problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
We plotted the graph of the function \( f(x) = \frac{1}{x} \) in the coordinate system. How should we choose the new, still equal units on the axes, if we want the curve to become the graph of the function \( g(x) = \frac{2}{x} \)? | \frac{\sqrt{2}}{2} |
If $S=1!+2!+3!+\cdots +99!$, then the units' digit in the value of S is: | 3 |
Given \( x = -2272 \), \( y = 10^3 + 10^2 c + 10 b + a \), and \( z = 1 \), which satisfy the equation \( a x + b y + c z = 1 \), where \( a \), \( b \), \( c \) are positive integers and \( a < b < c \). Find \( y \). | 1987 |
In the regular quadrangular pyramid \(P-ABCD\), \(M\) and \(N\) are the midpoints of \(PA\) and \(PB\) respectively. If the tangent of the dihedral angle between a side face and the base is \(\sqrt{2}\), find the cosine of the angle between skew lines \(DM\) and \(AN\). | 1/6 |
Find all positive integers $a,b$ for which $a^4+4b^4$ is a prime number. | (1, 1) |
The device consists of three independently operating elements. The probabilities of failure-free operation of the elements (over time $t$) are respectively: $p_{1}=0.7$, $p_{2}=0.8$, $p_{3}=0.9$. Find the probabilities that over time $t$, the following will occur:
a) All elements operate without failure;
b) Two ele... | 0.006 |
John draws a regular five pointed star in the sand, and at each of the 5 outward-pointing points and 5 inward-pointing points he places one of ten different sea shells. How many ways can he place the shells, if reflections and rotations of an arrangement are considered equivalent? | 362880 |
Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c,$ where $a, b,$ and $c$ are integers in $\{-20,-19,-18,\ldots,18,19,20\},$ such that there is a unique integer $m \not= 2$ with $p(m) = p(2).$ | 738 |
Find all three-digit numbers \( \overline{\mathrm{MGU}} \) consisting of distinct digits \( M, \Gamma, \) and \( U \) for which the equality \( \overline{\mathrm{MGU}} = (M + \Gamma + U) \times (M + \Gamma + U - 2) \) holds. | 195 |
John hits 70% of his free throws. For every foul he gets 2 shots. He gets fouled 5 times a game. How many free throws does he get if he plays in 80% of the 20 games the team plays? | John plays in 20*.8=<<20*.8=16>>16 games
That means he gets fouled 16*5=<<16*5=80>>80 fouls
So he gets 80*2=<<80*2=160>>160 foul shots
So he makes 160*.7=<<160*.7=112>>112 free throws
#### 112 |
Granger went to the grocery store. He saw that the Spam is $3 per can, the peanut butter is $5 per jar, and the bread is $2 per loaf. If he bought 12 cans of spam, 3 jars of peanut butter, and 4 loaves of bread, how much is the total amount he paid? | The 12 cans of Spam costs $3 x 12 = $<<3*12=36>>36.
The total amount of 3 jars of peanut butter is $5 x 3 = $<<5*3=15>>15.
And the total amount of the 4 loaves of bread is $2 x 4 = $<<2*4=8>>8.
Therefore the total amount he will have to pay is $36 + $15 + $8 = $<<36+15+8=59>>59.
#### 59 |
There exist two distinct unit vectors $\mathbf{v}$ such that the angle between $\mathbf{v}$ and $\begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix}$ is $45^\circ,$ and the angle between $\mathbf{v}$ and $\begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$ is $60^\circ.$ Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be these vectors. Find $\... | \sqrt{2} |
Square $PQRS$ has sides of length 1. Points $M$ and $N$ are on $\overline{QR}$ and $\overline{RS},$ respectively, so that $\triangle PMN$ is equilateral. A square with vertex $Q$ has sides that are parallel to those of $PQRS$ and a vertex on $\overline{PM}.$ The length of a side of this smaller square is $\frac{d-\sqrt... | 12 |
Given the rectangular coordinate system xOy, establish a polar coordinate system with O as the pole and the non-negative semi-axis of the x-axis as the polar axis. The line l passes through point P(-1, 2) with an inclination angle of $\frac{2π}{3}$, and the polar coordinate equation of circle C is $ρ = 2\cos(θ + \frac{... | 6 + 2\sqrt{3} |
Find $2 \cdot 5^{-1} + 8 \cdot 11^{-1} \pmod{56}$.
