problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
There are $10$ seats in each of $10$ rows of a theatre and all the seats are numbered. What is the probablity that two friends buying tickets independently will occupy adjacent seats? | \dfrac{1}{55} |
Joey has 30 pages to read for history class tonight. He decided that he would take a break when he finished reading 70% of the pages assigned. How many pages must he read after he takes a break? | Joey will take a break after 30 * 0.7 = <<30*0.7=21>>21 pages
Joey will have to read 30 - 21 = <<30-21=9>>9 pages more.
#### 9 |
Tetrahedron $PQRS$ is such that $PQ=6$, $PR=5$, $PS=4\sqrt{2}$, $QR=3\sqrt{2}$, $QS=5$, and $RS=4$. Calculate the volume of tetrahedron $PQRS$.
**A)** $\frac{130}{9}$
**B)** $\frac{135}{9}$
**C)** $\frac{140}{9}$
**D)** $\frac{145}{9}$ | \frac{140}{9} |
Four vehicles were traveling on the highway at constant speeds: a car, a motorcycle, a scooter, and a bicycle. The car passed the scooter at 12:00, encountered the bicyclist at 14:00, and met the motorcyclist at 16:00. The motorcyclist met the scooter at 17:00 and caught up with the bicyclist at 18:00.
At what time did the bicyclist meet the scooter? | 15:20 |
Haruto has tomato plants in his backyard. This year the plants grew 127 tomatoes. Birds had eaten 19 of the tomatoes. He picked the rest. If Haruto gave half of his tomatoes to his friend, how many tomatoes does he have left? | After the birds ate 19 of the tomatoes Haruto has 127 - 19 = <<127-19=108>>108 tomatoes left.
After giving half of his tomatoes to his friend, Haruto had 108 / 2 = <<108/2=54>>54 tomatoes.
#### 54 |
In triangle \(ABC\), the height \(BD\) is equal to 11.2 and the height \(AE\) is equal to 12. Point \(E\) lies on side \(BC\) and \(BE : EC = 5 : 9\). Find side \(AC\). | 15 |
What is the sum of the eight terms in the arithmetic sequence $-2, 3, \dots, 33$? | 124 |
Given the function $f(x) = (m^2 - m - 1)x^{m^2 - 2m - 1}$, if it is a power function and is an increasing function on the interval $(0, +\infty)$, determine the real number $m$. | -1 |
A factory produces a certain product with an annual fixed cost of $250 million. When producing $x$ million units, an additional cost of $C(x)$ million is required. When the annual production volume is less than 80 million units, $C(x)= \frac {1}{3}x^{2}+10x$; when the annual production volume is not less than 80 million units, $C(x)=51x+ \frac {10000}{x}-1450$. Assume that each unit of the product is sold for $50 million and all produced units can be sold that year.
1. Write out the annual profit $L(x)$ (million) as a function of the annual production volume $x$ (million units).
2. What is the production volume when the factory's profit from producing this product is the highest? What is the maximum profit? | 1000 |
There are $2$ boys for every $3$ girls in Ms. Johnson's math class. If there are $30$ students in her class, what percent of them are boys? | 40\% |
Uri buys two burgers and a soda for $\$2.10$, and Gen buys a burger and two sodas for $\$2.40$. How many cents does a soda cost? | 90 |
Given the hyperbola $$E: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$$ with left and right vertices A and B, respectively. Let M be a point on the hyperbola such that ∆ABM is an isosceles triangle, and the area of its circumcircle is 4πa², then the eccentricity of the hyperbola E is _____. | \sqrt{2} |
Let $ABCD$ be a convex quadrilateral, and let $G_A,$ $G_B,$ $G_C,$ $G_D$ denote the centroids of triangles $BCD,$ $ACD,$ $ABD,$ and $ABC,$ respectively. Find $\frac{[G_A G_B G_C G_D]}{[ABCD]}.$
[asy]
unitsize(0.6 cm);
pair A, B, C, D;
pair[] G;
A = (0,0);
B = (7,1);
C = (5,-5);
D = (1,-3);
G[1] = (B + C + D)/3;
G[2] = (A + C + D)/3;
G[3] = (A + B + D)/3;
G[4] = (A + B + C)/3;
draw(A--B--C--D--cycle);
draw(G[1]--G[2]--G[3]--G[4]--cycle,red);
label("$A$", A, W);
label("$B$", B, NE);
label("$C$", C, SE);
label("$D$", D, SW);
dot("$G_A$", G[1], SE);
dot("$G_B$", G[2], W);
dot("$G_C$", G[3], NW);
dot("$G_D$", G[4], NE);
[/asy] | \frac{1}{9} |
Pima invested $400 in Ethereum. In the first week, it gained 25% in value. In the second week, it gained an additional 50% on top of the previous gain. How much is her investment worth now? | For the first week, her investment gain is $400 * 25% = $<<400*25*.01=100>>100.
