problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $$(a-1)(b-1)(c-1) \neq 0$$ | \frac{125}{216} |
Suppose $f(x), g(x), h(x), p(x)$ are linear equations where $f(x) = x + 1$, $g(x) = -x + 5$, $h(x) = 4$, and $p(x) = 1$. Define new functions $j(x)$ and $k(x)$ as follows:
$$j(x) = \max\{f(x), g(x), h(x), p(x)\},$$
$$k(x)= \min\{f(x), g(x), h(x), p(x)\}.$$
Find the length squared, $\ell^2$, of the graph of $y=k(x)$ fro... | 64 |
For a positive integer \( k \), find the greatest common divisor (GCD) \( d \) of all positive even numbers \( x \) that satisfy the following conditions:
1. Both \( \frac{x+2}{k} \) and \( \frac{x}{k} \) are integers, and the difference in the number of digits of these two numbers is equal to their difference;
2. The ... | 1998 |
6 boys and 4 girls are each assigned as attendants to 5 different buses, with 2 attendants per bus. Assuming that boys and girls are separated, and the buses are distinguishable, how many ways can the assignments be made? | 5400 |
Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\sqrt{21}$ and $\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. A parallelepiped is a sol... | 125 |
Given an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ (where $a>0$, $b>0$) with eccentricity $\frac{\sqrt{6}}{3}$, the distance from the origin O to the line passing through points A $(0, -b)$ and B $(a, 0)$ is $\frac{\sqrt{3}}{2}$. Further, the line $y=kx+m$ ($k \neq 0$, $m \neq 0$) intersects the ellipse a... | \frac{5}{4} |
In a theater performance of King Lear, the locations of Acts II-V are drawn by lot before each act. The auditorium is divided into four sections, and the audience moves to another section with their chairs if their current section is chosen as the next location. Assume that all four sections are large enough to accommo... | 1/2 |
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a fifty-cent piece. What is the probability that at least 40 cents worth of coins land on heads? | \frac{1}{2} |
Simplify $(r^2 + 3r - 2) - (r^2 + 7r - 5)$. | -4r+3 |
In the complex plane, the complex numbers $\frac {1}{1+i}$ and $\frac {1}{1-i}$ (where $i$ is the imaginary unit) correspond to points A and B, respectively. If point C is the midpoint of line segment AB, determine the complex number corresponding to point C. | \frac {1}{2} |
Determine the range of $w(w + x)(w + y)(w + z)$, where $x, y, z$, and $w$ are real numbers such that
\[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\] | 0 |
The corner of a unit cube is chopped off such that the cut runs through the three vertices adjacent to the vertex of the chosen corner. What is the height of the remaining cube when the freshly-cut face is placed on a table? | \frac{2\sqrt{3}}{3} |
Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $P(-1) = -9$?
$\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D) } 220 \qquad \textbf{(E) } 286$
| 220 |
At the circus, the clown has 3 dozen balloons on a string in his hand. 3 boys and 12 girls buy a balloon each. How many balloons is the clown still holding? | The clown is holding 3 x 12 = <<3*12=36>>36 balloons.
This many children buy a balloon: 3 + 12 = <<3+12=15>>15 kids.
After the children buy the balloons, the clown is holding this many in his hand 36 - 15 = <<36-15=21>>21 balloons.