Express your answer as an integer from $0$ to $55$, inclusive. | 50 |
Express $\sin (a + b) - \sin (a - b)$ as the product of trigonometric functions. | 2 \sin b \cos a |
A cylindrical tank with radius 6 feet and height 7 feet is lying on its side. The tank is filled with water to a depth of 3 feet. Find the volume of water in the tank, in cubic feet. | 84\pi - 63\sqrt{3} |
Find the number of addition problems in which a two digit number is added to a second two digit number to give a two digit answer, such as in the three examples: \[\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}2342\hline 65\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}3636\hline 72\end{tabular... | 3240 |
Suppose that $x$ is a positive multiple of $3$. If $x$ cubed is less than $1000$, what is the greatest possible value of $x$? | 9 |
Coach Randall is preparing a 6-person starting lineup for her soccer team, the Rangers, which has 15 players. Among the players, three are league All-Stars (Tom, Jerry, and Spike), and they are guaranteed to be in the starting lineup. Additionally, the lineup must include at least one goalkeeper, and there is only one ... | 55 |
Every day Janet spends 8 minutes looking for her keys and another 3 minutes complaining after she finds them. If Janet stops losing her keys, how many minutes will she save every week? | First find the total time Janet spends looking for her key and complaining each day: 8 minutes/day + 3 minutes/day = <<8+3=11>>11 minutes/day
Then multiply that number by the number of days in a week to find the weekly time spent: 11 minutes/day * 7 days/week = <<11*7=77>>77 minutes
#### 77 |
There are 12 different-colored crayons in a box. How many ways can Karl select four crayons if the order in which he draws them out does not matter? | 495 |
For positive integers $a, b, a \uparrow \uparrow b$ is defined as follows: $a \uparrow \uparrow 1=a$, and $a \uparrow \uparrow b=a^{a \uparrow \uparrow(b-1)}$ if $b>1$. Find the smallest positive integer $n$ for which there exists a positive integer $a$ such that $a \uparrow \uparrow 6 \not \equiv a \uparrow \uparrow 7... | 283 |
Let's define a calendar week as even or odd according to whether the sum of the day numbers within the month in that week is even or odd. Out of the 52 consecutive weeks starting from the first Monday of January, how many can be even? | 30 |
For any real number $x$, the symbol $\lfloor x \rfloor$ represents the integer part of $x$, that is, $\lfloor x \rfloor$ is the largest integer not exceeding $x$. Calculate the value of $\lfloor \log_{2}1 \rfloor + \lfloor \log_{2}2 \rfloor + \lfloor \log_{2}3 \rfloor + \lfloor \log_{2}4 \rfloor + \ldots + \lfloor \log... | 8204 |
Find all positive integers $k<202$ for which there exists a positive integer $n$ such that $$\left\{\frac{n}{202}\right\}+\left\{\frac{2 n}{202}\right\}+\cdots+\left\{\frac{k n}{202}\right\}=\frac{k}{2}$$ where $\{x\}$ denote the fractional part of $x$. | k \in\{1,100,101,201\} |
Let $A_1,A_2,A_3,\cdots,A_{12}$ be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set $\{A_1,A_2,A_3,\cdots,A_{12}\} ?$ | 183 |
If $a \lt 0$, the graph of the function $f\left(x\right)=a^{2}\sin 2x+\left(a-2\right)\cos 2x$ is symmetric with respect to the line $x=-\frac{π}{8}$. Find the maximum value of $f\left(x\right)$. | 4\sqrt{2} |
