problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
How many positive integers $n$ satisfy \[(n + 8)(n - 3)(n-12)<0\] | 8 |
In the trapezoid \( ABCD \) with bases \( AD \) and \( BC \), the side \( AB \) is equal to 2. The angle bisector of \( \angle BAD \) intersects the line \( BC \) at point \( E \). A circle is inscribed in triangle \( ABE \), touching side \( AB \) at point \( M \) and side \( BE \) at point \( H \). Given that \( MH ... | 120 |
A circle has a radius of 3 units. There are many line segments of length 4 units that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.
A) $3\pi$
B) $5\pi$
C) $4\pi$
D) $7\pi$
E) $6\pi$ | 4\pi |
Vanessa wants to buy a dress she saw at the mall, which costs $80, and she already has $20 in savings. Her parents give her $30 every week, but she also spends $10 each weekend at the arcades. How many weeks will she have to wait until she can gather enough money to buy the dress? | Vanessa needs $80 – $20 = $<<80-20=60>>60 to buy the dress.
She manages to gather $30 - $10 = $<<30-10=20>>20 each week
The number of weeks she has to wait is 60 ÷ 20 = <<60/20=3>>3 weeks.
#### 3 |
A number is considered a visible factor number if it is divisible by each of its non-zero digits. For example, 204 is divisible by 2 and 4 and is therefore a visible factor number. Determine how many visible factor numbers exist from 200 to 250, inclusive. | 16 |
The ratio of the sums of the first n terms of the arithmetic sequences {a_n} and {b_n} is given by S_n / T_n = (2n) / (3n + 1). Find the ratio of the fifth terms of {a_n} and {b_n}. | \dfrac{9}{14} |
Andy started out the year weighing 156 pounds. He then grew 3 inches and gained 36 pounds. Andy wasn't happy with his weight and decided to exercise. Over the next 3 months, he lost an eighth of his weight every month. How much less does Andy weigh now than at the beginning of the year? | Andy weighed 156 + 36 = <<156+36=192>>192 pounds after growing taller.
An eighth of his weight is 1/8 x 192 = <<1/8*192=24>>24 pounds.
Over the next 3 months, Andy lost 24 x 3 = <<24*3=72>>72 pounds.
He now weighs 192 - 72 = <<192-72=120>>120 pounds.
Andy weighs 156 - 120 = <<156-120=36>>36 pounds less than at the begi... |
Stella collects stamps. She counted 50 pages in her collector's album. There are 5 rows of 30 stamps in each of the first 10 pages of her album. The rest of the pages each have 50 stamps. How many stamps are in Stella's album? | There are 30 x 5 = <<30*5=150>>150 stamps on each of the first 10 pages.
So she has a total of 150 x 10 = <<150*10=1500>>1500 stamps in the first 10 pages.
There are 50 - 10 = <<50-10=40>>40 pages that have 50 stamps each.
There is a total of 40 x 50 = <<40*50=2000>>2000 stamps in the remaining 40 pages.
Therefore, Ste... |
The hyperbola $M$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ has left and right foci $F_l$ and $F_2$. The parabola $N$: $y^{2} = 2px (p > 0)$ has a focus at $F_2$. Point $P$ is an intersection point of hyperbola $M$ and parabola $N$. If the midpoint of $PF_1$ lies on the $y$-axis, calculate the ecc... | \sqrt{2} + 1 |
There are 20 dolphins in the aqua park. One-fourth of the dolphins are fully trained. Two-third of the remaining dolphins are currently in training and the rest will be trained next month. How many dolphins will be trained next month? | 20 x 1/4 = <<20*1/4=5>>5 dolphins are fully trained.
So, 20 - 5 = <<20-5=15>>15 dolphins are not yet fully trained.
15 x 2/3 = <<15*2/3=10>>10 are currently in training.
Therefore, 15 - 10 = <<15-10=5>>5 dolphins will be trained next month.
#### 5 |
A 26-mile circular marathon has four checkpoints inside it. The first is one mile from the start line, and the last checkpoint is one mile from the finish line. The checkpoints have equal spacing between them. How many miles apart are each of the consecutive checkpoints between the start-finish line? | The checkpoints are 1 mile from the start and finish, so they are spaced along 26 - 1 - 1 = 24 miles.
There are 4 equally spaced checkpoints, so the checkpoints are 24 / 4 = <<24/4=6>>6 miles apart.