This leaves her investment with a value of $400 + $100 = $<<400+100=500>>500.
For the second week, her investment increased by $500 * 50% = $<<500*50*.01=250>>250.
Pima's investment in Ethereum is now worth a total of $500 + $250 = $<<500+250=750>>750.
#### 750 |
In the trapezoid \(ABCD \) with \( AD \parallel BC \), the angle \( \angle ADB \) is twice the angle \( \angle ACB \). It is known that \( BC = AC = 5 \) and \( AD = 6 \). Find the area of the trapezoid. | 22 |
Let $\mathbf{a} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 0 \\ -1 \end{pmatrix}.$ Find the vector $\mathbf{v}$ that satisfies $\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}.$ | \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix} |
Given an even function $f(x)$ defined on $\mathbb{R}$, for $x \geq 0$, $f(x) = x^2 - 4x$
(1) Find the value of $f(-2)$;
(2) For $x < 0$, find the expression for $f(x)$;
(3) Let the maximum value of the function $f(x)$ on the interval $[t-1, t+1]$ (where $t > 1$) be $g(t)$, find the minimum value of $g(t)$. | -3 |
The sum of all numbers of the form $2k + 1$, where $k$ takes on integral values from $1$ to $n$ is: | $n(n+2)$ |
Xiao Hong asked Da Bai: "Please help me calculate the result of $999 \quad 9 \times 999 \quad 9$ and determine how many zeros appear in it."
2019 nines times 2019 nines
Da Bai quickly wrote a program to compute it. Xiao Hong laughed and said: "You don't need to compute the exact result to know how many zeros there are. I'll tell you it's....."
After calculating, Da Bai found Xiao Hong's answer was indeed correct. Xiao Hong's answer is $\qquad$. | 2018 |
Let \( A \) and \( B \) be the endpoints of a semicircular arc of radius \( 3 \). The arc is divided into five congruent arcs by four equally spaced points \( C_1, C_2, C_3, C_4 \). All chords of the form \( \overline{AC_i} \) or \( \overline{BC_i} \) are drawn. Find the product of the lengths of these eight chords. | 32805 |
What is the volume of a cube whose surface area is twice that of a cube with volume 1? | 2\sqrt{2} |
Given $x, y, z \in \mathbb{R}$ and $x^{2}+y^{2}+z^{2}=25$, find the maximum and minimum values of $x-2y+2z$. | -15 |
The quartic (4th-degree) polynomial P(x) satisfies $P(1)=0$ and attains its maximum value of $3$ at both $x=2$ and $x=3$ . Compute $P(5)$ . | -24 |
Express as a common fraction: $0.\overline5+0.\overline1-0.\overline3$ | \frac 13 |
Given a line and a point at a distance of $1 \mathrm{~cm}$ from the line. What is the volume of a cube where the point is one of the vertices and the line is one of the body diagonals? | \frac{3 \sqrt{6}}{4} |
A parallelogram-shaped paper WXYZ with an area of 7.17 square centimeters is placed on another parallelogram-shaped paper EFGH, as shown in the diagram. The intersection points A, C, B, and D are formed, and AB // EF and CD // WX. What is the area of the paper EFGH in square centimeters? Explain the reasoning. | 7.17 |
Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $f$ is a root of $g,$ and the $x$-coordinate of the vertex of $g$ is a root of $f,$ and both $f$ and $g$ have the same minimum value. If the graphs of the two polynomials intersect at the point $(100,-100),$ what is the value of $a + c$? | -400 |
For how many real numbers $a^{}_{}$ does the quadratic equation $x^2 + ax^{}_{} + 6a=0$ have only integer roots for $x^{}_{}$? | 10 |
Let \(p\) and \(q\) be relatively prime positive integers such that \(\dfrac pq = \dfrac1{2^1} + \dfrac2{4^2} + \dfrac3{2^3} + \dfrac4{4^4} + \dfrac5{2^5} + \dfrac6{4^6} + \cdots\), where the numerators always increase by 1, and the denominators alternate between powers of 2 and 4, with exponents also increasing by 1 for each subsequent term. Compute \(p+q\). | 169 |
