problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Albert wants a paintbrush that costs $1.50, a set of paints that costs $4.35, and a wooden easel that costs $12.65. Albert already has $6.50. How much more money does Albert need? | The total cost of the paintbrush, the paints, and the wooden easel is $1.50 + $4.35 + $12.65 = $<<1.5+4.35+12.65=18.50>>18.50.
Albert needs $18.50 − $6.50 = $12 more.
#### 12 |
There are three times as many hogs as cats in King Henry's kingdom. If there are 75 hogs, what's 5 less than 60% of the number of cats in King Henry's kingdom? | There are three times as many hogs as cats in King Henry's kingdom, meaning there are 75/3=<<75/3=25>>25 cats
60% of the number of cats in King Henry's kingdom is 60/100*25=<<60/100*25=15>>15 cats
5 less than 60% of the number of cats in King Henry's kingdom is 15-5=<<15-5=10>>10 cats
#### 10 |
Evaluate $\lfloor 3.2\rfloor$. | 3 |
Let $P(x)$ be a polynomial such that
\[P(x) = P(0) + P(1) x + P(2) x^2\]and $P(-1) = 1.$ Find $P(x).$ | x^2 - x - 1 |
Given $m=(\sqrt{3}\sin \omega x,\cos \omega x)$, $n=(\cos \omega x,-\cos \omega x)$ ($\omega > 0$, $x\in\mathbb{R}$), $f(x)=m\cdot n-\frac{1}{2}$ and the distance between two adjacent axes of symmetry on the graph of $f(x)$ is $\frac{\pi}{2}$.
$(1)$ Find the intervals of monotonic increase for the function $f(x)$;
$(... | \frac{3\sqrt{3}}{4} |
A domino is a rectangular tile composed of two squares. An integer is represented on both squares, and each integer 0-9 is paired with every integer 0-9 exactly once to form a complete set. A $\textit{double}$ is a domino that has the same integer on both of its squares. What is the probability that a domino randomly s... | \frac{2}{11} |
Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. | 42 |
Given that $F$ is the right focus of the ellipse $C:\frac{x^2}{4}+\frac{y^2}{3}=1$, $P$ is a point on the ellipse $C$, and $A(1,2\sqrt{2})$, find the maximum value of $|PA|+|PF|$. | 4 + 2\sqrt{3} |
Given vectors $\overrightarrow{m}=( \sqrt {3}\sin x-\cos x,1)$ and $\overrightarrow{n}=(\cos x, \frac {1}{2})$, and the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$,
(1) Find the interval(s) where the function $f(x)$ is monotonically increasing;
(2) If $a$, $b$, $c$ are the sides opposite to angles $A$, ... | 2 \sqrt {3} |
Find all real numbers $k$ such that
\[\left\| k \begin{pmatrix} 2 \\ -3 \end{pmatrix} - \begin{pmatrix} 4 \\ 7 \end{pmatrix} \right\| = 2 \sqrt{13}.\]Enter all the solutions, separated by commas. | -1 |
Team Soccer Stars plays in a soccer league. They played a total of 20 games, and won 14 games, lost 2, and drew the rest of the matches. They earn 3 points for a win, 1 point for a draw, and nothing for a loss. How many points did they have at the end of the season? | Soccer Stars won and lost a total of 14+2 = <<14+2=16>>16 games.
This means they drew 20 - 16 = <<20-16=4>>4 games.
The total points from the wins are 14*3 = <<14*3=42>>42 points.
The total points from draws are 4*1 = <<4*1=4>>4 points.
The total points from the losses are 2 * 0 = <<2*0=0>>0 points.
They had 42+4+0 = <... |
Consider a 4-by-4 grid where each of the unit squares can be colored either purple or green. Each color choice is equally likely independent of the others. Compute the probability that the grid does not contain a 3-by-3 grid of squares all colored purple. Express your result in the form $\frac{m}{n}$, where $m$ and $n$... | 255 |
Elias uses a bar of soap every month. If each bar of soap costs $4, how much does he spend on bars of soap in two years? | A year is 12 months, so Elias spends 4 * 12 = $<<4*12=48>>48 per year on bars of soap.
In two years, he spends 2 * 48 = $<<2*48=96>>96 on bars of soap.