#### 6 |
A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. In meters, what is the perimeter of the parking lot? | 62 |
There is only one set of five prime numbers that form an arithmetic sequence with a common difference of 6. What is the sum of those five prime numbers? | 85 |
Let \(a\) and \(b\) be real numbers such that
\[
\frac{a}{2b} + \frac{a}{(2b)^2} + \frac{a}{(2b)^3} + \dots = 6.
\]
Find
\[
\frac{a}{a + 2b} + \frac{a}{(a + 2b)^2} + \frac{a}{(a + 2b)^3} + \dots.
\] | \frac{3}{4} |
Let $f(x) = x^2 + px + q$ and $g(x) = x^2 + rx + s$ 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 $f$. If both $f$ and $g$ have the same minimum value and the graphs of the two polynomials i... | -200 |
Consider the function $f(x) = 2x^2 - 4x + 9$. Evaluate $2f(3) + 3f(-3)$. | 147 |
A certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x$, and that $f(x) = 1 - |x - 2|$ for $1\leq x \leq 3$. Find the smallest $x$ for which $f(x) = f(2001)$. | 429 |
Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? Express your answer as a common fraction. | \frac{1}{8} |
Let $\mathbb{R}$ be the set of real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that for all real numbers $x$ and $y$, we have $$f\left(x^{2}\right)+f\left(y^{2}\right)=f(x+y)^{2}-2 x y$$ Let $S=\sum_{n=-2019}^{2019} f(n)$. Determine the number of possible values of $S$. | 2039191 |
How many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive? | 58 |
The sum $2\frac{1}{7} + 3\frac{1}{2} + 5\frac{1}{19}$ is between | $10\frac{1}{2} \text{ and } 11$ |
A deck of cards now contains 54 cards, including two jokers, one being a red joker and the other a black joker, along with the standard 52 cards. In how many ways can we pick two different cards such that at least one of them is a joker? (Order matters.) | 210 |
Find the number of five-digit palindromes. | 900 |
Let $a$ and $b$ be randomly selected numbers from the set $\{1,2,3,4\}$. Given the line $l:y=ax+b$ and the circle $O:x^{2}+y^{2}=1$, the probability that the line $l$ intersects the circle $O$ is ______. The expected number of intersection points between the line $l$ and the circle $O$ is ______. | \frac{5}{4} |
How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit? | 296 |
There are 15 tables in the school's cafeteria. Each table can seat 10 people. Usually, only 1/10 of the seats are left unseated. How many seats are usually taken? | The cafeteria can seat 15 x 10 = <<15*10=150>>150 people.
Usually, 150 x 1/10 = <<150*1/10=15>>15 seat are unseated.
Thus, 150 - 15 = <<150-15=135>>135 seats are usually taken.
#### 135 |
A jacket was originally priced $\textdollar 100$ . The price was reduced by $10\%$ three times and increased by $10\%$ four times in some order. To the nearest cent, what was the final price? | 106.73 |
James can buy a new computer for $600 and have it last 6 years, or buy 2 used computers for $200 each that each last 3 years. How much money will he save by picking the cheaper option? | First find the cost of the two used computers: $200/computer * 2 computers = $<<200*2=400>>400
Then subtract that cost from the cost of the new computer to find the savings: $600 - $400 = $<<600-400=200>>200
#### 200 |
Alice needs to replace a light bulb located $10$ centimeters below the ceiling in her kitchen. The ceiling is $2.4$ meters above the floor. Alice is $1.5$ meters tall and can reach $46$ centimeters above the top of her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in cen... | 34 |
Joe had walked half way from home to school when he realized he was late. He ran the rest of the way to school. He ran 3 times as fast as he walked. Joe took 6 minutes to walk half way to school. How many minutes did it take Joe to get from home to school? | 8 |
Find the smallest natural number \( n \) such that both \( n^2 \) and \( (n+1)^2 \) contain the digit 7. | 27 |
A magician has a top hat with 20 red marbles and a top hat with 30 blue marbles. If he takes away 3 red marbles and four times as many blue marbles as red marbles (without looking), how many marbles in total does he have left? | He had 20 red marbles and took away 3 leaving 20-3 = <<20-3=17>>17 red marbles
He took 4 times as many blue marbles as red marbles which is 4*3 = <<4*3=12>>12 blue marbles
He took 12 blue marbles from 30 leaving 30-12 = 18 blue marbles
He now has 17+18 = <<17+18=35>>35 marbles left
#### 35 |
Find the number of positive integers $n$ such that
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) > 0.\] | 24 |
What is the probability that Fatima gets fewer heads than tails if she flips 10 coins? | \dfrac{193}{512} |
Find the value of $x$ for the following expressions.