The national student loan is a credit loan subsidized by the finance department, aimed at helping college students from families with financial difficulties to pay for tuition, accommodation, and living expenses during their study period in college. The total amount applied for each year shall not exceed 6,000 yuan. A ... | 31 |
There are 5 balls numbered $(1)$, $(2)$, $(3)$, $(4)$, $(5)$ and 5 boxes numbered $(1)$, $(2)$, $(3)$, $(4)$, $(5)$. Each box contains one ball. The number of ways in which at most two balls have the same number as their respective boxes is $\_\_\_\_\_\_$. | 109 |
The height of a right truncated quadrilateral pyramid is 3 cm, its volume is 38 cm³, and the areas of its bases are in the ratio 4:9. Determine the lateral surface area of the truncated pyramid. | 10 \sqrt{19} |
Given that the domains of functions f(x) and g(x) are both $\mathbb{R}$, and $f(x) + g(2-x) = 5$, $g(x) - f(x-4) = 7$. If the graph of $y = g(x)$ is symmetric about the line $x = 2$, $g(2) = 4$, calculate the value of $\sum _{k=1}^{22}f(k)$. | -24 |
I randomly pick an integer $p$ between $1$ and $10$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 4p - 2q = 2$? Express your answer as a common fraction. | \frac{2}{5} |
Today is 17.02.2008. Natasha noticed that in this date, the sum of the first four digits is equal to the sum of the last four digits. When will this coincidence happen for the last time this year? | 25.12.2008 |
Given vectors $\overrightarrow{a}=(\cos α,\sin α)$, $\overrightarrow{b}=(\cos β,\sin β)$, and $|\overrightarrow{a}- \overrightarrow{b}|= \frac {4 \sqrt {13}}{13}$.
(1) Find the value of $\cos (α-β)$;
(2) If $0 < α < \frac {π}{2}$, $- \frac {π}{2} < β < 0$, and $\sin β=- \frac {4}{5}$, find the value of $\sin α$. | \frac {16}{65} |
$ABCDEFGH$ shown below is a cube with volume 1. Find the volume of pyramid $ABCH$.
[asy]
import three;
triple A,B,C,D,EE,F,G,H;
A = (0,0,0);
B = (1,0,0);
C = (1,1,0);
D= (0,1,0);
EE = (0,0,1);
F = B+EE;
G = C + EE;
H = D + EE;
draw(B--C--D);
draw(B--A--D,dashed);
draw(EE--F--G--H--EE);
draw(A--EE,dashed);
draw(B--F);... | \frac16 |
A bag contains 20 candies: 4 chocolate, 6 mint, and 10 butterscotch. Candies are removed randomly from the bag and eaten. What is the minimum number of candies that must be removed to be certain that at least two candies of each flavor have been eaten? | 18 |
Given that the terminal side of $\alpha$ passes through the point $(a, 2a)$ (where $a < 0$),
(1) Find the values of $\cos\alpha$ and $\tan\alpha$.
(2) Simplify and find the value of $$\frac {\sin(\pi-\alpha)\cos(2\pi-\alpha)\sin(-\alpha+ \frac {3\pi}{2})}{\tan(-\alpha-\pi)\sin(-\pi-\alpha)}$$. | \frac{1}{10} |
In $\triangle ABC$, if $|\overrightarrow{AB}|=2$, $|\overrightarrow{AC}|=3$, $|\overrightarrow{BC}|=4$, and $O$ is the incenter of $\triangle ABC$, and $\overrightarrow{AO}=\lambda \overrightarrow{AB}+\mu \overrightarrow{BC}$, calculate the value of $\lambda+\mu$. | \frac{7}{9} |
The nonzero roots of the equation $x^2 + 6x + k = 0$ are in the ratio $2:1$. What is the value of $k$? | 8 |
For a finite graph $G$, let $f(G)$ be the number of triangles and $g(G)$ the number of tetrahedra formed by edges of $G$. Find the least constant $c$ such that \[g(G)^3\le c\cdot f(G)^4\] for every graph $G$.
