problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
There were 90 jellybeans in a jar. Samantha snuck 24 jellybeans out of the jar, without being seen. Shelby ate 12 jellybeans from the jar. Their mom, Shannon, refilled the jar with half as much as Samantha and Shelby took out. How many jellybeans are in the jar now? | There were 90-24 = <<90-24=66>>66 jellybeans in the jar after Samantha snuck some.
There were 66-12 = <<66-12=54>>54 jellybean in the jar after Shelby ate some.
Samantha and Shelby took 24+12 = <<24+12=36>>36 jellybeans from the jar.
Shelby refilled the jar with 36/2 = <<36/2=18>>18 jellybeans.
There are 54+18 = <<54+1... |
In $\triangle{ABC}$ with side lengths $AB = 13$, $AC = 12$, and $BC = 5$, let $O$ and $I$ denote the circumcenter and incenter, respectively. A circle with center $M$ is tangent to the legs $AC$ and $BC$ and to the circumcircle of $\triangle{ABC}$. What is the area of $\triangle{MOI}$?
$\textbf{(A)}\ 5/2\qquad\textbf{(... | \frac{7}{2} |
Let vector $a = (\cos 25^\circ, \sin 25^\circ)$, $b = (\sin 20^\circ, \cos 20^\circ)$. If $t$ is a real number, and $u = a + tb$, then the minimum value of $|u|$ is \_\_\_\_\_\_\_\_. | \frac{\sqrt{2}}{2} |
Using the six digits 0, 1, 2, 3, 4, 5,
(1) How many distinct three-digit numbers can be formed?
(2) How many distinct three-digit odd numbers can be formed? | 48 |
Given that the magnitude of the star Altair is $0.75$ and the magnitude of the star Vega is $0$, determine the ratio of the luminosity of Altair to Vega. | 10^{-\frac{3}{10}} |
Find all real numbers $x$ so that the product $(x + i)((x + 1) + i)((x + 2) + i)$ is pure imaginary. Enter all the solutions, separated by commas. | -3,-1,1 |
The product of the digits of a 5 -digit number is 180 . How many such numbers exist? | 360 |
Let $S_1 = \{(x, y)|\log_{10}(1 + x^2 + y^2) \le 1 + \log_{10}(x+y)\}$ and $S_2 = \{(x, y)|\log_{10}(2 + x^2 + y^2) \le 2 + \log_{10}(x+y)\}$. What is the ratio of the area of $S_2$ to the area of $S_1$? | 102 |
A number of linked rings, each $1$ cm thick, are hanging on a peg. The top ring has an outside diameter of $20$ cm. The outside diameter of each of the outer rings is $1$ cm less than that of the ring above it. The bottom ring has an outside diameter of $3$ cm. What is the distance, in cm, from the top of the top ring ... | 173 |
The set $\{[x]+[2x]+[3x] \mid x \in \mathbf{R}\} \bigcap \{1, 2, \cdots, 100\}$ contains how many elements, where $[x]$ represents the greatest integer less than or equal to $x$. | 67 |
In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is known that $A=\frac{\pi}{4}$ and $b=\frac{\sqrt{2}}{2}a$.