1. $4x^2 = 9$
2. $(1 - 2x)^3 = 8$ | -\frac{1}{2} |
Let $z$ be a complex number. If the equation \[x^3 + (4-i)x^2 + (2+5i)x = z\] has two roots that form a conjugate pair, find the absolute value of the real part of $z$ .
*Proposed by Michael Tang* | 423 |
Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\frac{n}{d}$ where $n$ and $d$ are integers with $1 \le d \le 5$. What is the probability that \[(\text{cos}(a\pi)+i\text{sin}(b\pi))^4\]is a real number? | \frac{6}{25} |
The perimeter of a particular square and the circumference of a particular circle are equal. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of $\pi$. | \frac{\pi}{4} |
Bridge and Sarah have $3 between them. If Bridget has 50 cents more than Sarah, how many cents does Sarah have? | Let X be the amount of money Sarah has, so Bridget has X + 50 cents
We know that Sarah and Bridget combined have $3, or 300 cents, so X + X + 50 = 300
We therefore know that 2X = 250
So Sarah has 250 / 2 = <<250/2=125>>125 cents
#### 125 |
For some positive integer \( n \), the number \( 150n^3 \) has \( 150 \) positive integer divisors, including \( 1 \) and the number \( 150n^3 \). How many positive integer divisors does the number \( 108n^5 \) have? | 432 |
Given an ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, the distance from a point $M$ on the ellipse to the left focus $F_1$ is 8. Find the distance from $M$ to the right directrix. | \frac{5}{2} |
Let $x$ be a positive real number. Find the minimum value of $4x^5 + 5x^{-4}.$ | 9 |
What three-digit number with units digit 2 and hundreds digit 4 is divisible by 9? | 432 |
Marge planted 23 seeds in her garden. Five of the seeds never grew into plants. A third of the remaining seeds grew, but the plants were eaten by squirrels and rabbits. A third of the number of uneaten plants were strangled by weeds. Marge pulled two weeds, but liked the flowers on one weed and let the plant grow as pa... | Marge had 23 - 5 = <<23-5=18>>18 seeds grow into plants.
Rabbits ate 18 / 3 = <<18/3=6>>6 plants.
There were 18 - 6 = <<18-6=12>>12 plants left.
Weeds strangled 12 / 3 = <<12/3=4>>4 plants.
She had 12 - 4 = <<12-4=8>>8 plants left.
Marge kept 1 weed, so she ended up with 8 + 1 = <<8+1=9>>9 plants.
#### 9 |
Theo and Tia are buying food for their picnic basket. They invited two of their friends. They buy individual sandwiches and individual fruit salads. They buy two sodas per person and 3 bags of snacks to share for their friends and themselves. Sandwiches are $5 each. Fruit salad is $3 each. Sodas are $2 each. The sn... | The sandwiches are $5 x 4 = $<<5*4=20>>20.
The fruit salads are $3 x 4 = $<<3*4=12>>12.
The sodas are $2 x 4 x 2 = $<<2*4*2=16>>16.
The snacks are $4 x 3 = $<<4*3=12>>12.
For all the food, they spend $20 + $12 + $16 + $12 = $<<20+12+16+12=60>>60.