The image of the point with coordinates $(-3,-1)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(5,3)$. Find $m+b$. | 1 |
Given that \( 990 \times 991 \times 992 \times 993 = \overline{966428 A 91 B 40} \), find the values of \( \overline{A B} \). | 50 |
Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area? | 8 |
In a polar coordinate system, the polar equation of curve C is $\rho=2\cos\theta+2\sin\theta$. Establish a Cartesian coordinate system with the pole as the origin and the positive x-axis as the polar axis. The parametric equation of line l is $\begin{cases} x=1+t \\ y= \sqrt{3}t \end{cases}$ (t is the parameter). Find the length of the chord that curve C cuts off on line l. | \sqrt{7} |
Find all values of the parameter \(a\) for which the quadratic trinomial \(\frac{1}{3} x^2 + \left(a+\frac{1}{2}\right) x + \left(a^2 + a\right)\) has two roots, the sum of the cubes of which is exactly 3 times their product. In your answer, specify the largest of such \(a\). | -1/4 |
In \\(\triangle ABC\\), \\(AB=BC\\), \\(\cos B=-\dfrac{7}{18}\\). If an ellipse with foci at points \\(A\\) and \\(B\\) passes through point \\(C\\), find the eccentricity of the ellipse. | \dfrac{3}{8} |
In a physical education class, students line up in four rows to do exercises. One particular class has over 30 students, with three rows having the same number of students and one row having one more student than the other three rows. What is the smallest possible class size for this physical education class? | 33 |
Given that points \( B \) and \( C \) are in the fourth and first quadrants respectively, and both lie on the parabola \( y^2 = 2px \) where \( p > 0 \). Let \( O \) be the origin, and \(\angle OBC = 30^\circ\) and \(\angle BOC = 60^\circ\). If \( k \) is the slope of line \( OC \), find the value of \( k^3 + 2k \). | \sqrt{3} |
Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius $20$ feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point $10$ vertical feet above the bottom? | 10 |
The parabolas defined by the equations $y=2x^2-4x+4$ and $y=-x^2-2x+4$ intersect at points $(a,b)$ and $(c,d)$, where $c\ge a$. What is $c-a$? Express your answer as a common fraction. | \frac{2}{3} |
Find the number of subsets $S$ of $\{1,2, \ldots, 48\}$ satisfying both of the following properties: - For each integer $1 \leq k \leq 24$, exactly one of $2 k-1$ and $2 k$ is in $S$. - There are exactly nine integers $1 \leq m \leq 47$ so that both $m$ and $m+1$ are in $S$. | 177100 |
What is the smallest multiple of 7 that is greater than -50? | -49 |
The vertical axis indicates the number of employees, but the scale was accidentally omitted from this graph. What percent of the employees at the Gauss company have worked there for $5$ years or more? | 30 \% |
You have a whole pizza in the refrigerator. On your first trip to the refrigerator, you eat half the pizza. On each successive trip, you eat half of the remaining pizza. After five trips to the refrigerator, what fractional part of the pizza have you eaten? | \frac{31}{32} |
The numbers \( a, b, c, d \) belong to the interval \([-7, 7]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \). | 210 |
Suppose we flip four coins simultaneously: a penny, a nickel, a dime, and a quarter. What is the probability that the penny and dime both come up the same? | \dfrac{1}{2} |
Let $\star (x)$ be the sum of the digits of a positive integer $x$. $\mathcal{S}$ is the set of positive integers such that for all elements $n$ in $\mathcal{S}$, we have that $\star (n)=12$ and $0\le n< 10^{7}$. If $m$ is the number of elements in $\mathcal{S}$, compute $\star(m)$.