#### 21 |
A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether? | 15 |
Given the function $f\left(x\right)=x^{3}+ax^{2}+x+1$ achieves an extremum at $x=-1$. Find:<br/>$(1)$ The equation of the tangent line to $f\left(x\right)$ at $\left(0,f\left(0\right)\right)$;<br/>$(2)$ The maximum and minimum values of $f\left(x\right)$ on the interval $\left[-2,0\right]$. | -1 |
What is the $y$-coordinate of the point on the $y$-axis that is equidistant from points $A( -2, 0)$ and $B(-1,4)$? | \frac{13}{8} |
In the staircase-shaped region below, all angles that look like right angles are right angles, and each of the eight congruent sides marked with a tick mark have length 1 foot. If the region has area 53 square feet, what is the number of feet in the perimeter of the region? [asy]
size(120);
draw((5,7)--(0,7)--(0,0)--(... | 32 |
The mean, median, and mode of the five numbers 12, 9, 11, 16, x are all equal. Find the value of x. | 12 |
What is the greatest four-digit number which is a multiple of 17? | 9996 |
Two different numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6, 7\}$. What is the probability that the positive difference between the two numbers is $2$ or greater? Express your answer as a common fraction. | \frac{5}{7} |
Calculate the area bounded by the graph of $y = \arcsin(\cos x)$ and the $x$-axis over the interval $0 \leq x \leq 2\pi$. | \pi^2 |
Determine all real values of the parameter $a$ for which the equation
\[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\]
has exactly four distinct real roots that form a geometric progression. | $\boxed{a=170}$ |
In the diagram below, \( \triangle ABC \) is a triangle with \( AB = 39 \text{ cm}, BC = 45 \text{ cm}, \) and \( CA = 42 \text{ cm} \). The tangents at \( A \) and \( B \) to the circumcircle of \( \triangle ABC \) meet at the point \( P \). The point \( D \) lies on \( BC \) such that \( PD \) is parallel to \( AC \)... | 168 |
Suppose that $ABCD$ is a rectangle with sides of length $12$ and $18$ . Let $S$ be the region of points contained in $ABCD$ which are closer to the center of the rectangle than to any of its vertices. Find the area of $S$ . | 54 |
There are \( n \) distinct lines in the plane. One of these lines intersects exactly 5 of the \( n \) lines, another one intersects exactly 9 of the \( n \) lines, and yet another one intersects exactly 11 of them. Which of the following is the smallest possible value of \( n \)? | 12 |
A certain school bought 10 cases of bottled water for their athletes. There are 20 bottles in each case. Seventy bottles of water were used during the first game. After the second game, only 20 bottles of water were left. How many bottles of water were used during the second game? | The school bought 10 x 20 = <<10*20=200>>200 bottled water.
There were 200 - 70 = <<200-70=130>>130 bottled water left after the first game.
So, there were 130 - 20 = <<130-20=110>>110 bottled water used during the second game.
#### 110 |
Let $n$ be the answer to this problem. An urn contains white and black balls. There are $n$ white balls and at least two balls of each color in the urn. Two balls are randomly drawn from the urn without replacement. Find the probability, in percent, that the first ball drawn is white and the second is black. | 19 |
Carter can read half as many pages as Lucy in 1 hour. Lucy can read 20 more pages than Oliver in 1 hour. Oliver can read 40 pages. How many pages can Carter read in 1 hour? | Lucy can read 20 more pages than Oliver who can read 40 pages so Lucy can read 40+20 = <<20+40=60>>60 pages
Carter can read half as many pages as Lucy who can read 60 pages so he can read 60/2 = <<30=30>>30 pages in 1 hour
#### 30 |
Given $f(x)=\sin 2x+\cos 2x$.
$(1)$ Find the period and the interval of monotonic increase of $f(x)$.
$(2)$ Find the maximum and minimum values of the function $f(x)$ on $[0,\frac{π}{2}]$. | -1 |
We have a cube with 4 blue faces and 2 red faces. What's the probability that when it is rolled, a blue face will be facing up? | \frac{2}{3} |
A cylinder has a radius of 5 cm and a height of 10 cm. What is the longest segment, in centimeters, that would fit inside the cylinder? | 10\sqrt{2} |
Each bird eats 12 beetles per day, each snake eats 3 birds per day, and each jaguar eats 5 snakes per day. If there are 6 jaguars in a forest, how many beetles are eaten each day? | First find the total number of snakes eaten: 5 snakes/jaguar * 6 jaguars = <<5*6=30>>30 snakes
Then find the total number of birds eaten per day: 30 snakes * 3 birds/snake = <<30*3=90>>90 snakes
Then multiply the number of snakes by the number of beetles per snake to find the total number of beetles eaten per day: 90 s... |
Find the $x$-intercept point of the line $3x+5y=20$. Provide your answer as an ordered pair. Express the $x$ and $y$ coordinates as common fractions as needed. | \left(\frac{20}{3},0\right) |
Given that Ron mistakenly reversed the digits of the two-digit number $a$, and the product of $a$ and $b$ was mistakenly calculated as $221$, determine the correct product of $a$ and $b$. | 527 |
Given $X \sim N(5, 4)$, find $P(1 < X \leq 7)$. | 0.9759 |
If we want to write down all the integers from 1 to 10,000, how many times do we have to write a digit, for example, the digit 5? | 4000 |
Compute the smallest positive integer such that, no matter how you rearrange its digits (in base ten), the resulting number is a multiple of 63. | 111888 |
Given the function $f(x)=-x^{3}+ax^{2}+bx$ in the interval $(-2,1)$. The function reaches its minimum value when $x=-1$ and its maximum value when $x=\frac{2}{3}$.
