number such that \( |z| = 2 \). Find the maximum value of
\[
|(z - 2)(z + 2)^2|.
\] | 16 \sqrt{2} |
Given points P(-2,-3) and Q(5,3) in the xy-plane; point R(x,m) is such that x=2 and PR+RQ is a minimum. Find m. | \frac{3}{7} |
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