#### 60 |
Given that the product of Kiana's age and the ages of her two older siblings is 256, and that they have distinct ages, determine the sum of their ages. | 38 |
Olaf is sailing across the ocean with 25 men, including himself. He needs 1/2 a gallon of water per day per man. The boat can go 200 miles per day and he needs to travel 4,000 miles. How many gallons of water does he need? | He needs 12.5 gallons per day because 25 x .5 = <<25*.5=12.5>>12.5
It will take 20 days of travel because 4,000 / 200 = <<4000/200=20>>20
He will need 250 gallons of water because 20 x 12.5 = <<20*12.5=250>>250
#### 250 |
What is the value of $c$ if $x\cdot(3x+1)<c$ if and only when $x\in \left(-\frac{7}{3},2\right)$? | 14 |
How many rectangles are there whose four vertices are points on this grid? [asy]
size(50);
dot((0,0));
dot((5,0));
dot((10,0));
dot((0,5));
dot((0,10));
dot((5,5));
dot((5,10));
dot((10,5));
dot((10,10));
[/asy] | 10 |
On a luxurious ocean liner, 3000 adults consisting of men and women embark on a voyage. If 55% of the adults are men and 12% of the women as well as 15% of the men are wearing sunglasses, determine the total number of adults wearing sunglasses. | 409 |
Consider an isosceles triangle $T$ with base 10 and height 12. Define a sequence $\omega_{1}, \omega_{2}, \ldots$ of circles such that $\omega_{1}$ is the incircle of $T$ and $\omega_{i+1}$ is tangent to $\omega_{i}$ and both legs of the isosceles triangle for $i>1$. Find the total area contained in all the circles. | \frac{180 \pi}{13} |
Let $a$ and $b$ be real numbers such that $a + 4i$ and $b + 5i$ are the roots of
\[z^2 - (10 + 9i) z + (4 + 46i) = 0.\]Enter the ordered pair $(a,b).$ | (6,4) |
In the Cartesian coordinate plane $(xOy)$, if the line $ax + y - 2 = 0$ intersects the circle centered at $C$ with the equation $(x - 1)^2 + (y - a)^2 = 16$ at points $A$ and $B$, and $\triangle ABC$ is a right triangle, then the value of the real number $a$ is _____. | -1 |
Solve the inequality
\[\frac{x^2 - 25}{x + 5} < 0.\] | (-\infty,-5) \cup (-5,5) |
There are 3414 yellow balloons and there are 1762 more black balloons than yellow balloons. If the balloons will be evenly divided among 10 schools, how many balloons will one school receive? | Black balloons: 3414 + 1762 = <<3414+1762=5176>>5176
Total balloons: 3414 + 5176 = <<3414+5176=8590>>8590
8590/10 = <<8590/10=859>>859
Each school will receive 859 balloons.
#### 859 |
Given that \( f(x) \) and \( g(x) \) are two quadratic functions both with a leading coefficient of 1, where \( g(6) = 35 \) and \( \frac{f(-1)}{g(-1)} = \frac{f(1)}{g(1)} = \frac{21}{20} \), what is \( f(6) \)? | 35 |
How many ways are there to insert +'s between the digits of 111111111111111 (fifteen 1's) so that the result will be a multiple of 30? | 2002 |
Given are real numbers $x, y$. For any pair of real numbers $a_{0}, a_{1}$, define a sequence by $a_{n+2}=x a_{n+1}+y a_{n}$ for $n \geq 0$. Suppose that there exists a fixed nonnegative integer $m$ such that, for every choice of $a_{0}$ and $a_{1}$, the numbers $a_{m}, a_{m+1}, a_{m+3}$, in this order, form an arithme... | \[ y = 0, 1, \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2} \] |
The first term of an arithmetic series of consecutive integers is $k^2 + 1$. The sum of $2k + 1$ terms of this series may be expressed as: | $k^3 + (k + 1)^3$ |
Given that $α$ is an angle in the first quadrant, it satisfies $\sin α - \cos α = \frac{\sqrt{10}}{5}$. Find $\cos 2α$. | - \frac{4}{5} |
In triangle $ABC$, $AB = BC$, and $\overline{BD}$ is an altitude. Point $E$ is on the extension of $\overline{AC}$ such that $BE =
10$. The values of $\tan \angle CBE$, $\tan \angle DBE$, and $\tan \angle ABE$ form a geometric progression, and the values of $\cot \angle DBE$, $\cot \angle CBE$, $\cot \angle DBC$ form ... | \frac{50}{3} |
Rationalize the denominator of $\frac{\sqrt{8}+\sqrt{3}}{\sqrt{2}+\sqrt{3}}$. Express your answer in simplest form. | \sqrt{6}-1 |
Suppose that $k \geq 2$ is a positive integer. An in-shuffle is performed on a list with $2 k$ items to produce a new list of $2 k$ items in the following way: - The first $k$ items from the original are placed in the odd positions of the new list in the same order as they appeared in the original list. - The remaining... | 83 |
Given $0 \leq a_k \leq 1$ for $k=1,2,\ldots,2020$, and defining $a_{2021}=a_1, a_{2022}=a_2$, find the maximum value of $\sum_{k=1}^{2020}\left(a_{k}-a_{k+1} a_{k+2}\right)$. | 1010 |
Solve the system of equations: $20=4a^{2}+9b^{2}$ and $20+12ab=(2a+3b)^{2}$. Find $ab$. | \frac{20}{3} |
$3^n = 3 \cdot 9^3 \cdot 81^2$. What is the value of $n$? | 15 |
Sam has 19 dimes and 6 quarters. She buys 4 candy bars for 3 dimes each and 1 lollipop for 1 quarter. How much money, in cents, does she have left? | The candy bars cost 4*3=<<4*3=12>>12 dimes.