(1) Find the equation of the tangent line to the function $y=f(x)$ at $x=-2$.
(2) Find the maximum and minimum values of the function $f(x)$ in the interval... | -\frac{3}{2} |
On a redesigned dartboard, the outer circle radius is increased to $8$ units and the inner circle has a radius of $4$ units. Additionally, two radii divide the board into four congruent sections, each labeled inconsistently with point values as follows: inner sections have values of $3$ and $4$, and outer sections have... | \frac{9}{1024} |
What numeral is in the 100th decimal place in the decimal representation of $\frac{6}{7}$? | 1 |
Find the greatest common divisor of 91 and 72. | 1 |
Let $z$ be a non-real complex number with $z^{23}=1$. Compute $$ \sum_{k=0}^{22} \frac{1}{1+z^{k}+z^{2 k}} $$ | 46 / 3 |
If 3913 were to be expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 47 |
Given a geometric sequence $\{a_n\}$ with a common ratio $q$, and it satisfies $|q| > 1$, $a_2 + a_7 = 2$, and $a_4 \cdot a_5 = -15$, find the value of $a_{12}$. | -\frac{25}{3} |
Find the smallest positive integer $b$ for which $x^2+bx+2008$ factors into a product of two binomials, each having integer coefficients. | 259 |
Let $a$, $b$, and $c$ be positive real numbers. What is the smallest possible value of $(a+b+c)\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+c}\right)$? | \frac{9}{2} |
The sequence $(a_n)$ satisfies $a_0=0$ and $a_{n + 1} = \frac85a_n + \frac65\sqrt {4^n - a_n^2}$ for $n\geq 0$. Find the greatest integer less than or equal to $a_{10}$. | 983 |
A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimetres and its base has radius $30$ centimetres, find the volume of the largest box that can be co... | 108000 |
There is a beach soccer tournament with 17 teams, where each team plays against every other team exactly once. A team earns 3 points for a win in regular time, 2 points for a win in extra time, and 1 point for a win in a penalty shootout. The losing team earns no points. What is the maximum number of teams that can eac... | 11 |
If
\[\mathbf{A} = \begin{pmatrix} 1 & 3 \\ 2 & 1 \end{pmatrix},\]then compute $\det (\mathbf{A}^2 - 2 \mathbf{A}).$ | 25 |
The product $8 \times .25 \times 2 \times .125 =$ | $\frac{1}{2}$ |
Find $x$ if $\log_9(2x-7) = \dfrac{3}{2}$. | 17 |
Let $T$ be the set of points $(x, y)$ in the Cartesian plane that satisfy
\[\big|\big| |x|-3\big|-1\big|+\big|\big| |y|-3\big|-1\big|=2.\]
What is the total length of all the lines that make up $T$? | 32\sqrt{2} |
Compute the domain of the function $$f(x)=\frac{1}{\lfloor x^2-7x+13\rfloor}.$$ | (-\infty,3] \cup [4,\infty) |
The consecutive angles of a particular trapezoid form an arithmetic sequence. If the largest angle measures $120^{\circ}$, what is the measure of the smallest angle? | 60^\circ |
Convert the point $(\sqrt{2},-\sqrt{2})$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$ | \left( 2, \frac{7 \pi}{4} \right) |
Three dice are thrown, and the sums of the points that appear on them are counted. In how many ways can you get a total of 5 points and 6 points? | 10 |
Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in fee... | 8 |
A collector has \( N \) precious stones. If he takes away the three heaviest stones, then the total weight of the stones decreases by \( 35\% \). From the remaining stones, if he takes away the three lightest stones, the total weight further decreases by \( \frac{5}{13} \). Find \( N \). | 10 |
What is the value of $525^2 - 475^2$? | 50000 |
Given rational numbers $x$ and $y$ satisfy $|x|=9$, $|y|=5$.
