problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
On a 6 by 6 square grid, each dot is 1 cm from its nearest horizontal and vertical neighbors. What is the product of the value of the area of square $EFGH$ (in cm$^2$) and the value of the perimeter of square $EFGH$ (in cm)? Express your answer in simplest radical form.
[asy]
unitsize(1cm);
defaultpen(linewidth(0.7));
for(int i = 0; i <= 5; ++i) {
for(int j = 0; j <= 5; ++j) {
dot((i,j));
}
}
draw((1,5)--(5,5)--(5,1)--(1,1)--cycle);
label("$E$",(1,5),W);
label("$F$",(5,5),N);
label("$G$",(5,1),E);
label("$H$",(1,1),S);
[/asy]
|
256
| 0.75 |
The ''roundness'' of an integer greater than 1 is defined as the sum of the exponents in the prime factorization of the number. Determine the roundness of 1728.
|
9
| 0.916667 |
Genevieve decides to create a new design for her kite display. In her revised design, the small kite has vertices at points on a grid where the distance between each point is one inch. The vertices are at the following coordinates on this new grid:
- (0,6), (4,8), (8,6), (4,1).
For her larger kite, Genevieve doubles the height and quadruples the width of the grid for the entire kite compared to the small kite.
Calculate the number of square inches in the area of the newly designed small kite.
|
28 \text{ square inches}
| 0.833333 |
Find the sum of $753_8$ and $326_8$ in base $8$.
|
1301_8
| 0.833333 |
Given vectors $\mathbf{v}$ and $\mathbf{w}$ such that $\|\mathbf{v}\| = 5$, $\|\mathbf{w}\| = 8$, and $\mathbf{v} \cdot \mathbf{w} = 20$, find $\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\|$.
|
2.5
| 0.083333 |
Amanda took a loan of $\$x$ from her friend to purchase a laptop. Her friend agreed to let Amanda pay off the debt by pet sitting. The payment structure for pet sitting is cyclic every 4 hours: the first hour she earns $\$2$, the second hour $\$3$, the third hour $\$2$, the fourth hour $\$3$, and then the cycle repeats. If Amanda works for 50 hours to clear her debt, how much did she borrow?
|
\$125
| 0.916667 |
Jackie and Phil each has one fair coin and one biased coin. The biased coin shows heads with a probability of $\frac{2}{5}$. They each flip both coins once. Find the probability that both Jackie and Phil get the same number of heads, and if $\frac{p}{q}$ is this simplified probability (where p and q are coprime integers), compute $p + q$.
|
69
| 0.416667 |
Triangle $PQR$ has side-lengths $PQ = 20, QR = 40,$ and $PR = 30.$ The line through the incenter of $\triangle PQR$ parallel to $\overline{QR}$ intersects $\overline{PQ}$ at $X$ and $\overline{PR}$ at $Y.$ What is the perimeter of $\triangle PXY?$
|
50
| 0.25 |
In quadrilateral $ABCD$, sides $\overline{AB}$ and $\overline{BC}$ both have length 12, sides $\overline{CD}$ and $\overline{DA}$ both have length 20, and the measure of angle $ADC$ is $45^\circ$. What is the length of diagonal $\overline{AC}$?
|
20\sqrt{2 - \sqrt{2}}
| 0.75 |
Compute $\frac{x^6 - 54x^3 + 729}{x^3 - 27}$ when $x = 3$.
|
0
| 0.916667 |
The expression \(x^2 + 15x + 36\) can be written as \((x + a)(x + b)\), and the expression \(x^2 + 7x - 60\) written as \((x + b)(x - c)\), where \(a\), \(b\), and \(c\) are integers. What is the value of \(a + b + c\)?
|
20
| 0.25 |
A circular pie has a diameter of 18 cm and is cut into four equal-sized sector-shaped pieces. Let \( l \) be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces. What is \( l^2 \)?
|
162
| 0.583333 |
Three faces of a right rectangular prism have areas of 56, 63, and 72 square units. The area of the face with an area of 72 square units is represented as twice a product of its dimensions. What is the volume of the prism, in cubic units?