| 26 |
A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. | 6 + 4\sqrt{2} |
Three people, A, B, and C, visit three tourist spots, with each person visiting only one spot. Let event $A$ be "the three people visit different spots," and event $B$ be "person A visits a spot alone." Then, the probability $P(A|B)=$ ______. | \dfrac{1}{2} |
On Tuesday, Max's mom gave him $8 dollars for a hot dog at the Grand Boulevard park. On Wednesday, his mom gave him 5 times as much money as she gave him on Tuesday. On Thursday, his mom gave him $9 more in money than she gave him on Wednesday. How much more money did his mom give him on Thursday than she gave him on Tuesday? | On Wednesday Max was given 8*5 = <<8*5=40>>40 dollars
On Thursday Max was given 40+9 = <<40+9=49>>49 dollars
Max’s mom gave him 49-8 = <<49-8=41>>41 more dollars on Thursday than she gave him on Tuesday.
#### 41 |
Borris liquor store uses 90 kilograms of grapes every 6 months. He is thinking of increasing his production by twenty percent. How many grapes does he need in a year after increasing his production? | Borris uses 90 x 2 = <<90*2=180>>180 kilograms of grapes per year.
The increase of kilograms of grapes he needed per year is 180 x 0.20 = <<180*0.20=36>>36.
Therefore, Borris needs 180 + 36 = <<180+36=216>>216 kilograms of grapes in a year.
#### 216 |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} x=1+\cos \alpha \\ y=\sin \alpha\end{cases}$ ($\alpha$ is the parameter), and in the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of line $l$ is $\rho\sin (\theta+ \dfrac {\pi}{4})=2 \sqrt {2}$.
(Ⅰ) Convert the parametric equation of curve $C$ and the polar equation of line $l$ into ordinary equations in the Cartesian coordinate system;
(Ⅱ) A moving point $A$ is on curve $C$, a moving point $B$ is on line $l$, and a fixed point $P$ has coordinates $(-2,2)$. Find the minimum value of $|PB|+|AB|$. | \sqrt {37}-1 |
Let
\[\mathbf{M} = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{pmatrix}.\]If $\mathbf{M} \mathbf{M}^T = 9 \mathbf{I},$ then enter the ordered pair $(a,b).$
Note: For a matrix $\mathbf{A},$ $\mathbf{A}^T$ is the transpose of $\mathbf{A},$ which is generated by reflecting the matrix $\mathbf{A}$ over the main diagonal, going from the upper-left to the lower-right. So here,
\[\mathbf{M}^T = \begin{pmatrix} 1 & 2 & a \\ 2 & 1 & 2 \\ 2 & -2 & b \end{pmatrix}.\] | (-2,-1) |
How many 3-term geometric sequences $a$ , $b$ , $c$ are there where $a$ , $b$ , and $c$ are positive integers with $a < b < c$ and $c = 8000$ ? | 39 |
The curve $y = \sin x$ cuts the line whose equation is $y = \sin 70^\circ$ into segments having the successive ratios
\[\dots p : q : p : q \dots\]with $p < q.$ Compute the ordered pair of relatively prime positive integers $(p,q).$ | (1,8) |
For a school fundraiser, Chandler needs to sell 12 rolls of wrapping paper. So far, he has sold 3 rolls to his grandmother, 4 rolls to his uncle, and 3 rolls to a neighbor. How many more rolls of wrapping paper does Chandler need to sell? | Chandler has sold 3 rolls + 4 rolls + 3 rolls = <<3+4+3=10>>10 rolls.
Chandler still needs to sell 12 rolls – 10 rolls = <<12-10=2>>2 rolls.