#### 96 |
$a,b,c$ - are sides of triangle $T$ . It is known, that if we increase any one side by $1$ , we get new
a) triangle
b)acute triangle
Find minimal possible area of triangle $T$ in case of a) and in case b) | \frac{\sqrt{3}}{4} |
Wanda weighs 30 pounds more than Yola currently. Wanda also weighs 80 pounds more than Yola did 2 years ago. How much did Yola weigh 2 years ago if she currently weighs 220 pounds? | If Yola currently weighs 220 pounds, then Wanda, who weighs 30 pounds more than Yola, weighs 220+30 = <<220+30=250>>250 pounds.
Wanda also weighs 80 pounds more than Yola did 2 years ago, meaning Yola weighed 250-80=<<250-80=170>>170 pounds 2 years ago.
#### 170 |
A regular hexagon `LMNOPQ` has sides of length 4. Find the area of triangle `LNP`. Express your answer in simplest radical form. | 8\sqrt{3} |
The teacher decided to rewards his students with extra recess on report card day if they got good grades. Students normally get 20 minutes for recess. He told the students that every A got them 2 extra minutes of recess. Every B got them one extra minute. Every C got them zero extra minutes, but every D got them 1 less... | The students have 20 minutes to start.
They get 20 minutes added for the As because 10 times 2 equals <<10*2=20>>20
They get 12 minutes added for the Bs because 12 times 1 equals <<12*1=12>>12
They get no minutes added or subtracted for the Cs because 14 times 0 equals 0.
They get 5 minutes taken away for the Ds becaus... |
Twenty-seven increased by twice a number is 39. What is the number? | 6 |
To calculate $41^2$, David mentally figures the value $40^2$ and adds 81. David subtracts a number from $40^2$ to calculate $39^2$. What number does he subtract? | 79 |
Given a function $f(x)$ $(x \in \mathbb{R})$ that satisfies the equation $f(-x) = 8 - f(4 + x)$, and another function $g(x) = \frac{4x + 3}{x - 2}$. If the graph of $f(x)$ has 168 intersection points with the graph of $g(x)$, denoted as $P_i(x_i, y_i)$ $(i = 1,2, \dots, 168)$, calculate the value of $(x_{1} + y_{1}) + ... | 1008 |
What is the greatest integer less than 150 for which the greatest common divisor of that integer and 18 is 6? | 144 |
The largest divisor of a natural number \( N \), smaller than \( N \), was added to \( N \), producing a power of ten. Find all such \( N \). | 75 |
Compute the value of the infinite series \[
\sum_{n=2}^{\infty} \frac{n^4+3n^2+10n+10}{2^n \cdot \left(n^4+4\right)}
\] | \frac{11}{10} |
Let $P$ be the point on line segment $\overline{AB}$ such that $AP:PB = 3:2.$ Then
\[\overrightarrow{P} = t \overrightarrow{A} + u \overrightarrow{B}\]for some constants $t$ and $u.$ Enter the ordered pair $(t,u).$
[asy]
unitsize(1 cm);
pair A, B, P;
A = (0,0);
B = (5,1);
P = interp(A,B,3/5);
draw(A--B);
dot("$A... | \left( \frac{2}{5}, \frac{3}{5} \right) |
Suppose that $\alpha$ is inversely proportional to $\beta$. If $\alpha = -3$ when $\beta = -6$, find $\alpha$ when $\beta = 8$. Express your answer as a fraction. | \frac{9}{4} |
Lydia is planning a road trip with her family and is trying to plan a route. She has 60 liters of fuel and she will use all of this during her trip. She isn't sure how much fuel she is going to use in the first third of the trip but she knows she will need a third of all of her fuel for the second third of the trip, an... | In the second third of the trip, Lydia is going to use 60 liters of fuel / 3 = <<60/3=20>>20 liters of fuel.
In the final third of the trip, she needs half of this amount so she is going to be using 20 liters / 2 = <<20/2=10>>10 liters of fuel.
