instruction_seed stringlengths 0 21.2k | source stringclasses 4
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Convert the point $(\rho,\theta,\phi) = \left( 2, \frac{2 \pi}{3}, \frac{\pi}{4} \right)$ in spherical coordinates to rectangular coordinates. | tiger_math |
Two sectors of a circle of radius $10$ overlap as shown, with $P$ and $R$ as the centers of the respective circles. Determine the area of the shaded region.
[asy]
draw((0,0)--(8.6603,-5)--(17.3205,0)--(8.6603,5)--cycle,black+linewidth(1));
filldraw((8.6603,5)..(10,0)..(8.6603,-5)--cycle,gray,black+linewidth(1));
filld... | tiger_math |
The areas of two similar regular hexagons are in the ratio $49:81$. What is the ratio of their perimeters? Express your answer in the form $a:b$. | tiger_math |
A hexagon is inscribed in a circle. Five of the sides have length $100$ and the sixth, denoted by $\overline{AB}$, has length $20$. Find the sum of the lengths of the three diagonals that can be drawn from $A$. | tiger_math |
Below is a portion of the graph of a quadratic function, $y=q(x)=ax^2+bx+c$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.99,xmax=10.5,ymin=-5.5,ymax=5.5;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 ... | tiger_math |
Three dice with faces numbered 1 through 6 are stacked as shown. Six of the eighteen faces are visible, leaving twelve faces hidden (back, bottom, between). What is the total number of dots NOT visible in this view? [asy]
/* AMC8 2000 #8 Problem */
draw((0,0)--(1,0)--(1.5,0.66)--(1.5,3.66)--(.5,3.66)--(0,3)--cycle);
dr... | tiger_math |
How many factors of 8000 are perfect cubes? | tiger_math |
For real numbers $x > 4,$ find the minimum value of
\[\frac{x + 10}{\sqrt{x - 4}}.\] | tiger_math |
If $y = \displaystyle\frac{1}{3x+1}$, what is the value of $x$ when $y = 2$? | tiger_math |
Suppose $\cos Q = 0.4$ in the diagram below. What is the length of the hypotenuse of the triangle?
[asy]
pair P,Q,R;
P = (0,0);
Q = (6,0);
R = (0,6*tan(acos(0.4)));
draw(P--Q--R--P);
draw(rightanglemark(Q,P,R,18));
label("$P$",P,SW);
label("$Q$",Q,SE);
label("$R$",R,N);
label("$12$",Q/2,S);
[/asy] | tiger_math |
Five toppings for a pizza are available at Polina's Pizza. How many combinations of at most two different toppings are possible? | tiger_math |
What number, when divided by 3, gives the same result as when 3 is subtracted from it? | tiger_math |
Angles $P$ and $Q$ are supplementary. If the measure of angle $P$ is $5$ times angle $Q$ minus $10$ degrees, what is the measure of angle P? | tiger_math |
What is the coefficient of $x^{-1}$ in the expansion of $\left(\sqrt{x}+\dfrac5x\right)^{9}$? | tiger_math |
Find the area of the ellipse given by $x^2 + 6x + 4y^2 - 8y + 13 = 0.$ | tiger_math |
Quantities $x$ and $y$ vary inversely. When $x$ is $2500,$ $y$ is $0.84.$ What is the value of $y$ when $x$ is $1250$? Express your answer as a decimal to the nearest thousandths. | tiger_math |
If the point $(-2,5)$ is on the graph of $y=g(x)$, and $h(x)=g(x+3)$ for all $x$, then there is one point that must be on the graph of $y=h(x)$. What is the sum of the coordinates of that point? | tiger_math |
Factor the quadratic $9y^2-30y+25$ into the form $(ay+b)^2$. | tiger_math |
Evaluate $\sqrt{16 +\!\sqrt{16 + \!\sqrt{16 + \!\sqrt{16 + \cdots}}}}$. | tiger_math |
If $a,$ $b,$ $x,$ and $y$ are real numbers such that $ax+by=3,$ $ax^2+by^2=7,$ $ax^3+by^3=16,$ and $ax^4+by^4=42,$ find $ax^6+by^6.$ | tiger_math |
Find the point in the plane $3x - 4y + 5z = 30$ that is closest to the point $(0,0,0).$ | tiger_math |
The function $f(x)$ satisfies
\[f(xy) = \frac{f(x)}{y}+x\]for all positive real numbers $x$ and $y.$ If $f(30) = 20,$ find $f(40).$ | tiger_math |
Suppose $77\cdot 88\cdot 99 \equiv n\pmod{25}$, where $0\le n<25$.