[i] | \frac{3}{32} |
Rectangle $ABCD$ has $AB = 8$ and $BC = 13$ . Points $P_1$ and $P_2$ lie on $AB$ and $CD$ with $P_1P_2 \parallel BC$ . Points $Q_1$ and $Q_2$ lie on $BC$ and $DA$ with $Q_1Q_2 \parallel AB$ . Find the area of quadrilateral $P_1Q_1P_2Q_2$ . | 52 |
Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0)$ , $(2,0)$ , $(2,1)$ , and $(0,1)$ . $R$ can be divided into two unit squares, as shown. [asy]size(120); defaultpen(linewidth(0.7));
draw(origin--(2,0)--(2,1)--(0,1)--cycle^^(1,0)--(1,1));[/asy] Pro selects a point $P$ at random in the i... | 3/4 |
\(A, B, C, D\) are consecutive vertices of a parallelogram. Points \(E, F, P, H\) lie on sides \(AB\), \(BC\), \(CD\), and \(AD\) respectively. Segment \(AE\) is \(\frac{1}{3}\) of side \(AB\), segment \(BF\) is \(\frac{1}{3}\) of side \(BC\), and points \(P\) and \(H\) bisect the sides they lie on. Find the ratio of t... | 37/72 |
How many children did my mother have?
If you asked me this question, I would only tell you that my mother dreamed of having at least 19 children, but she couldn't make this dream come true; however, I had three times more sisters than cousins, and twice as many brothers as sisters. How many children did my mother have... | 10 |
A parallelogram is defined by the vectors $\begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$. Determine the cosine of the angle $\theta$ between the diagonals of this parallelogram. | \frac{-11}{3\sqrt{69}} |
Two circles of radius \( r \) are externally tangent to each other and internally tangent to the ellipse \( x^2 + 4y^2 = 8 \). Find \( r \). | \frac{\sqrt{6}}{2} |
Calculate the probability that the numbers 1, 1, 2, 2, 3, 3 can be arranged into two rows and three columns such that no two identical numbers appear in the same row or column. | \frac{2}{15} |
Given $-765^\circ$, convert this angle into the form $2k\pi + \alpha$ ($0 \leq \alpha < 2\pi$), where $k \in \mathbb{Z}$. | -6\pi + \frac{7\pi}{4} |
The graph of the equation \[\sqrt{x^2+y^2} + |y-1| = 3\]consists of portions of two different parabolas. Compute the distance between the vertices of the parabolas. | 3 |
A baker bakes 5 loaves of bread an hour in one oven. He has 4 ovens. From Monday to Friday, he bakes for 5 hours, but on Saturday and Sunday, he only bakes for 2 hours. How many loaves of bread does the baker bake in 3 weeks? | In an hour, the baker bakes 5 x 4 = <<5*4=20>>20 loaves of bread.
From Monday to Friday, he bakes 5 x 20 = <<5*20=100>>100 loaves of bread per day.
From Monday to Friday, he bakes a total of 100 x 5 = <<100*5=500>>500 loaves of bread.
On Saturday and Sunday, he bakes 2 x 20 = <<2*20=40>>40 loaves of bread per day.
On S... |
Given $\boldsymbol{a} = (\cos \alpha, \sin \alpha)$ and $\boldsymbol{b} = (\cos \beta, \sin \beta)$, the relationship between $\boldsymbol{a}$ and $\boldsymbol{b}$ is given by $|k \boldsymbol{a} + \boldsymbol{b}| - \sqrt{3}|\boldsymbol{a} - k \boldsymbol{b}|$, where $k > 0$. Find the minimum value of $\boldsymbol{a} \... | \frac{1}{2} |
Given the following matrix $$ \begin{pmatrix}
11& 17 & 25& 19& 16
24 &10 &13 & 15&3
12 &5 &14& 2&18
23 &4 &1 &8 &22
6&20&7 &21&9
\end{pmatrix}, $$ choose five of these elements, no two from the same row or column, in such a way that the minimum of these elements is as large as possible. | 17 |
Express the given data "$20$ nanoseconds" in scientific notation. | 2 \times 10^{-8} |
In the Cartesian coordinate system $xOy$, it is known that the distance from any point on curve $C$ to point $M(0, \frac{1}{2})$ is equal to its distance to the line $y = -\frac{1}{2}$.