#### 2 |
Find the value of $k$ for which $kx^2 -5x-12 = 0$ has solutions $x=3$ and $ x = -\frac{4}{3}$. | 3 |
Krista put 1 cent into her new bank on a Sunday morning. On Monday she put 2 cents into her bank. On Tuesday she put 4 cents into her bank, and she continued to double the amount of money she put into her bank each day for two weeks. On what day of the week did the total amount of money in her bank first exceed $\$2$? | \text{Sunday} |
Find the rational number that is the value of the expression
$$
\cos ^{6}(3 \pi / 16)+\cos ^{6}(11 \pi / 16)+3 \sqrt{2} / 16
$$ | 5/8 |
Lightning McQueen, the race car, cost 140000$. Mater only cost 10 percent of that, and Sally McQueen cost triple what Mater costs. How much does Sally McQueen cost? | Mater:140000(.10)=14000$
Sally:14000(3)=42000$
#### 42000 |
The simple quadrilateral \(ABCD\) has sides \(AB\), \(BC\), and \(CD\) with lengths 4, 5, and 20, respectively. If the angles \(B\) and \(C\) are obtuse, and \(\sin C = -\cos B = \frac{3}{5}\), calculate the length of the side \(AD\). | 25 |
Let A and B be fixed points in the plane with distance AB = 1. An ant walks on a straight
line from point A to some point C in the plane and notices that the distance from itself to B
always decreases at any time during this walk. Compute the area of the region in the plane
containing all points where point C could possibly be located. | \frac{\pi}{4} |
In the Cartesian coordinate system, points whose x and y coordinates are both integers are called lattice points. How many lattice points \((x, y)\) satisfy the inequality \((|x|-1)^{2}+(|y|-1)^{2}<2\)? | 16 |
Cirlce $\Omega$ is inscribed in triangle $ABC$ with $\angle BAC=40$ . Point $D$ is inside the angle $BAC$ and is the intersection of exterior bisectors of angles $B$ and $C$ with the common side $BC$ . Tangent form $D$ touches $\Omega$ in $E$ . FInd $\angle BEC$ . | 110 |
What is $0.1 \div 0.004$? | 25 |
The Gropkas of Papua New Guinea have ten letters in their alphabet: A, E, G, I, K, O, R, U, and V. Suppose license plates of four letters use only the letters in the Gropka alphabet. How many possible license plates are there of four letters that begin with either A or E, end with V, cannot contain P, and have no letters that repeat? | 84 |
Starting with the number 100, Shaffiq repeatedly divides his number by two and then takes the greatest integer less than or equal to that number. How many times must he do this before he reaches the number 1? | 6 |
Given a number $x$ is randomly selected from the interval $[-1,1]$, calculate the probability that the value of $\sin \frac{\pi x}{4}$ falls between $-\frac{1}{2}$ and $\frac{\sqrt{2}}{2}$. | \frac{5}{6} |
How many whole numbers between 1 and 2000 do not contain the digit 2? | 6560 |
How many positive integers less that $200$ are relatively prime to either $15$ or $24$ ? | 120 |
Compute \[\sum_{n=1}^{1000} \frac{1}{n^2 + n}.\] | \frac{1000}{1001} |
A geometric sequence starts $16$, $-24$, $36$, $-54$. What is the common ratio of this sequence? | -\frac{3}{2} |
In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let $N$ be the number of shadings with this property. Find the remainder when $N$ is divided by 1000. | 860 |
As $t$ takes on all real values, the set of points $(x,y)$ defined by
\begin{align*}
x &= t^2 - 2, \\
y &= t^3 - 9t + 5
\end{align*}forms a curve that crosses itself. Compute the ordered pair $(x,y)$ where this crossing occurs. | (7,5) |
Compute \[\lfloor 1 \rfloor + \lfloor 1.6 \rfloor + \lfloor 2.2 \rfloor + \lfloor 2.8 \rfloor + \dots + \lfloor 99.4 \rfloor + \lfloor 100 \rfloor,\]where the arguments of the floor functions are in arithmetic progression. | 8317 |
On the beach, there was a pile of apples belonging to 3 monkeys. The first monkey came, divided the apples into 3 equal piles with 1 apple remaining, then it threw the remaining apple into the sea and took one pile for itself. The second monkey came, divided the remaining apples into 3 equal piles with 1 apple remaining again, it also threw the remaining apple into the sea and took one pile. The third monkey did the same. How many apples were there originally at least? | 25 |
Which type of conic section is described by the equation \[|y+5| = \sqrt{(x-2)^2 + y^2}?\]Enter "C" for circle, "P" for parabola, "E" for ellipse, "H" for hyperbola, and "N" for none of the above. | \text{(P)} |
An ant starts at one vertex of an octahedron and moves along the edges according to a similar rule: at each vertex, the ant chooses one of the four available edges with equal probability, and all choices are independent. What is the probability that after six moves, the ant ends at the vertex exactly opposite to where it started?