|
504
| 0.916667 |
A square with sides 8 inches is shown. If $P$ is a point such that the segment $\overline{PA}$, $\overline{PB}$, $\overline{PC}$ are equal in length, and segment $\overline{PC}$ is perpendicular to segment $\overline{FD}$, what is the area, in square inches, of triangle $APB$? [asy]
pair A, B, C, D, F, P;
A = (0,0); B= (4,0); C = (2,4); D = (4,4); F = (0,4); P = (2,2);
draw(A--B--D--F--cycle);
draw(C--P); draw(P--A); draw(P--B);
label("$A$",A,SW); label("$B$",B,SE);label("$C$",C,N);label("$D$",D,NE);label("$P$",P,NW);label("$F$",F,NW);
label("$8''$",(2,0),S);
[/asy]
|
12
| 0.083333 |
Tom's favorite number is between $100$ and $150$. It is a multiple of $13$, but not a multiple of $3$. The sum of its digits is a multiple of $4$. What is Tom's favorite number?
|
143
| 0.166667 |
A bridge is constructed by linking a wooden plank between two triangular supports, as shown in the diagram below:
```
[Drawing not to scale]
A D
/\ /\
/ \ / \
/ \ / \
B------C E------F
```
Both triangles are isosceles with equal heights, such that AB = AC and DE = DF. The apex angles of the triangles are $\angle BAC = 25^\circ$ and $\angle EDF = 35^\circ$. Determine the sum $\angle DAC + \angle ADE$.
|
150^\circ
| 0.333333 |
A map of a region is depicted by the Cartesian plane. John is located at $(3, -15)$ and Linda is located at $(-2, 20)$. They decide to meet at the closest point equidistant from both their starting points, and then walk upwards together to the location of their friend Maria at $\left(0.5, 5\right)$. How many units do John and Linda walk upwards together to get to Maria's location?
|
2.5
| 0.5 |
Rationalize the denominator of \(\frac{1}{\sqrt[3]{5} - \sqrt[3]{3}}\) and express your answer in the form \(\frac{\sqrt[3]{X} + \sqrt[3]{Y} + \sqrt[3]{Z}}{W}\), where the fraction is in lowest terms. What is \(X + Y + Z + W\)?
|
51
| 0.833333 |
How many integers $-10 \leq n \leq 12$ satisfy $(n-1)(n+3)(n+7) < 0$?
|
6
| 0.833333 |
Let $\mathbf{u},$ $\mathbf{v},$ $\mathbf{w}$ be vectors in $\mathbb{R}^3$, and let $E$ be the determinant of the matrix whose column vectors are $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}.$ Find the determinant of the matrix whose column vectors are $\mathbf{u} \times \mathbf{v},$ $\mathbf{v} \times \mathbf{w},$ and $\mathbf{w} \times \mathbf{u}$ and express it in terms of $E$.
|
E^2
| 0.75 |
Two diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that they intersect inside the nonagon?
|
\dfrac{14}{39}
| 0.833333 |
The angles of quadrilateral $EFGH$ satisfy $\angle E = 3\angle F = 2\angle G = 6\angle D$. What is the degree measure of $\angle E$, rounded to the nearest whole number?
|
180
| 0.916667 |
Define the operation $a \oslash b = (\sqrt{3a - b})^4$. If $10 \oslash y = 81$, find the value of $y$.
|
21
| 0.916667 |
When \(x\) is divided by each of \(3\), \(7\), and \(8\), remainders of \(2\), \(6\), and \(7\) (respectively) are obtained. What is the smallest possible positive integer value of \(x\)?
|
167
| 0.75 |
Determine the product $ABC$ for the partial fraction decomposition of
\[\frac{x^2 - 19}{x^3 - 3x^2 - 4x + 12}\]
into the form
\[\frac{A}{x - 1} + \frac{B}{x + 3} + \frac{C}{x - 4}.\]
|
\frac{15}{196}
| 0.416667 |
A prism has 18 edges. How many faces does the prism have?
|
8
| 0.916667 |
Solve for $z$ in the following equation: $5 - 3iz = 2 + 5iz$.
|
z = -\frac{3i}{8}
| 0.916667 |
Find the sum of the coefficients in the polynomial \(2(4x^{8} + 7x^6 - 9x^3 + 3) + 6(x^7 - 2x^4 + 8x^2 - 2)\) when it is fully simplified.
|
40
| 0.916667 |
Evaluate the product \[ (a-5) \cdot (a-4) \cdot \cdot \ldots \cdot (a-1) \cdot a, \] where \( a = 7 \).
|
5040
| 0.916667 |
Find the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$ in the diagram if $CX$ bisects $\angle ACB$. [asy]
import markers;
real t=32/(32+35);
pair A=(-16.57,0);
pair B=(9.43,0);
pair C=(0,30.65);
pair X=t*A+(1-t)*B;
draw(C--A--B--C--X);
label("$A$",A,SW);
label("$B$",B,E);
label("$C$",C,N);
label("$X$",X,NE);
label("$28$",.5*(B+A),S);
label("$32$",.5*(B+C),NE);
label("$35$",.5*(A+C),NW);
[/asy]
|
\frac{32}{35}
| 0.5 |
What is the sum of all positive integer solutions less than or equal to $30$ to the congruence $7(5x-3) \equiv 14 \pmod{10}$?
|
225
| 0.833333 |
For how many integer values of $n$ between 1 and 299 inclusive does the decimal representation of $\frac{n}{450}$ terminate?
|
33
| 0.916667 |
In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled, with $\angle AEB = 45^\circ$, $\angle BEC = 60^\circ$, and $\angle CED = 60^\circ$, and $AE=30$.