So in the first third, Lydia used the remaining amount of fuel which is 60 ... |
The line $x = k$ intersects the graph of the parabola $x = -2y^2 - 3y + 5$ at exactly one point. What is $k$? | \frac{49}{8} |
For each positive integer $n$ let $a_n$ be the least positive integer multiple of $23$ such that $a_n \equiv 1 \pmod{2^n}.$ Find the number of positive integers $n$ less than or equal to $1000$ that satisfy $a_n = a_{n+1}.$ | 363 |
Determine all such pairs pf positive integers $(a, b)$ such that $a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \cdot lcm(a -1, b)$, where $lcm (a, b)$ denotes the smallest common multiple, and $gcd (a, b)$ denotes the greatest common divisor of numbers $a, b$. | (2, 3) \text{ and } (6, 15) |
A five-digit number is called a "hill" if its first three digits are in ascending order and its last three digits are in descending order. For example, 13760 and 28932 are hills, whereas 78821 and 86521 are not hills. How many hills exist that are greater than the number 77777? | 36 |
When $x$ is divided by each of $4$, $5$, and $6$, remainders of $3$, $4$, and $5$ (respectively) are obtained. What is the smallest possible positive integer value of $x$? | 59 |
Six people are arranged in a row. In how many ways can the three people A, B, and C be arranged such that they are not adjacent to each other? | 144 |
In $\triangle ABC$, $\overrightarrow {AD}=3 \overrightarrow {DC}$, $\overrightarrow {BP}=2 \overrightarrow {PD}$, if $\overrightarrow {AP}=λ \overrightarrow {BA}+μ \overrightarrow {BC}$, then $λ+μ=\_\_\_\_\_\_$. | - \frac {1}{3} |
The first term of a sequence is 934. Each subsequent term is equal to the sum of the digits of the previous term multiplied by 13. Find the 2019th term of the sequence. | 130 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $c = a \cos B + 2b \sin^2 \frac{A}{2}$.
(1) Find angle $A$.
(2) If $b=4$ and the length of median drawn to side $AC$ is $\sqrt{7}$, find $a$. | \sqrt{13} |
How many positive two-digit integers are there in which each of the two digits is prime? | 16 |
Let $M = 36 \cdot 36 \cdot 77 \cdot 330$. Find the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$. | 1 : 62 |
Find the maximum value of the function
$$
f(x)=\sin (x+\sin x)+\sin (x-\sin x)+\left(\frac{\pi}{2}-2\right) \sin (\sin x)
$$ | \frac{\pi - 2}{\sqrt{2}} |
The expression $\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{4}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{6}\right)\left(1+\frac{1}{7}\right)\left(1+\frac{1}{8}\right)\left(1+\frac{1}{9}\right)$ is equal to what? | 5 |
What is the units digit of the product of all the odd positive integers between 10 and 110? | 5 |
Let $m$ and $n$ be positive integers with $m\le 2000$ and $k=3-\frac{m}{n}$. Find the smallest positive value of $k$. | $\boxed{ \frac{1}{667}} .$ |
In $\triangle ABC$, $D$ is a point on side $\overline{AC}$ such that $BD=DC$ and $\angle BCD$ measures $70^\circ$. What is the degree measure of $\angle ADB$? | 140 |
What is the smallest positive integer that can be written in the form $2002m + 44444n$, where $m$ and $n$ are integers? | 2 |
Given positive numbers $m$ and $n$ that satisfy $m^2 + n^2 = 100$, find the maximum or minimum value of $m + n$. | 10\sqrt{2} |
The measures of the interior angles of a convex polygon of $n$ sides are in arithmetic progression. If the common difference is $5^{\circ}$ and the largest angle is $160^{\circ}$, then $n$ equals: | 16 |
Sixteen wooden Cs are placed in a 4-by-4 grid, all with the same orientation, and each is to be colored either red or blue. A quadrant operation on the grid consists of choosing one of the four two-by-two subgrids of Cs found at the corners of the grid and moving each C in the subgrid to the adjacent square in the subg... | 1296 |
A cube, all of whose surfaces are painted, is cut into $1000$ smaller cubes of the same size. Find the expected value $E(X)$, where $X$ denotes the number of painted faces of a small cube randomly selected. | \frac{3}{5} |
Compute $\frac{x^8+12x^4+36}{x^4+6}$ when $x=5$. | 631 |
A certain electronic device contains three components, with probabilities of failure for each component being $0.1, 0.2, 0.3$, respectively. If the probabilities of the device failing when one, two, or three components fail are $0.25, 0.6, 0.9$, respectively, find the probability that the device fails. | 0.1601 |