(Ⅰ) Find the magnitude of $B$;
(Ⅱ) If $a=\sqrt{2}$, find the area of $\Delta ABC$. | \frac{\sqrt{3}+1}{4} |
If $\frac{b}{a} = 2$ and $\frac{c}{b} = 3$, what is the ratio of $a + b$ to $b + c$? | \frac{3}{8} |
12 real numbers x and y satisfy \( 1 + \cos^2(2x + 3y - 1) = \frac{x^2 + y^2 + 2(x+1)(1-y)}{x-y+1} \). Find the minimum value of xy. | \frac{1}{25} |
Calculate the value of $8\cos ^{2}25^{\circ}-\tan 40^{\circ}-4$. | \sqrt{3} |
The number $695$ is to be written with a factorial base of numeration, that is, $695=a_1+a_2\times2!+a_3\times3!+ \ldots a_n \times n!$ where $a_1, a_2, a_3 ... a_n$ are integers such that $0 \le a_k \le k,$ and $n!$ means $n(n-1)(n-2)...2 \times 1$. Find $a_4$ | 3 |
If $n$ is an integer, what is the remainder when the sum of $7 - n$ and $n + 3$ is divided by $7$? | 3 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it satisfies $(2a-c)\cos B = b\cos C$;
(1) Find the magnitude of angle $B$;
(2) Let $\overrightarrow{m}=(\sin A, \cos 2A), \overrightarrow{n}=(4k,1)$ ($k>1$), and the maximum value of $\overrightarr... | \frac{3}{2} |
On 40 squares of an $8 \times 8$ chessboard, a stone was placed on each square. The product of the number of stones on white squares and the number of stones on black squares was calculated. Find the minimum possible value of this product. | 256 |
Find the smallest three-digit number in a format $abc$ (where $a, b, c$ are digits, $a \neq 0$) such that when multiplied by 111, the result is not a palindrome. | 105 |
Rectangle $ABCD$ is 8 cm by 4 cm. $M$ is the midpoint of $\overline{BC}$ , and $N$ is the midpoint of $\overline{CD}$. What is the number of square centimeters in the area of region $AMCN$?
[asy]
draw((0,0)--(32,0)--(32,16)--(0,16)--cycle);
draw((0,16)--(16,0)--(32,8)--cycle);
label("$A$",(0,16),N);
label("$B$",(32,16... | 16 |
You can lower the price by 20% if you buy more than fifteen units of iPhone cases. If you pay $500 to buy 18 units, what is the original price? | If you buy 18 units, the percentage of the total price drops to 100% - 20% = 80%
If 80% is equal to $500, then 100% is equals 100%/80% * $500 = $625
#### 625 |
A parallelogram in the coordinate plane has vertices at points (2,1), (7,1), (5,6), and (10,6). Calculate the sum of the perimeter and the area of the parallelogram. | 35 + 2\sqrt{34} |
On each of 7 Mondays, it rained 1.5 centimeters. On each of 9 Tuesdays it rained 2.5 centimeters. How many more centimeters did it rain on Tuesdays than Mondays? | Mondays: 7 * 1.5 = <<7*1.5=10.5>>10.5
Tuesdays: 9 * 2.5 = <<9*2.5=22.5>>22.5
22.5 - 10.5 = <<22.5-10.5=12>>12 cm
It rained 12 centimeters more on Tuesdays than on Mondays.
#### 12 |
The positive difference between the two roots of the quadratic equation $3x^2 - 7x - 8 = 0$ can be written as $\frac{\sqrt{m}}{n}$, where $n$ is an integer and $m$ is an integer not divisible by the square of any prime number. Find $m + n$. | 148 |
Bob Barker went back to school for a PhD in math, and decided to raise the intellectual level of The Price is Right by having contestants guess how many objects exist of a certain type, without going over. The number of points you will get is the percentage of the correct answer, divided by 10, with no points for going... | 292864 |
Cyclic quadrilateral $ABCD$ satisfies $\angle ABD = 70^\circ$ , $\angle ADB=50^\circ$ , and $BC=CD$ . Suppose $AB$ intersects $CD$ at point $P$ , while $AD$ intersects $BC$ at point $Q$ . Compute $\angle APQ-\angle AQP$ . | 20 |
Natasha and Inna each bought the same box of tea bags. It is known that one tea bag is enough for either two or three cups of tea. This box lasted Natasha for 41 cups of tea, and Inna for 58 cups of tea. How many tea bags were in the box? | 20 |
Find the smallest possible sum of two perfect squares such that their difference is 175 and both squares are greater or equal to 36. | 625 |
The sum of the first n terms of the sequence $\{a_n\}$ is $S_n$. If the terms of the sequence $\{a_n\}$ are arranged according to the following rule: $$\frac {1}{2}, \frac {1}{3}, \frac {2}{3}, \frac {1}{4}, \frac {2}{4}, \frac {3}{4}, \frac {1}{5}, \frac {2}{5}, \frac {3}{5}, \frac {4}{5}, \ldots, \frac {1}{n}, \frac ... | \frac{6}{7} |