(Ⅰ) Find the equation of curve $C$;
(Ⅱ) Let $A_1(x_1, 0)$ and $A_2(x_2, 0)$ be two points on the x-axis ($x_1 + x_2 \neq 0$, $x_1x_... | \frac{6}{7} |
Let \( n \) be the smallest positive integer such that the sum of its digits is 2011. How many digits does \( n \) have? | 224 |
The store bought a batch of New Year cards at 0.21 yuan each and sold them for a total of 14.57 yuan. If each card was sold at the same price and did not exceed twice the purchase price, how much profit did the store make? | 4.7 |
How many positive integers less than 60 have an even number of positive divisors? | 52 |
Find the larger of the two distinct solutions to the equation $$x^2 - 11x - 42 = 0.$$ | 14 |
Among all pairs of real numbers $(x, y)$ such that $\cos \sin x = \cos \sin y$ with $-\frac{15\pi}{2} \le x, y \le \frac{15\pi}{2}$, Ana randomly selects a pair $(X, Y)$. Compute the probability that $X = Y$. | \frac{1}{4} |
The three different points \(A(x_1, y_1)\), \(B\left(4, \frac{9}{5}\right)\), and \(C(x_2, y_2)\) on the ellipse \(\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1\), along with the focus \(F(4,0)\) have distances that form an arithmetic sequence. If the perpendicular bisector of line segment \(AC\) intersects the x-axis at poin... | 5/4 |
There are 3 numbers that are consecutive integers. Together they have a sum of 18. What is the largest of the 3 numbers? | Let N = smallest number
N + 1 = next number
N + 2 = largest number
N + (N + 1) + (N + 2) = 18
3N + 3 = 18
3N = <<3*5=15>>15
N = <<5=5>>5
The largest number is <<7=7>>7.
#### 7 |
Find the number of collections of $16$ distinct subsets of $\{1,2,3,4,5\}$ with the property that for any two subsets $X$ and $Y$ in the collection, $X \cap Y \not= \emptyset.$ | 081 |
A bowling ball is a solid ball with a spherical surface and diameter 30 cm. To custom fit a bowling ball for each bowler, three holes are drilled in the ball. Bowler Kris has holes drilled that are 8 cm deep and have diameters of 2 cm, 2 cm, and 3 cm. Assuming the three holes are right circular cylinders, find the numb... | 4466\pi |
Compute $\left(\sqrt{625681 + 1000} - \sqrt{1000}\right)^2$. | 626681 - 2 \cdot \sqrt{626681} \cdot 31.622776601683793 + 1000 |
An athlete's heart beats an average of 150 times per minute while running. How many times does the athlete's heart beat during a 26-mile race if the athlete runs at a pace of 5 minutes per mile? | 19500 |
Find the number of subsets $S$ of $\{1,2, \ldots 63\}$ the sum of whose elements is 2008. | 66 |
Emily is 20 years old and her older sister, Rachel, is 24 years old. How old is Rachel when Emily is half her age? | The difference in age between the two sisters is 24 - 20 = <<24-20=4>>4 years.
When Rachel is twice Emily’s age, she is 4 x 2 = <<2*4=8>>8 years old.
#### 8 |
Let $A$ and $B$ be the endpoints of a semicircular arc of radius $2$. The arc is divided into seven congruent arcs by six equally spaced points $C_1$, $C_2$, $\dots$, $C_6$. All chords of the form $\overline {AC_i}$ or $\overline {BC_i}$ are drawn. Find the product of the lengths of these twelve chords. | 28672 |
Felix is chopping down trees in his backyard. For every 13 trees he chops down he needs to get his axe resharpened. It cost him $5 to sharpen his axe. If he spends $35 on axe sharpening, at least how many trees has he chopped down? | He has sharpened in 7 times because 35 / 5 = <<35/5=7>>7
He has chopped down 91 trees because 7 x 13 = <<7*13=91>>91
#### 91 |
Five positive integers (not necessarily all different) are written on five cards. Boris calculates the sum of the numbers on every pair of cards. He obtains only three different totals: 57, 70, and 83. What is the largest integer on any card?