$(1)$ If $x \lt 0$, $y \gt 0$, find the value of $x+y$;
$(2)$ If $|x+y|=x+y$, find the value of $x-y$. | 14 |
What is $8^{15} \div 64^3$? | 8^9 |
A rectangular pool table has vertices at $(0,0)(12,0)(0,10)$, and $(12,10)$. There are pockets only in the four corners. A ball is hit from $(0,0)$ along the line $y=x$ and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before entering a pocket. | 9 |
Let the function $f(x)= \frac{ \sqrt{3}}{2}- \sqrt{3}\sin^2 \omega x-\sin \omega x\cos \omega x$ ($\omega > 0$) and the graph of $y=f(x)$ has a symmetry center whose distance to the nearest axis of symmetry is $\frac{\pi}{4}$.
$(1)$ Find the value of $\omega$; $(2)$ Find the maximum and minimum values of $f(x)$ in the... | -1 |
A regular football match is being played. A draw is possible. The waiting time for the next goal is independent of previous events in the match. It is known that the expected total number of goals in football matches of these teams is 2.8. Find the probability that an even number of goals will be scored during the matc... | 0.502 |
Sally needs to make a tablecloth that measures 102 inches by 54 inches. She also needs to make 8 napkins that are 6 by 7 inches. How many square inches of material will Sally need to make the tablecloth and the 8 napkins? | Tablecloth = 102 * 54 = <<102*54=5508>>5508 square inches
Napkins = 8 * (6 * 7) = <<8*(6*7)=336>>336 square inches
5508 + 336 = <<5508+336=5844>>5844 square inches
Sally will need 5844 square inches of material to make the tablecloth and 8 napkins.
#### 5844 |
In Pascal's triangle, compute the seventh element in Row 20. Afterward, determine how many times greater this element is compared to the third element in the same row. | 204 |
In triangle $PQR$, let the side lengths be $PQ = 7,$ $PR = 8,$ and $QR = 5$. Calculate:
\[\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.\] | \frac{16}{7} |
The notation $[x]$ stands for the greatest integer that is less than or equal to $x$. Calculate $[-1.2]$. | -2 |
A cafe has 3 tables and 5 individual counter seats. People enter in groups of size between 1 and 4, inclusive, and groups never share a table. A group of more than 1 will always try to sit at a table, but will sit in counter seats if no tables are available. Conversely, a group of 1 will always try to sit at the counte... | 16 |
In a cultural performance, there are already 10 programs arranged in the program list. Now, 3 more programs are to be added, with the requirement that the relative order of the originally scheduled 10 programs remains unchanged. How many different arrangements are there for the program list? (Answer with a number). | 1716 |
A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F,$ in such a way that t... | 512 |
Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | 24 |
A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether? | 15 |
Irene shares half of a small apple with her dog every day. A small apple weighs about 1/4 of a pound. She can currently buy apples for $2.00 a pound. How much will she spend so that she and her dog have enough apples to last for 2 weeks? | There are 7 days in a week and she needs apples for 2 weeks so that's 7*2 = <<7*2=14>>14 days
She shares an apple with her dog every day so for 14 days so she needs 1*14 = <<1*14=14>>14 apples
Each apple weighs 1/4 of a pound and she needs 14 apples so that's .25*14 = <<1/4*14=3.5>>3.5 pounds of apples
She needs 3.5 po... |
What is the value of $2468 + 8642 + 6824 + 4286$? | 22220 |
Given the function f(x) = 2x^3 - ax^2 + 1, where a ∈ R.
(I) When a = 6, the line y = -6x + m is tangent to f(x). Find the value of m.
(II) If the function f(x) has exactly one zero in the interval (0, +∞), find the monotonic intervals of the function.
(III) When a > 0, if the sum of the maximum and minimum values of th... | \frac{1}{2} |
Let $ABCD$ be a square of side length $1$ , and let $P$ be a variable point on $\overline{CD}$ . Denote by $Q$ the intersection point of the angle bisector of $\angle APB$ with $\overline{AB}$ . The set of possible locations for $Q$ as $P$ varies along $\overline{CD}$ is a line segment; what is the l... | 3 - 2\sqrt{2} |
The total average age of three friends is 40. Jared is ten years older than Hakimi, and Molly's age is 30. How old is Hakimi? | The total age for the three friends is 40*3 = <<40*3=120>>120
If Molly's age is 30, then Jared and Hakimi have a total age of 120-30 = 90.
Let's say the age of Hakimi is x.
Since Jared is 10 years older than Hakimi, Jared is x+10 years old.