The integer $x$ has 12 positive factors. The numbers 12 and 15 are factors of $x$. What is $x$? | 60 |
Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done? | 20 |
Given that events A and B are independent, and both are mutually exclusive with event C. It is known that $P(A) = 0.2$, $P(B) = 0.6$, and $P(C) = 0.14$. Find the probability that at least one of A, B, or C occurs, denoted as $P(A+B+C)$. | 0.82 |
When the vectors $\begin{pmatrix} -5 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ are both projected onto the same vector $\mathbf{v},$ the result is $\mathbf{p}$ in both cases. Find $\mathbf{p}.$ | \begin{pmatrix} -34/53 \\ 119/53 \end{pmatrix} |
Emmett does 12 jumping jacks, 8 pushups, and 20 situps. What percentage of his exercises were pushups? | First find the total number of exercises Emmett does: 12 jumping jacks + 8 pushups + 20 situps = <<12+8+20=40>>40 exercises
Then divide the number of pushups by the total number of exercises and multiply by 100% to express the answer as a percentage: 8 pushups / 40 exercises * 100% = 20%
#### 20 |
Given $\cos\left(\alpha + \frac{\pi}{6}\right) = \frac{1}{3}$, where $\alpha$ is in the interval $\left(0, \frac{\pi}{2}\right)$, find the values of $\sin\alpha$ and $\sin\left(2\alpha + \frac{5\pi}{6}\right)$. | -\frac{7}{9} |
Ryosuke is picking up his friend from work. The odometer reads 74,568 when he picks his friend up, and it reads 74,592 when he drops his friend off at his house. Ryosuke's car gets 28 miles per gallon and the price of one gallon of gas is $\$4.05$. What was the cost of the gas that was used for Ryosuke to drive his fri... | \$3.47 |
The numbers \(a, b, c, d\) belong to the interval \([-12.5, 12.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | 650 |
Six distinguishable players are participating in a tennis tournament. Each player plays one match of tennis against every other player. There are no ties in this tournament; each tennis match results in a win for one player and a loss for the other. Suppose that whenever $A$ and $B$ are players in the tournament such t... | 2048 |
Four students are admitted to three universities. Find the probability that each university admits at least one student. | \frac{4}{9} |
Compute $\cos 72^\circ.$ | \frac{-1 + \sqrt{5}}{4} |
Henry drinks 15 bottles of kombucha every month. Each bottle costs $3.00 and is eligible for a cash refund of $0.10 per bottle when he takes it to a recycling center. After 1 year, how many bottles of kombucha will he be able to buy after he receives his cash refund? | He drinks 15 bottles every month and there are 12 months in 1 year so that's 15*12 = <<15*12=180>>180 bottles
He earns $0.10 on every bottle that he recycles so that's .10*180 = $<<.10*180=18.00>>18.00
He has $18.00 and each bottle costs $3.00 so he can buy 18/3 = <<18/3=6>>6 bottles
#### 6 |
Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | 0.296 |
Find all primes $p$ and $q$ such that $3p^{q-1}+1$ divides $11^p+17^p$ | (3, 3) |
Let \( f(x) = a^x - 1 \). Find the largest value of \( a > 1 \) such that if \( 0 \leq x \leq 3 \), then \( 0 \leq f(x) \leq 3 \). | \sqrt[3]{4} |
A fair coin is tossed 4 times. What is the probability of at least two consecutive heads? | \frac{5}{8} |
Given that $(1+\sin t)(1+\cos t)=5/4$ and
$(1-\sin t)(1-\cos t)=\frac mn-\sqrt{k},$
where $k, m,$ and $n$ are positive integers with $m$ and $n$ relatively prime, find $k+m+n.$
| 27 |
Let $T = \{9^k : k ~ \mbox{is an integer}, 0 \le k \le 4000\}$. Given that $9^{4000}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of $T$ have 9 as their leftmost digit?
| 184 |
It takes Omar 12 minutes to raise his kite 240 feet into the air. Jasper can raise his kite at three times the rate of speed as Omar can raise his kite. If Jasper raises his kite to a height of 600 feet, how many minutes will it take? | Jasper raises his kite at a rate of 240 feet in 12 minutes, or 240/12=<<240/12=20>>20 feet per minute.