What is the value of $n$? | tiger_math |
Suppose that all four of the numbers \[2 - \sqrt{3}, \;4+\sqrt{8}, \;14 - 2\sqrt{5}, \;-\sqrt{2}\]are roots of the same nonzero polynomial with integer coefficients. What is the smallest possible degree of the polynomial? | tiger_math |
How many positive integers less than or equal to 60 have an even number of positive divisors? | tiger_math |
In the diagram below, triangle $ABC$ has been reflected over its median $\overline{AM}$ to produce triangle $AB'C'$. If $AE = 8$, $EC =16$, and $BD = 12$, then find $AB$.
[asy]
size(250);
pair A,B,C,D,M,BB,CC,EE;
B = (0,0);
D = (12,0);
M = (18,0);
C=2*M;
A = D + (scale(1.2)*rotate(aCos((225-144-25)/120))*(M-D));
CC =... | tiger_math |
Calculate $\frac{3}{5}+\frac{1}{4}$. | tiger_math |
Find the distance between the points (-2,1) and (1,5). | tiger_math |
We have a rectangle of dimensions $2x - 1$ by $x + 3$ such that its area is $10x - 5$. What is the value of $x$? | tiger_math |
A supermarket display of oranges is arranged in layers, forming an arithmetic sequence. The bottom layer contains 32 oranges, and the next layer contains 28 oranges. This pattern continues until there is a single orange at the top of the display. Determine the total number of oranges in the display. | tiger_math |
The equation $x^2 + 2x + 5 = i$ has two complex solutions. Determine the product of their real parts. | tiger_math |
Is the function $f(x) = \lfloor x \rfloor + \frac{1}{3}$ even, odd, or neither? | tiger_math |
A right triangle with legs $12$ and $9$ is similar to another right triangle with corresponding legs $x$ and $6$, respectively. What is the value of $x$? Express your answer as a decimal to the nearest hundredth. | tiger_math |
Determine the number of ways to arrange the letters of the word MADAME, where the two E's must be together. | tiger_math |
A plane intersects a right circular cylinder of radius $r$ forming an ellipse. If the major axis of the ellipse of $50\%$ longer than the minor axis, the length of the major axis is
$\textbf{(A)}\ r\qquad \textbf{(B)}\ \frac{3r}{2}\qquad \textbf{(C)}\ 2r\qquad \textbf{(D)}\ \frac{9r}{4}\qquad \textbf{(E)}\ 3r$ | tiger_math |
A standard six-sided die is rolled, and $P$ is the product of the four numbers that are visible. What is the largest number that is certain to divide $P$? | tiger_math |
The reciprocal of a number is increased by 1, resulting in $\frac{7}{3}$. Find the original number as a common fraction. | tiger_math |
Find the polynomial $p(x)$ such that
\[p(p(x) - 1) = xp(x) + x^2.\] | tiger_math |
The equation $x^2+10x=84$ has two solutions. The negative solution has the form $-\sqrt{a}-b$ for positive natural numbers $a$ and $b$. What is $a+b$? | tiger_math |
In tetrahedron $ABCD$, edge $AB$ has length 5 cm. The area of face $ABC$ is $15\mbox{cm}^2$ and the area of face $ABD$ is $12 \mbox { cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\mbox{cm}^3$. | tiger_math |
The asymptotes of a hyperbola are $y = x - 1$ and $y = 5 - x.$ Also, the hyperbola passes through $(2,2).$ Find the distance between the foci of the hyperbola. | tiger_math |
How many two-digit primes have a tens digit of 1? | tiger_math |
I have eleven different books, of which I want to bring two to read on vacation. How many different pairs can I choose if the order in which I choose the books does not matter? | tiger_math |
If $f(x)$ is a monic quartic polynomial such that $f(-1)=1$, $f(2)=4$, $f(-3)=9$, and $f(4)=16$, find $f(1)$. | tiger_math |
Let $z$ be a complex number such that $z^4 = 1$ and $z \neq 1.$ Compute
\[(z - 1)(z^2 - 1)(z^3 - 1).\] | tiger_math |