- A) 35
- B) 42
- C) 48
- D) 53
- E) 82 | 48 |
A ray of light originates from point $A$ and travels in a plane, being reflected $n$ times between lines $AD$ and $CD$ before striking a point $B$ (which may be on $AD$ or $CD$) perpendicularly and retracing its path back to $A$ (At each point of reflection the light makes two equal angles as indicated in the adjoining... | 10 |
How many integers between $3250$ and $3500$ have four distinct digits arranged in increasing order? | 20 |
Katya is passing time while her parents are at work. On a piece of paper, she absentmindedly drew Cheburashkas in two rows (at least one Cheburashka was drawn in each row).
Afterwards, she drew a Crocodile Gena between every two adjacent Cheburashkas in both rows. Then she drew an Old Lady Shapoklyak to the left of ea... | 11 |
Given the function \( f(x) = x^3 + ax^2 + bx + c \) where \( a, b, \) and \( c \) are nonzero integers, if \( f(a) = a^3 \) and \( f(b) = b^3 \), what is the value of \( c \)? | 16 |
A die is rolled twice continuously, resulting in numbers $a$ and $b$. What is the probability $p$, in numerical form, that the cubic equation in $x$, given by $x^{3}-(3 a+1) x^{2}+(3 a+2 b) x-2 b=0$, has three distinct real roots? | 3/4 |
Carolyn and Paul are playing a game starting with a list of the integers $1$ to $n.$ The rules of the game are:
$\bullet$ Carolyn always has the first turn.
$\bullet$ Carolyn and Paul alternate turns.
$\bullet$ On each of her turns, Carolyn must remove one number from the list such that this number has at least o... | 12 |
Find the maximum natural number which is divisible by 30 and has exactly 30 different positive divisors. | 11250 |
Given the function f(x) = 2cos^2(x) - 2$\sqrt{3}$sin(x)cos(x).
(I) Find the monotonically decreasing interval of the function f(x);
(II) Find the sum of all the real roots of the equation f(x) = $- \frac{1}{3}$ in the interval [0, $\frac{\pi}{2}$]. | \frac{2\pi}{3} |
Hilary is collecting her toenails in a jar to gross out her sister. She can fit 100 toenails in the jar, unless they are from her two big toes, which are twice as big as the rest. She has already filled it with 20 big toenails and 40 regular toenails. How many regular toenails can she fit into the remainder of the jar? | The 20 big toenails take up as much space as 40 regular toenails because 20 x 2 = <<20*2=40>>40
She has filled up 80 toenails worth because 40 + 40 = <<40+40=80>>80
She can fit 20 regular toenails in the jar because 100 - 80 = <<100-80=20>>20
#### 20 |
Point $D$ is on side $AC$ of triangle $ABC$, $\angle ABD=15^{\circ}$ and $\angle DBC=50^{\circ}$. What is the measure of angle $BAD$, in degrees?