Let $ABC$ be a right triangle where $\measuredangle A = 90^\circ$ and $M\in (AB)$ such that $\frac{AM}{MB}=3\sqrt{3}-4$ . It is known that the symmetric point of $M$ with respect to the line $GI$ lies on $AC$ . Find the measure of $\measuredangle B$ . | 30 |
Maurice travels to work either by his own car (and then due to traffic jams, he is late in half the cases) or by subway (and then he is late only one out of four times). If on a given day Maurice arrives at work on time, he always uses the same mode of transportation the next day as he did the day before. If he is late... | 2/3 |
Consider the sequence defined recursively by $u_1 = a > 0$ and
\[u_{n + 1} = -\frac{1}{u_n + 1}\]for $n \ge 1.$ Express $u_{16}$ in terms of $a.$ | a |
For a nonnegative integer $n$, let $r_7(3n)$ represent the remainder when $3n$ is divided by $7$. Determine the $22^{\text{nd}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\le 4~.$$ | 29 |
Nicky is trading baseball cards with Jill. If Nicky trades two cards worth $8 each for 1 card worth $21, how many dollars of profit does he make? | First find the total cost of the two $8 cards: $8/card * 2 cards = $<<8*2=16>>16
Then subtract the cost of the cards Nicky gave away from the cost of the card he received to find his profit: $21 - $16 = $<<21-16=5>>5
#### 5 |
There is a list of seven numbers. The average of the first four numbers is $5$, and the average of the last four numbers is $8$. If the average of all seven numbers is $6\frac{4}{7}$, then the number common to both sets of four numbers is | 6 |
Given the set $A=\{2,3,4,8,9,16\}$, if $a\in A$ and $b\in A$, the probability that the event "$\log_{a}b$ is not an integer but $\frac{b}{a}$ is an integer" occurs is $\_\_\_\_\_\_$. | \frac{1}{18} |
A box contains 5 white balls and 6 black balls. Five balls are drawn out of the box at random. What is the probability that they all are white? | \dfrac{1}{462} |
The amount $2.5$ is split into two nonnegative real numbers uniformly at random, for instance, into $2.143$ and $.357$, or into $\sqrt{3}$ and $2.5-\sqrt{3}$. Then each number is rounded to its nearest integer, for instance, $2$ and $0$ in the first case above, $2$ and $1$ in the second. What is the probability that th... | \frac{3}{5} |
On the sides \(A B, B C, C D\) and \(A D\) of the convex quadrilateral \(A B C D\) are points \(M, N, K\) and \(L\) respectively, such that \(A M: M B = 3: 2\), \(C N: N B = 2: 3\), \(C K = K D\) and \(A L: L D = 1: 2\). Find the ratio of the area of the hexagon \(M B N K D L\) to the area of the quadrilateral \(A B C ... | 4/5 |
A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$ . In what ratio does the fold line divide side $BC$ ? | 5/3 |
A circle with center $O$ has radius $8$ units and circle $P$ has radius $2$ units. The circles are externally tangent to each other at point $Q$. Segment $TS$ is the common external tangent to circle $O$ and circle $P$ at points $T$ and $S$, respectively. What is the length of segment $OS$? Express your answer in s... | 8\sqrt{2} |
Two equal circles in the same plane cannot have the following number of common tangents. | 1 |
All positive integers whose digits add up to 11 are listed in increasing order: $29, 38, 47, ...$. What is the eleventh number in that list? | 137 |
Let $\triangle ABC$ have sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$ respectively, given that $a^{2}+2b^{2}=c^{2}$, then $\dfrac {\tan C}{\tan A}=$ ______ ; the maximum value of $\tan B$ is ______. | \dfrac { \sqrt {3}}{3} |
In a trapezoid $ABCD$ with bases $\overline{AB} \parallel \overline{CD}$ and $\overline{BC} \perp \overline{CD}$, suppose that $CD = 10$, $\tan C = 2$, and $\tan D = 1$. Calculate the length of $AB$ and determine the area of the trapezoid. | 300 |
Alice drew a regular $2021$-gon in the plane. Bob then labeled each vertex of the $2021$-gon with a real number, in such a way that the labels of consecutive vertices differ by at most $1$. Then, for every pair of non-consecutive vertices whose labels differ by at most $1$, Alice drew a diagonal connecting them. Let $d... | 2018 |
Player A and player B are two basketball players shooting from the same position independently, with shooting accuracies of $\dfrac{1}{2}$ and $p$ respectively, and the probability of player B missing both shots is $\dfrac{1}{16}$.
- (I) Calculate the probability that player A hits at least one shot in two attempts.