Jasper raises his kite at three times the speed Omar does, or 3*20=<<3*20=60>>60 feet per minute.
To raise his kite 600 feet, it will take Jasper 600/60=<<600/60=10>>10 minutes.
#### 10 |
Suppose $a$, $b$, $c$ and $d$ are integers satisfying: $a-b+c=5$, $b-c+d=6$, $c-d+a=3$, and $d-a+b=2$. What is the value of $a+b+c+d$? | 16 |
Mark bought a shirt, pants, and shoes for $340. What is the price of the pants knowing that the price of a shirt is three-quarters of the price of the pants and that the price of a shoe is ten dollars more than the price of the pants? | Let X be the price of the pants. The price of the shirt is 3/4*X. The price of the shoes is X+10.
The total amount paid is X + 3/4*X + X+10 = $340.
Combining like terms we get X*11/4 + 10 = $340.
Subtracting 10 from both sides we get X*11/4 = $330.
Dividing both sides by 11/4 we get X = $120.
#### 120 |
Every day, Billie bakes 3 pumpkin pies for 11 days and puts them in the refrigerator. It takes 2 cans of whipped cream to cover 1 pie. If Tiffany comes over and eats 4 pies, how many cans of whipped cream does Billie need to buy to cover the remaining pies? | In total Billie bakes 3 x 11 = <<3*11=33>>33 pies.
After Tiffany eats the pies, this many pies are in the refrigerator: 33 - 4 = <<33-4=29>>29 pies.
She needs to buy this many cans of whipped cream 29 x 2 = <<29*2=58>>58 cans.
#### 58 |
Given a structure formed by joining eight unit cubes where one cube is at the center, and each face of the central cube is shared with one additional cube, calculate the ratio of the volume to the surface area in cubic units to square units. | \frac{4}{15} |
A parametric graph is defined by:
\[
x = \cos t + \frac{t}{3}, \quad y = \sin t.
\]
Determine the number of times the graph intersects itself between \(x = 3\) and \(x = 45\). | 12 |
What is the area of the region enclosed by $x^2 + y^2 = |x| - |y|$? | \frac{\pi}{2} |
In the diagram, rectangle $PQRS$ is divided into three identical squares. If $PQRS$ has perimeter 120 cm, what is its area, in square centimeters? [asy]
size(4cm);
pair p = (0, 1); pair q = (3, 1); pair r = (3, 0); pair s = (0, 0);
draw(p--q--r--s--cycle);
draw(shift(1) * (p--s)); draw(shift(2) * (p--s));
label("$... | 675 |
Find the smallest solution to the equation \[\lfloor x^2 \rfloor - \lfloor x \rfloor^2 = 19.\] | \sqrt{109} |
When simplified, what is the value of $\sqrt{3} \times 3^{\frac{1}{2}} + 12 \div 3 \times 2 - 4^{\frac{3}{2}}$? | 3 |
The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$? | 7 |
Given a pedestrian signal light that alternates between red and green, with the red light lasting for 50 seconds, calculate the probability that a student needs to wait at least 20 seconds for the green light to appear. | \dfrac{3}{5} |
Express $\frac{165_7}{11_2}+\frac{121_6}{21_3}$ in base 10. | 39 |
Vasya wrote a note on a piece of paper, folded it in four, and labeled the top with "MAME". Then he unfolded the note, added something else, folded it along the creases in a random manner (not necessarily the same as before), and left it on the table with a random side up. Find the probability that the inscription "MAM... | 1/8 |
Find the area of the parallelogram generated by $\begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ -3 \\ 4 \end{pmatrix}.$
[asy]
unitsize(0.4 cm);
pair A, B, C, D;
A = (0,0);
B = (7,2);
C = (1,3);
D = B + C;
draw(A--B,Arrow(6));
draw(A--C,Arrow(6));
draw(B--D--C);
[/asy] | 10 \sqrt{3} |
Thomas has 25 drawings to display. 14 of the drawings were made using colored pencils. 7 of the drawings were made using blending markers. The rest of the drawings were made with charcoal. How many are charcoal drawings? | Thomas made 14 + 7 = <<14+7=21>>21 colored pencil and blending marker drawings.