Sam has 19-12=<<19-12=7>>7 dimes left.
Sam has 6-1=<<6-1=5>>5 quarters left.
Sam has 7*10+5*25=<<7*10+5*25=195>>195 cents left.
#### 195 |
Given $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $\overrightarrow{a}⊥(\overrightarrow{a}+\overrightarrow{b})$, determine the projection of $\overrightarrow{a}$ onto $\overrightarrow{b}$. | -\frac{1}{2} |
The decreasing sequence $a, b, c$ is a geometric progression, and the sequence $19a, \frac{124b}{13}, \frac{c}{13}$ is an arithmetic progression. Find the common ratio of the geometric progression. | 247 |
Find the last three digits of $9^{105}.$ | 049 |
The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n\ge3$, $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outside square. For example, figure $F_3$ has $... | 761 |
Two concentric circles have radii $r$ and $R>r$. Three new circles are drawn so that they are each tangent to the big two circles and tangent to the other two new circles. Find $\frac{R}{r}$. | 3 |
While waiting for their next class on Killian Court, Alesha and Belinda both write the same sequence $S$ on a piece of paper, where $S$ is a 2020-term strictly increasing geometric sequence with an integer common ratio $r$. Every second, Alesha erases the two smallest terms on her paper and replaces them with their geo... | \[
\boxed{2018}
\] |
Given that \( a, b, c \) are positive integers, and the parabola \( y = ax^2 + bx + c \) intersects the x-axis at two distinct points \( A \) and \( B \). If the distances from \( A \) and \( B \) to the origin are both less than 1, find the minimum value of \( a + b + c \). | 11 |
There exists \( x_{0} < 0 \) such that \( x^{2} + |x - a| - 2 < 0 \) (where \( a \in \mathbb{Z} \)) is always true. Find the sum of all values of \( a \) that satisfy this condition. | -2 |
In the figure, the visible gray area within the larger circle is equal to three times the area of the white circular region. What is the ratio of the radius of the small circle to the radius of the large circle? Express your answer as a common fraction.
[asy]size(101);
filldraw(Circle((0,0),2)^^Circle((.8,-.3),1),gray... | \frac{1}{2} |
Compute the sum of $x^2+y^2$ over all four ordered pairs $(x,y)$ of real numbers satisfying $x=y^2-20$ and $y=x^2+x-21$ .
*2021 CCA Math Bonanza Lightning Round #3.4* | 164 |
Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 12n = 2012$. | 34 |
The following image is 1024 pixels by 1024 pixels, and each pixel is either black or white. The border defines the boundaries of the image, but is not part of the image. Let $a$ be the proportion of pixels that are black. Estimate $A=\lfloor 10000 a\rfloor$. An estimate of $E$ will earn $\left\lfloor 20 \min \left(\fra... | 3633 |
Find the smallest positive number $\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$ ,if $z_1+z_2+z_3=0$, then $$\left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda .$$ | 1 |
The lifespan of a hamster is 6 years less than that of a bat. The lifespan of a frog is 4 times that of a hamster. Altogether, the lifespan of the animals is 30 years. What is the lifespan of the bat? | Let’s set up an equation by calling the lifespan of a bat x, and therefore the lifespan of the hamster x – 6, and the lifespan of the frog 4 (x – 6) which add together to equal 30, like this: x + x – 6 + 4 (x – 6) = 30.
That gives us 6x – 30 = 30 and we should add 30 to each side to try to isolate the x.