Find $x$ if $\log_x64 = \dfrac{3}{2}$. | tiger_math |
Spinners $A$ and $B$ are spun. On Spinner $A$, the arrow is equally likely to land on each of the numbers $1, 2, 3, 4$. On Spinner $B$, the arrow is equally likely to land on each of the numbers $1, 3, 5$. What is the probability that the product of the two spinners' numbers is even? Express your answer as a common fr... | tiger_math |
A solid right prism $ABCDEF$ has a height of 16, as shown. Also, its bases are equilateral triangles with side length 12. Points $X$, $Y$, and $Z$ are the midpoints of edges $AC$, $BC$, and $DC$, respectively. If the lateral area of the prism is 720, determine the perimeter of triangle $XYZ$. | tiger_math |
A hemisphere-shaped bowl with radius 10 inches is half-filled with chocolate. All of the chocolate is then evenly distributed between 8 congruent, smaller hemisphere-shaped molds. What is the radius of each of the smaller molds, in inches? | tiger_math |
Compute $\tan \left (\operatorname{arccot} \frac{x}{y} \right),$ where $x$ and $y$ are positive. | tiger_math |
A box of 30 chocolate candies costs $\$7$. How many dollars does it cost to buy 500 chocolate candies? | tiger_math |
Jeff will pick a card at random from ten cards numbered 1 through 10. The number on this card will indicate his starting point on the number line shown below. He will then spin the fair spinner shown below (which has four congruent sectors) and follow the instruction indicated by his spin. From this new point he wil... | tiger_math |
A Senate committee has 10 Republicans and 8 Democrats. In how many ways can we form a subcommittee with 4 Republicans and 3 Democrats? | tiger_math |
Simplify $\dfrac{5-2i}{2+3i}$. Your answer should be of the form $a+bi$, where $a$ and $b$ are both real numbers and written as improper fractions (if necessary). | tiger_math |
A right triangle has legs with lengths of 30 centimeters and 40 centimeters. Find the area of the triangle. | tiger_math |
One leg of a right triangle is 15 inches, and the measure of the angle opposite that leg is $30^\circ$. What is the number of inches in the hypotenuse of the triangle? | tiger_math |
The equations of the asymptotes of a hyperbola are $y = 2x+5$ and $y = -2x+1.$ If the hyperbola passes through the point $(1, 9),$ determine the equation of the hyperbola in standard form, where the standard form is given by \[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1,\]where $a,$ $b$, $h,$ and $k$ are constants wi... | tiger_math |
From the set of integers $\{1,2,3,\dots,2010\}$, choose $k$ pairs $\{a_i,b_i\}$ with $a_i<b_i$ so that no two pairs have a common element. Suppose that all the sums $a_i+b_i$ are distinct and less than or equal to $2010$. Find the maximum possible value of $k$. | tiger_math |
Ramon sells three enchiladas and three tacos for $\$$3.50 and he sells two enchiladas and four tacos for $\$$3.60. Assuming a fixed price per item, what is the cost, in dollars, of two enchiladas and three tacos? Express your answer as a decimal to the nearest hundredth. | tiger_math |
If $(x-2)(x+3)=14$, find the sum of the possible values of $x$. | tiger_math |
Marius is entering a wildlife photo contest, and wishes to arrange his eight snow leopards of different heights in a row. If the shortest two leopards have inferiority complexes and demand to be placed at the ends of the row, and the tallest leopard insists on being placed third from the left, how many ways can he line... | tiger_math |
How many different four-letter arrangements can be formed using the six letters $A, B, C, D, E$ and $F$, if the first letter must be $C$, one of the other letters must be $B$, and each letter can be used at most once in the arrangement? | tiger_math |
Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps half of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-third of them, and divides the rest equally between the other two. The third monkey take... | tiger_math |
Quadrilateral $ABCD$ is a parallelogram. The degree measure of angle $C$ is 60. What is the degree measure of angle $A$? | tiger_math |
I want to choose a license plate which is 3 characters long, where the first character is a letter, the last character is a digit, and the middle is either a letter or a digit. I also want there to be exactly one character on my license plate which is repeated. How many ways are there for me to choose a license plate w... | tiger_math |
A $\emph{convex}$ polygon is a polygon in which every interior angle is less than 180 degrees. A $\emph{diagonal}$ of a convex polygon is a line segment that connects two non-adjacent vertices. A convex polygon has 35 diagonals. How many sides does the polygon have? | tiger_math |
How many combinations of pennies (1 cent), nickels (5 cents), dimes (10 cents), and/or quarters (25 cents) are there with a total value of 50 cents? | tiger_math |
My co-worker Larry only likes numbers that are divisible by 4. He likes the number 20, and also the number 4,004. How many different two-digit endings are possible in numbers that Larry likes? | tiger_math |
The exact amount of fencing that enclosed the four congruent equilateral triangular corrals shown here is reused to form a single large square corral. What is the ratio of the total area of the four small corrals to the area of the new large corral? Express your answer as a common fraction.
[asy]
draw((0,0)--(1,sqrt(3... | tiger_math |
If I have four boxes arranged in a $2$ x $2$ grid, in how many distinct ways can I place the digits $1$, $2$, $3$, and $4$ in the boxes such that each box contains exactly one digit? | tiger_math |
What is the largest prime factor of $1424$? | tiger_math |
Compute $\sqrt{2^4\cdot 3^4}$. | tiger_math |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(xf(y) + x) = xy + f(x)\]for all $x,$ $y.$
If $f(1) = 2,$ find the sum of all possible values of $f(3).$ | tiger_math |
A cube 5 units on each side is composed of 125 unit cubes. Two faces of the larger cube that share an edge are painted blue, and the cube is disassembled into 125 unit cubes. Two of the unit cubes are selected uniformly at random. What is the probability that one of two selected unit cubes will have exactly two painted... | tiger_math |
Apples are sold at the farmer's market at a rate of $\$4$ per five pounds. How much would it cost to purchase 20 pounds of apples? | tiger_math |
For what values of $x$ is $x^3-8x^2>-15x$? | tiger_math |
The complex number $(-2 \operatorname{cis} 45^\circ)(3\operatorname{cis} 60^\circ)$ is expressed in polar form as $r \operatorname{cis} \theta,$ where $r > 0$ and $0^\circ \le \theta < 360^\circ.$ Enter the ordered pair $(r, \theta).$ | tiger_math |
If
\[\begin{pmatrix} 1 & 2 & a \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{pmatrix}^{2016} = \begin{pmatrix} 1 & 28 & 2020 \\ 0 & 1 & 44 \\ 0 & 0 & 1 \end{pmatrix},\]then find $a + 2016.$ | tiger_math |
Let $a$ and $b$ be nonzero real numbers such that
\[(3 - 5i)(a + bi)\]is pure imaginary. Find $\frac{a}{b}.$ | tiger_math |
Find the largest possible value of $k$ for which $2^{20}$ is expressible as the sum of $k$ consecutive odd positive integers. | tiger_math |
Three friends have a total of 6 identical pencils, and no friend can have zero pencils. In how many ways can this happen? | tiger_math |
Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A cone is formed by rolling the triangle along the sides of its midpoint triangle. What is the volume of this cone? | tiger_math |
A ball travels on a parabolic path in which the height (in feet) is given by the expression $-16t^2+80t+100$, where $t$ is the time after launch. What is the maximum height of the ball, in feet? | tiger_math |
June and Julia live 1 mile apart. It takes June 4 minutes to ride her bike directly to Julia's house. Bernard lives 3.5 miles away from June, but in the opposite direction of Julia. At the same rate, how many minutes would it take June to ride from her own house to Bernard's house? | tiger_math |
Given that $2+\sqrt{3}$ is a root of the equation \[x^3 + ax^2 + bx - 6 = 0\]and that $a$ and $b$ are rational numbers, find $b.$ | tiger_math |
Phil has 7 green marbles and 3 purple marbles in a bag. He removes a marble at random, records the color, puts it back, and then repeats this process until he has withdrawn 8 marbles. What is the probability that exactly four of the marbles that he removes are green? Express your answer as a decimal rounded to the near... | tiger_math |
Calculate $[(15^{10} \div 15^9)^2 \cdot 3^2] \div 3^4$. | tiger_math |
Find the value of the first term in the geometric sequence $a,b,c,16,32$. | tiger_math |
Let $a,$ $b,$ $c$ be distinct integers, and let $\omega$ be a complex number such that $\omega^3 = 1$ and $\omega \neq 1.$ Find the smallest possible value of
\[|a + b \omega + c \omega^2| - 1.\] | tiger_math |
Points $A(-2,3)$ and $B(4,9)$ are the endpoints of a diameter of a circle graphed in a coordinate plane. How many square units are in the area of the circle? Express your answer in terms of $\pi$. | tiger_math |
Simplify the expression
\[\frac{(\sqrt{3} - 1)^{1 - \sqrt{2}}}{(\sqrt{3} + 1)^{1 + \sqrt{2}}},\]writing your answer as $a - b \sqrt{c},$ where $a,$ $b,$ and $c$ are positive integers, and $c$ is not divisible by the square of a prime. | tiger_math |
A regular polygon has exterior angles of 12 degrees. How many sides does the polygon have? | tiger_math |
What is the sum of the digits of the base-3 expression for $222_{10}$? | tiger_math |
The expression $x^2-6x+13$ can be written as a combination of a square of a binomial and an integer. Find the integer. | tiger_math |
Convert $813_9$ to base 6. | tiger_math |
$\triangle DEF$ is inscribed inside $\triangle ABC$ such that $D,E,F$ lie on $BC, AC, AB$, respectively. The circumcircles of $\triangle DEC, \triangle BFD, \triangle AFE$ have centers $O_1,O_2,O_3$, respectively. Also, $AB = 13, BC = 15, AC=14$, and $\stackrel{\frown}{BF} = \stackrel{\frown}{EC},\ \stackrel{\frown}{AF... | tiger_math |
In a right-angled triangle, the sum of the squares of the three side lengths is 2000. If the length of one leg is 10, what is the length of the hypotenuse? | tiger_math |
Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four slips. Three slips are drawn from the hat at random and without replacement. Let $p$ be the probability that all three slips bear the same number. Let $q$ be the probability that two of the slips ... | tiger_math |
A bored student walks down a hall that contains a row of closed lockers, numbered $1$ to $2048$. The student follows the same pattern as before: he opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and star... | tiger_math |
How many integers between $100$ and $200$ have three different digits in decreasing order? One such integer is $195$. | tiger_math |
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