Jared and Hakimi's total age is x+(x+10) = 90
This translates to 2x=90-10
2x=80... |
Let $0 \le a,$ $b,$ $c \le 1.$ Find the maximum value of
\[\sqrt{abc} + \sqrt{(1 - a)(1 - b)(1 - c)}.\] | 1 |
Tommy has 3 toy cars. His neighbor, Jessie, has 3 cars too. Jessie's older brother has 5 more cars than Tommy and Jessie. How many cars do the three of them have altogether? | Tommy and Jessie have 3 + 3 = <<3+3=6>>6 cars.
Jessie's brother has 5 + 6 = <<5+6=11>>11 cars.
Altogether, they have 6 + 11 = <<6+11=17>>17 cars.
#### 17 |
Grandma Wang has 6 stools that need to be painted by a painter. Each stool needs to be painted twice. The first coat takes 2 minutes, but there must be a 10-minute wait before applying the second coat. How many minutes will it take to paint all 6 stools? | 24 |
Find the ordered pair $(a,b)$ of positive integers, with $a < b,$ for which
\[\sqrt{1 + \sqrt{21 + 12 \sqrt{3}}} = \sqrt{a} + \sqrt{b}.\] | (1,3) |
Find the only real number that can be expressed in the form \[(a + bi)^3 - 107i,\]where $i^2 = -1,$ and $a$ and $b$ are positive integers. | 198 |
If $2x+7$ is a factor of $6x^3+19x^2+cx+35$, find $c$. | 3 |
What is the total number of digits used when the first 2500 positive even integers are written? | 9449 |
Given integers $x$ and $y$ satisfy the equation $2xy + x + y = 83$, find the values of $x + y$. | -85 |
Let $x$ and $y$ be positive real numbers. Find the minimum value of
\[\left( x + \frac{1}{y} \right) \left( x + \frac{1}{y} - 1024 \right) + \left( y + \frac{1}{x} \right) \left( y + \frac{1}{x} - 1024 \right).\] | -524288 |
Two numbers whose sum is $6$ and the absolute value of whose difference is $8$ are roots of the equation: | x^2-6x-7=0 |
Let $AB$ be a segment of unit length and let $C, D$ be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set $\{A,B,C,D\}.$ | \frac{\sqrt{5}}{125} |
Perform the calculations:
3.21 - 1.05 - 1.95
15 - (2.95 + 8.37)
14.6 × 2 - 0.6 × 2
0.25 × 1.25 × 32 | 10 |
Jean has 60 stuffies. She keeps 1/3 of them and then gives away the rest. She gives 1/4 of what she gave away to her sister Janet. How many stuffies did Janet get? | Jean kept 60/3=<<60/3=20>>20 stuffies
That means she gave away 60-20=<<60-20=40>>40 stuffies
So Janet got 40/4=<<40/4=10>>10 stuffies
#### 10 |
An apple tree can fill 20 baskets. Each basket can be filled with 15 apples. How many apples can you get from 10 trees? | You can get 20 x 15 = <<20*15=300>>300 apples per tree.
So a total of 300 x 10 = <<300*10=3000>>3000 apples can be gotten from 10 trees.
#### 3000 |
How many distinguishable arrangements are there of $1$ brown tile, $1$ purple tile, $2$ green tiles, and $3$ yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.) | 420 |
$2016$ bugs are sitting in different places of $1$ -meter stick. Each bug runs in one or another direction with constant and equal speed. If two bugs face each other, then both of them change direction but not speed. If bug reaches one of the ends of the stick, then it flies away. What is the greatest number of cont... | 1008^2 |
Determine the radius of the sphere that touches the faces of the unit cube passing through vertex $A$ and the edges passing through vertex $B$. | 2 - \sqrt{2} |
If $\Phi$ and $\varphi$ are the two distinct solutions to the equation $x^2=x+1$, then what is the value of $(\Phi-\varphi)^2$? | 5 |
Bill's roof can bear 500 pounds of weight. If 100 leaves fall on his roof every day, and 1000 leaves weighs 1 pound, how many days will it take for his roof to collapse? | First, find how many leaves the roof can hold: 500 pounds * 1000 leaves/pound = <<500*1000=500000>>500000 leaves
Then divide that by the number of leaves that fall each day to find how many days before the roof collapses: 500000 leaves / 100 leaves/day = <<500000/100=5000>>5000 days
#### 5000 |
Given the set $\{a, \frac{b}{a}, 1\} = \{a^2, a+b, 0\}$, find the value of $a^{2015} + b^{2016}$. | -1 |
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