Thomas made 25 total drawings – 21 colored pencil and blending marker drawings = <<25-21=4>>4 charcoal drawings.
#### 4 |
Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed ... | (2n-1)!! |
Points \(A = (2,8)\), \(B = (2,2)\), and \(C = (6,2)\) lie in the first quadrant and are vertices of triangle \(ABC\). Point \(D=(a,b)\) is also in the first quadrant, and together with \(A\), \(B\), and \(C\), forms quadrilateral \(ABCD\). The quadrilateral formed by joining the midpoints of \(\overline{AB}\), \(\over... | 14 |
The coefficient of the $x^3$ term in the expansion of $(2-\sqrt{x})^8$ is $1120x^3$. | 112 |
Given that in the rectangular coordinate system $(xOy)$, the parametric equations of the curve $C$ are $ \begin{cases} x=2+2\cos θ \
y=2\sin θ\end{cases} $ for the parameter $(θ)$, and in the polar coordinate system $(rOθ)$ (with the same unit length as the rectangular coordinate system $(xOy)$, and the origin $O$ as t... | 2 \sqrt {2} |
Given that the function $F(x) = f(x) + x^2$ is an odd function, and $f(2) = 1$, find $f(-2) = ( \ )$. | -9 |
In a factory, there are 3 machines working 23 hours a day. The owner decided to buy a fourth machine, which works only 12 hours a day. One machine can produce 2 kg of material every hour. The factory sells the produced material for $50 per 1 kg. How much can this factory earn in one day? | The total runtime for the first three machines is 3 machines * 23 hours/day/machine = <<3*23=69>>69 hours/day.
So these machines could produce 69 hours/day * 2 kg/hour = <<69*2=138>>138 kg/day.
The fourth machine could produce 12 hours/day * 2 kg/hour = <<12*2=24>>24 kg/day of material.
In total all the machines would ... |
The function $f(x)$ satisfies $f(1) = 1$ and
\[f(x + y) = 3^y f(x) + 2^x f(y)\]for all real numbers $x$ and $y.$ Find the function $f(x).$ | 3^x - 2^x |
What is $6^{12} \div 36^5$? | 36 |
Find the nonconstant polynomial $P(x)$ such that
\[P(P(x)) = (x^2 + x + 1) P(x).\] | x^2 + x |
Let $C$ be a point not on line $AE$ and $D$ a point on line $AE$ such that $CD \perp AE.$ Meanwhile, $B$ is a point on line $CE$ such that $AB \perp CE.$ If $AB = 4,$ $CD = 8,$ and $AE = 5,$ then what is the length of $CE?$ | 10 |
In a certain region are five towns: Freiburg, Göttingen, Hamburg, Ingolstadt, and Jena.