[asy]draw((-43,0)--(43,0)--(43,40)--cycle);
draw((-4,0)--(43,40));
draw((39,4)--(39,0));
draw((39,4)--(43,4));
draw((-1,36)--(22.5,26),Arrow);
label("$15^{\circ}$",(-1,36),... | 25^\circ |
We drew the face diagonals of a cube with an edge length of one unit and drew a sphere centered around the cube's center. The sphere intersected the diagonals at the vertices of a convex polyhedron, all of whose faces are regular. What was the radius of the sphere? | 0.579 |
Define a sequence recursively by $f_1(x)=|x-1|$ and $f_n(x)=f_{n-1}(|x-n|)$ for integers $n>1$. Find the least value of $n$ such that the sum of the zeros of $f_n$ exceeds $500,000$. | 101 |
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \cos ^{8} x \, dx
$$ | \frac{35 \pi}{8} |
On the complex plane, the parallelogram formed by the points 0, $z,$ $\frac{1}{z},$ and $z + \frac{1}{z}$ has area $\frac{35}{37}.$ If the real part of $z$ is positive, let $d$ be the smallest possible value of $\left| z + \frac{1}{z} \right|.$ Compute $d^2.$ | \frac{50}{37} |
Someone collected data relating the average temperature x (℃) during the Spring Festival to the sales y (ten thousand yuan) of a certain heating product. The data pairs (x, y) are as follows: (-2, 20), (-3, 23), (-5, 27), (-6, 30). Based on the data, using linear regression, the linear regression equation between sales... | 34.4 |
Tom decides to start running 5 days a week to lose weight. He runs 1.5 hours each day. He runs at a speed of 8 mph. How many miles does he run a week? | Each day he runs 1.5 * 8 = <<1.5*8=12>>12 miles
So he runs 5 * 12 = <<5*12=60>>60 miles in the week
#### 60 |
In the expansion of $((x^2+1)^2(x-1)^6)$, find the coefficient of the $x^3$ term. | -32 |
What is the largest $n$ for which it is possible to construct two bi-infinite sequences $A$ and $B$ such that any subsequence of $B$ of length $n$ is contained in $A$, $A$ has a period of 1995, and $B$ does not have this property (is either non-periodic or has a period of a different length)? | 1995 |
In rectangle $ABCD$, $AB=1$, $BC=2$, and points $E$, $F$, and $G$ are midpoints of $\overline{BC}$, $\overline{CD}$, and $\overline{AD}$, respectively. Point $H$ is the midpoint of $\overline{GE}$. What is the area of the shaded region? | \frac{1}{6} |
How many even perfect square factors does $2^4 \cdot 7^9$ have? | 10 |
What is $\frac56$ of 30? | 25 |
Solve for $x$: $$\sqrt[3]{3-\frac{1}{x}}=-4$$ | x=\frac{1}{67} |
The school band has 30 songs in their repertoire. The band played 5 songs in their first set and 7 in their second set. The band will play 2 songs for their encore. Assuming the band plays through their entire repertoire, how many songs will they play on average in the third and fourth sets? | The band played 5 + 7 = <<5+7=12>>12 songs in their first two sets.
They have 30 - 12 = <<30-12=18>>18 songs left for the third set, fourth set and encore.
Since they played 2 songs in the encore, they have 18 - 2 = <<18-2=16>>16 songs to play on the third and fourth sets
They will play 16 / 2 = <<16/2=8>>8 songs on av... |
On a trip from the United States to Canada, Isabella took $d$ U.S. dollars. At the border she exchanged them all, receiving $10$ Canadian dollars for every $7$ U.S. dollars. After spending $60$ Canadian dollars, she had $d$ Canadian dollars left. What is the sum of the digits of $d$? | 5 |
On floor 0 of a weird-looking building, you enter an elevator that only has one button. You press the button twice and end up on floor 1. Thereafter, every time you press the button, you go up by one floor with probability $\frac{X}{Y}$, where $X$ is your current floor, and $Y$ is the total number of times you have pre... | 97 |
3 lions and 2 rhinos escape from the zoo. If it takes 2 hours to recover each animal how long did the zoo spend recovering animals? | They had to recover 3+2=<<3+2=5>>5 animals
So it took 5*2=<<5*2=10>>10 hours to recover everyone
#### 10 |
A regular hexagon has an area of $150\sqrt{3}$ cm². If each side of the hexagon is decreased by 3 cm, by how many square centimeters is the area decreased? | 76.5\sqrt{3} |
Some students want to buy pencils from a stationery shop. The price of one pencil is 20 cents. Tolu wants 3 pencils, Robert wants 5 pencils and Melissa wants 2 pencils. How much (in dollars) will the students spend altogether at the stationery shop? | The three students want a total of 3+2+5 = <<3+2+5=10>>10 pencils
Each pencil cost 20 cents so 10 pencils cost 20*10 = <<20*10=200>>200 cents
100 cents make a dollar so 200 cents make 200/100 = <<200/100=2>>2 dollars
#### 2 |
What percent of the positive integers less than or equal to $100$ have no remainders when divided by $5?$ | 20 |
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