A) $\frac{1}{64}$
B) $\frac{1}{128}$
C) $\frac{1}{256}$
D) $\frac{1}{512}$ | \frac{1}{128} |
Barbeck has two times as many guitars as Steve, but Davey has three times as many guitars as Barbeck. If there are 27 guitars altogether, how many guitars does Davey have? | Let x = the number of guitars Steve has.
Barbeck has 2x guitars, and Davey has 3 * 2x = 6x guitars.
2x + 6x + x = 27
9x = 27
x = 27 / 9 = <<27/9=3>>3 so Steve has 3 guitars.
Barbeck has 2 * 3 = <<2*3=6>>6 guitars.
And Davey has 3 * 6 = <<3*6=18>>18 guitars.
#### 18 |
The robot vacuum cleaner is programmed to move on the floor according to the law:
$$\left\{\begin{array}{l} x = t(t-6)^2 \\ y = 0, \quad 0 \leq t \leq 7 \\ y = (t-7)^2, \quad t \geq 7 \end{array}\right.$$
where the axes are chosen parallel to the walls and the movement starts from the origin. Time $t$ is measured in minutes, and coordinates are measured in meters. Find the distance traveled by the robot in the first 7 minutes and the absolute change in the velocity vector during the eighth minute. | \sqrt{445} |
What is the positive difference between the sum of the first 20 positive even integers and the sum of the first 15 positive odd integers? | 195 |
A dental office gives away 2 toothbrushes to every patient who visits. His 8 hour days are packed and each visit takes .5 hours. How many toothbrushes does he give in a 5 day work week? | Each day he does 8/.5=<<8/.5=16>>16 visits
So he does 16*5=<<16*5=80>>80 visits a week
That means he gives away 80*2=<<80*2=160>>160 toothbrushes a week
#### 160 |
Given the ellipse $\frac{x^2}{3} + y^2 = 1$ and the line $l: y = kx + m$ intersecting the ellipse at two distinct points $A$ and $B$.
(1) If $m = 1$ and $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$ ($O$ is the origin), find the value of $k$.
(2) If the distance from the origin $O$ to the line $l$ is $\frac{\sqrt{3}}{2}$, find the maximum area of $\triangle AOB$. | \frac{\sqrt{3}}{2} |
Evaluate the expression $3 + 2\sqrt{3} + \frac{1}{3 + 2\sqrt{3}} + \frac{1}{2\sqrt{3} - 3}$. | 3 + \frac{10\sqrt{3}}{3} |
Ben's new car cost twice as much as his old car. He sold his old car for $1800 and used that to pay off some of the cost of his new car. He still owes another $2000 on his new car. How much did his old car cost, in dollars? | His new car costs 1800+2000=<<1800+2000=3800>>3800 dollars.
His old car cost 3800/2=<<3800/2=1900>>1900 dollars.
#### 1900 |
A worker first receives a 25% cut in wages, then undergoes a 10% increase on the reduced wage. Determine the percent raise on his latest wage that the worker needs to regain his original pay. | 21.21\% |
Angus, Patrick, and Ollie went fishing for trout on Rainbow Lake. Angus caught 4 more fish than Patrick did, but Ollie caught 7 fewer fish than Angus. If Ollie caught 5 fish, how many fish did Patrick catch? | If Ollie caught 7 less than Angus, then Angus caught 5+7=12 fish.
And since Angus caught four more than Patrick, then Patrick caught 12-4=<<12-4=8>>8 fish.