On a certain day, 40 trains each made a journey, leaving one of these towns and arriving at one of the other towns. Ten trains traveled either from or to Freiburg. Ten trains traveled either from or to Göttingen. Ten trains travel... | 40 |
Given the function $f(x)=\sqrt{2}\sin(2\omega x-\frac{\pi}{12})+1$ ($\omega > 0$) has exactly $3$ zeros in the interval $\left[0,\pi \right]$, determine the minimum value of $\omega$. | \frac{5}{3} |
Find the largest integer value of $n$ such that $n^2-9n+18$ is negative. | 5 |
Consider a regular tetrahedron $ABCD$. Find $\sin \angle BAC$. | \frac{2\sqrt{2}}{3} |
Let $\omega$ be a nonreal root of $x^3 = 1.$ Compute
\[(1 - \omega + \omega^2)^4 + (1 + \omega - \omega^2)^4.\] | -16 |
Calvin has been saving his hair clippings after each haircut to make a wig for his dog. He has gotten 8 haircuts and knows that he needs 2 more to reach his goal. What percentage towards his goal is he? | His goal is 10 haircuts because 8 + 2 = <<8+2=10>>10
He is 80% there because (8 / 10) x 100 = <<(8/10)*100=80>>80
#### 80 |
Find the least positive integer $n$ such that $$\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.$$ | 1 |
[asy] draw(circle((0,6sqrt(2)),2sqrt(2)),black+linewidth(.75)); draw(circle((0,3sqrt(2)),sqrt(2)),black+linewidth(.75)); draw((-8/3,16sqrt(2)/3)--(-4/3,8sqrt(2)/3)--(0,0)--(4/3,8sqrt(2)/3)--(8/3,16sqrt(2)/3),dot); MP("B",(-8/3,16*sqrt(2)/3),W);MP("B'",(8/3,16*sqrt(2)/3),E); MP("A",(-4/3,8*sqrt(2)/3),W);MP("A'",(4/3,8*s... | 2\pi |
For how many $n=2,3,4,\ldots,99,100$ is the base-$n$ number $235236_n$ a multiple of $7$? | 14 |
Calculate $7 \cdot 9\frac{2}{5}$. | 65\frac{4}{5} |
Let $A$ denote the set of all integers $n$ such that $1 \leq n \leq 10000$, and moreover the sum of the decimal digits of $n$ is 2. Find the sum of the squares of the elements of $A$. | 7294927 |
Luke wants to fence a rectangular piece of land with an area of at least 450 square feet. The length of the land is 1.5 times the width. What should the width of the rectangle be if he wants to use the least amount of fencing? | 10\sqrt{3} |
In triangle $ABC$, $AB=10$, $BC=12$ and $CA=14$. Point $G$ is on $\overline{AB}$, $H$ is on $\overline{BC}$, and $I$ is on $\overline{CA}$. Let $AG=s\cdot AB$, $BH=t\cdot BC$, and $CI=u\cdot CA$, where $s$, $t$, and $u$ are positive and satisfy $s+t+u=3/4$ and $s^2+t^2+u^2=3/7$. The ratio of the area of triangle $GHI$ ... | 295 |
In an organization with 200 employees, those over the age of 50 account for 20%, those aged 40-50 make up 30%, and those under 40 account for 50%. If 40 employees are to be sampled, and the systematic sampling method is used—where all employees are randomly numbered 1-200 and evenly divided into 40 groups (numbers 1-5,... | 20 |
Evaluate: $(12345679^2 \times 81 - 1) \div 11111111 \div 10 \times 9 - 8$ in billions. (Answer in billions) | 10 |
How many minutes are needed at least to finish these tasks: washing rice for 2 minutes, cooking porridge for 10 minutes, washing vegetables for 3 minutes, and chopping vegetables for 5 minutes. | 12 |
For some constants $a$ and $c,$
\[\begin{pmatrix} a \\ -1 \\ c \end{pmatrix} \times \begin{pmatrix} 7 \\ 3 \\ 5 \end{pmatrix} = \begin{pmatrix} -11 \\ -16 \\ 25 \end{pmatrix}.\]Enter the ordered pair $(a,c).$ | (6,2) |
Given the function $f(x)$, for any $x \in \mathbb{R}$, it satisfies $f(x+6) + f(x) = 0$, and the graph of $y=f(x-1)$ is symmetric about the point $(1,0)$. If $f(2) = 4$, find the value of $f(2014)$. | -4 |
Driving along a highway, Megan noticed that her odometer showed $15951$ (miles). This number is a palindrome-it reads the same forward and backward. Then $2$ hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this $2$-hour period? | 55 |
The school band is going to a competition. Five members play the flute. There are three times as many members who play the trumpet. There are eight fewer trombone players than trumpeters, and eleven more drummers than trombone players. There are twice as many members that play the clarinet as members that play the flut... | There are 3 trumpets/flute x 5 flutes = <<3*5=15>>15 trumpet players
There are 15 players - 8 players = <<15-8=7>>7 trombone players
There are 7 players + 11 players = <<7+11=18>>18 drummers.