- ... | \dfrac{3}{8} |
The eccentricity of the ellipse $\frac {x^{2}}{9}+ \frac {y^{2}}{4+k}=1$ is $\frac {4}{5}$. Find the value of $k$. | 21 |
The probability of getting rain on any given day in August in Beach Town is \(\frac{1}{5}\). What is the probability that it rains on at most 3 days in the first week of August? | 0.813 |
Trapezoid $ABCD$ has $\overline{AB}\parallel\overline{CD}, BC=CD=43$, and $\overline{AD}\perp\overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$, wh... | 194 |
Find the product of all constants \(t\) such that the quadratic \(x^2 + tx + 12\) can be factored in the form \((x+a)(x+b)\), where \(a\) and \(b\) are integers. | 530816 |
The letters of the alphabet are given numeric values based on the two conditions below.
$\bullet$ Only the numeric values of $-2,$ $-1,$ $0,$ $1$ and $2$ are used.
$\bullet$ Starting with A and going through Z, a numeric value is assigned to each letter according to the following pattern: $$
1, 2, 1, 0, -1, -2, -1,... | -1 |
Troy bakes 2 pans of brownies, cut into 16 pieces per pan. The guests eat all of 1 pan, and 75% of the 2nd pan. One tub of ice cream provides 8 scoops. If all but 4 of the guests eat the brownies ala mode (with 2 scoops of vanilla ice cream), how many tubs of ice cream did the guests eat? | Troy bakes 2 x 16 = <<2*16=32>>32 brownie pieces.
75% of one pan is 16 x 0.75 = <<16*0.75=12>>12 pieces.
Guests eat 16 + 12 = <<16+12=28>>28 pieces.
This many guests eat the brownies ala mode 28 - 4 = <<28-4=24>>24.
This many scoops are eaten by the guests 24 x 2 = <<24*2=48>>48.
The guests eat this many tubs 48 / 8 = ... |
Two positive integers differ by 6 and their product is 135. What is the larger integer? | 15 |
John's piggy bank contains quarters, dimes, and nickels. He has three more dimes than quarters and 6 fewer nickels than quarters. If he has 63 coins, how many quarters does John have? | Let x represent the number of quarters John has
Nickels: x-6
Dimes: x+3
Total:x+x-6+x+3=63
3x-3=63
3x=66
x=<<22=22>>22 quarters
#### 22 |
Two real numbers $x$ and $y$ satisfy $x-y=4$ and $x^3-y^3=28$. Compute $xy$. | -3 |
The desired number is greater than 400 and less than 500. Find it if the sum of its digits is 9 and it is equal to 47/36 of the number obtained by reversing its digits. | 423 |
Carlson bought land that cost $8000 and additional land that cost $4000. He initially owned 300 square meters of land. If the land he bought costs $20 per square meter, how big is his land after buying the new land? | Carlson bought land that cost $20 per square meter, therefore the size of the land he bought is 8000/20 = 400.
And the size of the other land he bought is 4000/20 = <<4000/20=200>>200.
Therefore the total size of the land he owns is 400+200+300 = <<400+200+300=900>>900.
#### 900 |
The distance between two vectors is the magnitude of their difference. Find the value of $t$ for which the vector
\[\bold{v} = \begin{pmatrix} 2 \\ -3 \\ -3 \end{pmatrix} + t \begin{pmatrix} 7 \\ 5 \\ -1 \end{pmatrix}\]is closest to
\[\bold{a} = \begin{pmatrix} 4 \\ 4 \\ 5 \end{pmatrix}.\] | \frac{41}{75} |
The trip from Carville to Nikpath requires $4\frac 12$ hours when traveling at an average speed of 70 miles per hour. How many hours does the trip require when traveling at an average speed of 60 miles per hour? Express your answer as a decimal to the nearest hundredth. | 5.25 |
For a real number $y$, find the maximum value of
\[
\frac{y^6}{y^{12} + 3y^9 - 9y^6 + 27y^3 + 81}.