Now we have 6x... |
Triangle $ABC$ has $AB=2 \cdot AC$. Let $D$ and $E$ be on $\overline{AB}$ and $\overline{BC}$, respectively, such that $\angle BAE = \angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose that $\triangle CFE$ is equilateral. What is $\angle ACB$? | 90^\circ |
"Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0\le x<m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $y\equiv x^2\pmod 5$ would consist of the points $(0,0)$, ... | 10 |
Let $m$ be the product of all positive integers less than $5!$ which are invertible modulo $5!$. Find the remainder when $m$ is divided by $5!$. | 119 |
Let $S(n)$ denote the sum of digits of a natural number $n$ . Find all $n$ for which $n+S(n)=2004$ . | 2001 |
Let $a_{1}, a_{2}, \ldots, a_{n}$ be a sequence of distinct positive integers such that $a_{1}+a_{2}+\cdots+a_{n}=2021$ and $a_{1} a_{2} \cdots a_{n}$ is maximized. If $M=a_{1} a_{2} \cdots a_{n}$, compute the largest positive integer $k$ such that $2^{k} \mid M$. | 62 |
Given the sequence $1990-1980+1970-1960+\cdots -20+10$, calculate the sum. | 1000 |
Mary told John her score on the American High School Mathematics Examination (AHSME), which was over $80$. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over $80$, John could not have determined this. What was Mary's score? (Recall that... | 119 |
Determine the largest value of \(x\) that satisfies the equation \(\sqrt{3x} = 5x^2\). Express your answer in simplest fractional form. | \left(\frac{3}{25}\right)^{\frac{1}{3}} |
There is very little car traffic on Happy Street. During the week, most cars pass it on Tuesday - 25. On Monday, 20% less than on Tuesday, and on Wednesday, 2 more cars than on Monday. On Thursday and Friday, it is about 10 cars each day. On the weekend, traffic drops to 5 cars per day. How many cars travel down Happy ... | On Monday there are 20/100 * 25 = <<20/100*25=5>>5 cars passing the street less than on Tuesday.
So on Monday, there are 25 - 5 = <<25-5=20>>20 cars on Happy Street.
On Wednesday, there are 20 + 2 = <<20+2=22>>22 cars on this street.
On Thursday and Friday, there is a total of 10 * 2 = <<10*2=20>>20 cars passing.
On th... |
In a new diagram below, we have $\cos \angle XPY = \frac{3}{5}$. A point Z is placed such that $\angle XPZ$ is a right angle. What is $\sin \angle YPZ$?
[asy]
pair X, P, Y, Z;
P = (0,0);
X = Rotate(-aCos(3/5))*(-2,0);
Y = (2,0);
Z = Rotate(-90)*(2,0);
dot("$Z$", Z, S);
dot("$Y$", Y, S);
dot("$X$", X, W);
dot("$P$", P,... | \frac{3}{5} |
How many four-digit numbers greater than 2999 can be formed such that the product of the middle two digits exceeds 5? | 4970 |
How many four-digit positive integers are there with thousands digit $2?$ | 1000 |
Mr. Manuel is a campsite manager who's been tasked with checking the number of tents set up in the recreation area. On a particular day, he counted 100 tents in the northernmost part of the campsite and twice that number on the east side of the grounds. The number of tents at the center of the camp was four times the n... | On the eastern part of the campsite, Mr Manuel counted 2*100 = <<2*100=200>>200 tents
The total number of tents in the eastern and the northern part of the campgrounds is 200+100 = <<200+100=300>>300
There are four times as many tents as the northernmost part in the central part of the campsite, which means there are 4... |
A) For a sample of size $n$ taken from a normal population with a known standard deviation $\sigma$, the sample mean $\bar{x}$ is found. At a significance level $\alpha$, it is required to find the power function of the test of the null hypothesis $H_{0}: a=a_{0}$ regarding the population mean $a$ with the hypothetical... | 0.8925 |
Given that $a$, $b$, $c$ form an arithmetic sequence in triangle $ABC$, $\angle B=30^{\circ}$, and the area of $\triangle ABC$ is $\frac{1}{2}$, determine the value of $b$. | \frac{3+ \sqrt{3}}{3} |
A right triangle has an area of 120 square units, and a leg length of 24 units. What is the perimeter of the triangle, in units? | 60 |
Eva learns for two semesters in a year. In 2019, she scored ten more marks in maths in the first semester than the second, 15 marks less in arts, and 1/3 marks less in science in the first semester than the second. If she got 80 marks in maths in the second semester, 90 marks in arts, and 90 in science, what's the tota... | In the first semester, she scored 80+10 = <<80+10=90>>90 marks in maths
She also scores 90-15 = <<90-15=75>>75 marks in arts in the first semester.
She scored 1/3 marks less in science in the first semester than the second, which is 1/3*90 = <<1/3*90=30>>30
Her score in science in the first semester is 90-30 = <<90-30=... |
Given the areas of the three squares in the figure, what is the area of the interior triangle? | 30 |
Danica drove her new car on a trip for a whole number of hours, averaging $55$ miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a $3$-digit number with $a\ge1$ and $a+b+c\le7$. At the end of the trip, the odometer showed $cba$ miles. What is $a^2+b^2+c^2$? | 37 |
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