There are 2 clarinets/flute x 5 flutes = 10 clarinet players
There are 7 players + 3 players = <<7+3=10>>10 French horn players
... |
A survey conducted at a conference found that 70% of the 150 male attendees and 75% of the 850 female attendees support a proposal for new environmental legislation. What percentage of all attendees support the proposal? | 74.2\% |
Given a $4 \times 4$ grid with 16 unit squares, each painted white or black independently and with equal probability, find the probability that the entire grid becomes black after a 90° clockwise rotation, where any white square landing on a place previously occupied by a black square is repainted black. | \frac{1}{65536} |
Eight numbers \( a_{1}, a_{2}, a_{3}, a_{4} \) and \( b_{1}, b_{2}, b_{3}, b_{4} \) satisfy the following equations:
$$
\left\{\begin{array}{c}
a_{1} b_{1}+a_{2} b_{3}=1 \\
a_{1} b_{2}+a_{2} b_{4}=0 \\
a_{3} b_{1}+a_{4} b_{3}=0 \\
a_{3} b_{2}+a_{4} b_{4}=1
\end{array}\right.
$$
It is known that \( a_{2} b_{3}=7 \). F... | -6 |
Simplify and evaluate
(Ⅰ) Evaluate \\( \dfrac{ \sqrt{3}\sin (- \dfrac{20}{3}\pi)}{\tan \dfrac{11}{3}\pi}-\cos \dfrac{13}{4}\pi\cdot\tan (- \dfrac{35}{4}\pi) \).
(Ⅱ) Evaluate: \\( \dfrac{\sqrt{1-2\sin {10}^{\circ }\cos {10}^{\circ }}}{\cos {10}^{\circ }-\sqrt{1-{\cos }^{2}{170}^{\circ }}} \)
(Ⅲ) If \\( \sin \theta, \... | - \dfrac{ \sqrt{7}}{2} |
Let \[\begin{aligned} a &= \sqrt{2}+\sqrt{3}+\sqrt{6}, \\ b &= -\sqrt{2}+\sqrt{3}+\sqrt{6}, \\ c&= \sqrt{2}-\sqrt{3}+\sqrt{6}, \\ d&=-\sqrt{2}-\sqrt{3}+\sqrt{6}. \end{aligned}\]Evaluate $\left(\frac1a + \frac1b + \frac1c + \frac1d\right)^2.$ | \frac{96}{529} |
A fair six-sided die is rolled twice, and the resulting numbers are denoted as $a$ and $b$.
(1) Find the probability that $a^2 + b^2 = 25$.
(2) Given three line segments with lengths $a$, $b$, and $5$, find the probability that they can form an isosceles triangle (including equilateral triangles). | \frac{7}{18} |
Brian can only hold his breath underwater for 10 seconds. He wants to get better, so he starts practicing. After a week, he's doubled the amount of time he can do it. After another week, he's doubled it again from the previous week. The final week, he's increased it by 50% from the previous week. How long can Bria... | At the end of his first week, Brian now has doubled his initial time of 10 seconds to 10*2=<<10*2=20>>20 seconds
At the end of his second week, Brian doubled the previous week's time of 20 seconds to 20*2=<<20*2=40>>40 seconds
At the end of his third week of practice, Brian adds another 50% of 40 seconds to his time so... |
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