#### 8 |
A bird discovered $543_{8}$ different ways to build a nest in each of its eight tree homes. How many ways are there in base 10? | 355 |
How many integers are there between 0 and \( 10^5 \) having the digit sum equal to 8? | 495 |
Given a function $f(x)$ that satisfies: For any $x \in (0, +\infty)$, it always holds that $f(2x) = 2f(x)$; (2) When $x \in (1, 2]$, $f(x) = 2 - x$. If $f(a) = f(2020)$, find the smallest positive real number $a$. | 36 |
Bernie loves eating chocolate. He buys two chocolates every week at the local store. One chocolate costs him $3. In a different store, there is a long-term promotion, during which each chocolate costs only $2. How much would Bernie save in three weeks, if he would buy his chocolates in this store instead of his local one? | During three weeks, Bernie buys 3 * 2 = <<3*2=6>>6 chocolates.
In the local store, he pays for them 6 * 3 = $<<6*3=18>>18.
If Bernie would buy the same chocolates in the different store, he would pay 6 * 2 = $<<6*2=12>>12.
So Bernie would be able to save 18 - 12 = $<<18-12=6>>6.
#### 6 |
Simplify $\sqrt{25000}$. | 50\sqrt{10} |
For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\]has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\]What is $f(1)$? | -7007 |
John and Anna bought some eBook readers. John bought 15 eBook readers less than Anna did. Unfortunately, John lost 3 eBook readers. If Anna bought 50 eBook readers, how many eBook readers do they have altogether? | John bought 50-15 = <<50-15=35>>35 eBook readers.
John has 35-3 = <<35-3=32>>32 eBook readers after losing some.
Together they have 50+32 = <<50+32=82>>82 eBook readers.
#### 82 |
Compute without using a calculator: $42!/40!$ | 1,\!722 |
The math questions in a contest are divided into three rounds: easy, average, and hard. There are corresponding points given for each round. That is 2, 3, and 5 points for every correct answer in the easy, average, and hard rounds, respectively. Suppose Kim got 6 correct answers in the easy; 2 correct answers in the average; and 4 correct answers in the difficult round, what are her total points in the contest? | Kim got 6 points/round x 2 round = <<6*2=12>>12 points in the easy round.
She got 2 points/round x 3 rounds = <<2*3=6>>6 points in the average round.
She got 4 points/round x 5 rounds = <<4*5=20>>20 points in the difficult round.
So her total points is 12 points + 6 points + 20 points = <<12+6+20=38>>38 points.
#### 38 |
Given $\triangle ABC$ and a point $P$ on one of its sides, call line $\ell$ the $\textit{splitting line}$ of $\triangle ABC$ through $P$ if $\ell$ passes through $P$ and divides $\triangle ABC$ into two polygons of equal perimeter. Let $\triangle ABC$ be a triangle where $BC = 219$ and $AB$ and $AC$ are positive integers. Let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{AC},$ respectively, and suppose that the splitting lines of $\triangle ABC$ through $M$ and $N$ intersect at $30^\circ.$ Find the perimeter of $\triangle ABC.$ | 459 |
The terms of the sequence $(b_i)$ defined by $b_{n + 2} = \frac {b_n + 2021} {1 + b_{n + 1}}$ for $n \ge 1$ are positive integers. Find the minimum possible value of $b_1 + b_2$. | 90 |
(The full score for this question is 12 points) Given $f(x+1)=x^2-1$,
(1) Find $f(x)$
(2) Find the maximum and minimum values of $f(x)$, and specify the corresponding values of $x$ | -1 |
Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number? | 12 |
The graph of $y = \frac{p(x)}{q(x)}$ is shown below, where $p(x)$ is linear and $q(x)$ is quadratic. (Assume that the grid lines are at integers.)
[asy]
unitsize(0.6 cm);
real func (real x) {
return (2*x/((x - 2)*(x + 3)));
}
int i;
for (i = -5; i <= 5; ++i) {
draw((i,-5)--(i,5),gray(0.7));
draw((-5,i)--(5,i),gray(0.7));
}
draw((-5,0)--(5,0));
draw((0,-5)--(0,5));
draw((-3,-5)--(-3,5),dashed);
draw((2,-5)--(2,5),dashed);
draw(graph(func,-5,-3.1),red);
draw(graph(func,-2.9,1.9),red);
draw(graph(func,2.1,5),red);
limits((-5,-5),(5,5),Crop);
[/asy]
Find $\frac{p(-1)}{q(-1)}.$ | \frac{1}{3} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.