\] | \frac{1}{27} |
Given $$a_{n}= \frac {n(n+1)}{2}$$, remove all the numbers in the sequence $\{a_n\}$ that can be divided by 2, and arrange the remaining numbers in ascending order to form the sequence $\{b_n\}$. Find the value of $b_{21}$. | 861 |
Nancy has six pairs of boots, nine more pairs of slippers than boots, and a number of pairs of heels equal to three times the combined number of slippers and boots. How many shoes (individual shoes, not pairs) does she have? | First find how many pairs of slippers Nancy has: 6 pairs + 9 pairs = <<6+9=15>>15 pairs
Then add that number to the number of pairs of boots she has: 15 pairs + 6 pairs = <<15+6=21>>21 pairs
Then triple that number to find how many pairs of heels she has: 21 pairs * 3 = <<21*3=63>>63 pairs
Then add the number of pairs ... |
Two different integers are randomly chosen from the set $$\{ -5, -8, 7, 4, -2 \}.$$ What is the probability that their product is negative? Express your answer as a common fraction. | \frac{3}{5} |
The increasing sequence \( T = 2, 3, 5, 6, 7, 8, 10, 11, \ldots \) consists of all positive integers which are not perfect squares. What is the 2012th term of \( T \)? | 2057 |
Given that $$(x+y+z)(xy+xz+yz)=25$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=7$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | 6 |
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 |
Every year, four clowns and thirty children go on a carousel. This year, the candy seller, at the carousel, had 700 candies. The candy seller then sold 20 candies, to each of the clowns and the children, who attended. How many candies did he have left? | Since there are four clowns at the carousel this year, the candy seller sold 20*4 = <<20*4=80>>80 candies to all the clowns.
The thirty children also bought 20*30 = <<20*30=600>>600 candies in total.
Altogether, the candy seller sold 600+80 = <<600+80=680>>680 candies to the clowns and children at the carousel.
The num... |
Given two circles $x^{2}+y^{2}=4$ and $x^{2}+y^{2}-2y-6=0$, find the length of their common chord. | 2\sqrt{3} |
In triangle $XYZ$, $E$ lies on $\overline{YZ}$ and $G$ lies on $\overline{XY}$. Let $\overline{XE}$ and $\overline{YG}$ intersect at $Q.$
If $XQ:QE = 5:2$ and $GQ:QY = 3:4$, find $\frac{XG}{GY}.$ | \frac{4}{3} |
If $|x| + x + y = 10$ and $x + |y| - y = 12,$ find $x + y.$ | \frac{18}{5} |
Bekah had to read 408 pages for history class. She read 113 pages over the weekend and has 5 days left to finish her reading. How many pages will she need to read each day for 5 days to complete her assignment? | Pages left to read: 408 - 113 = <<408-113=295>>295 pages
295/5 = <<295/5=59>>59 pages
Bekah needs to read 59 pages each day.
#### 59 |
If $y = -x^2 + 5$ and $x$ is a real number, then what is the maximum value possible for $y$? | 5 |
When a class of math students lined up for a class picture on Picture Day, they found that when they stood in rows of four there was one person left over. When they formed rows of five students, there were two extra students, and when they lined up in rows of seven, there were three students left over. What is the fewe... | 17 |
Carol sells tickets for an exhibition. During three days she sold tickets worth $960. One ticket costs $4. How many tickets on average did she sell during one of these three days? | Carol sold 960 / 4 = <<960/4=240>>240 tickets during three days.
So in one day, she sold on average 240 / 3 = <<240/3=80>>80 tickets.
#### 80 |
C is the complex numbers. \( f : \mathbb{C} \to \mathbb{R} \) is defined by \( f(z) = |z^3 - z + 2| \). What is the maximum value of \( f \) on the unit circle \( |z| = 1 \)? | \sqrt{13} |
Convert -630° to radians. | -\frac{7\pi}{2} |
Frankie and Carla played 30 games of ping pong against each other. Frankie won half as many games as did Carla. How many games did Carla win? | Let x be the number of games that Frankie won.
Then the number of games Carla won would be 2*x.
And the sum of all the games would be x+2*x=30 games.
Thus, the expression simplifies to 3*x=30.
And the value of x=<<10=10>>10 games.
Therefore, the number of Carla's wins would be 2x=20 games.
#### 20 |
June and Julia live 1 mile apart. It takes June 4 minutes to ride her bike directly to Julia's house. At the same rate, how many minutes would it take June to ride the 3.5 miles from her own house to Bernard's house? | 14 |
Let $\mathbf{u}$ and $\mathbf{v}$ be unit vectors, and let $\mathbf{w}$ be a vector such that $\mathbf{u} \times \mathbf{v} + \mathbf{u} = \mathbf{w}$ and $\mathbf{w} \times \mathbf{u} = \mathbf{v}.$ Compute $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}).$ | 1 |
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