Find the length of $CE.$
|
\frac{15\sqrt{2}}{2}
| 0.166667 |
The point $O$ is the center of the circle circumscribed about $\triangle ABC$, with $\angle BOC = 150^{\circ}$ and $\angle AOB = 130^{\circ}$. Calculate the degree measure of $\angle ABC$.
|
40^{\circ}
| 0.916667 |
Find \( X+Y \) (in base 10), given the following addition problem in base 7:
\[ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
& & & 5 & X & Y_{7} \\
&+& & & 5 & 2_{7} \\
\cline{2-6}
& & & 6 & 4 & X_{7} \\
\end{array} \]
|
10
| 0.333333 |
Factor $x^2 + 6x + 9 - 64x^4$ into two quadratic polynomials with integer coefficients. Submit your answer in the format $(ax^2 + bx + c)(dx^2 + ex + f)$, with $a < d$.
|
(-8x^2 + x + 3)(8x^2 + x + 3)
| 0.666667 |
The least common multiple of $x$, $15$, and $21$ is $105$. What is the greatest possible value of $x$?
|
105
| 0.916667 |
What is the greatest possible number of digits in the product of a 5-digit whole number and a 3-digit whole number?
|
8
| 0.75 |
Circle $D$ has a radius of 10 cm. How many square centimeters are in the area of the largest possible inscribed equilateral triangle in circle $D$?
|
75\sqrt{3}
| 0.916667 |
What is the sum of all integer solutions to $4 < (x - 3)^2 < 49$?
|
24
| 0.833333 |
Determine the value of \( k \) such that the set of vectors \( \left\{ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \begin{pmatrix} 4 \\ k \\ 6 \end{pmatrix} \right\} \) is linearly dependent.
|
8
| 0.916667 |
Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\]
|
75
| 0.916667 |
A cylinder has a volume of $72\pi$ cm³ and its height is twice its radius. Calculate the volume of a cone that has the same height and radius as the cylinder.
|
24\pi
| 0.666667 |
If $x^5 - x^4 + x^3 - px^2 + qx + 9$ is divisible by $(x + 3)(x - 2),$ find the ordered pair $(p,q).$
|
(-19.5, -55.5)
| 0.333333 |
Let
\[
\mathbf{B} = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}.
\]
Compute $\mathbf{B}^{98}$.
|
\mathbf{B}^{98} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}
| 0.916667 |
Compute \(\sin 300^\circ\).
|
-\frac{\sqrt{3}}{2}
| 0.916667 |
Four vertices of a rectangle are given as $(2, 10)$, $(12, 10)$, $(12, -5)$, and $(2, -5)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 2)^2 + (y + 5)^2 = 16$?
|
4\pi
| 0.25 |
Define a function \( f(x) \) for \( 0 \leq x \leq 1 \) with the following properties:
(i) \( f(0) = 0 \).
(ii) If \( 0 \leq x < y \leq 1 \), then \( f(x) \leq f(y) \).
(iii) \( f(1 - x) = \frac{3}{4} - \frac{f(x)}{2} \).
(iv) \( f \left( \frac{x}{3} \right) = \frac{f(x)}{3} \) for \( 0 \leq x \leq 1 \).
Determine the value of \( f \left( \frac{2}{9} \right) \).
|
\frac{5}{24}
| 0.25 |
Determine how many integers are in the list when Eleanor writes down the smallest positive multiple of 15 that is a perfect square, the smallest positive multiple of 15 that is a perfect cube, and all the multiples of 15 between them.
|
211
| 0.833333 |
Let $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = \|\mathbf{b}\| = 1,$ $\|\mathbf{a} - \mathbf{b}\| = 1,$ and
\[
\mathbf{c} - 2\mathbf{a} - 3\mathbf{b} = 4\mathbf{a} + 6\mathbf{b}.
\]
Find the value of $\mathbf{b} \cdot \mathbf{c}$.
|
12
| 0.916667 |
A frustum of a right circular cone is created by cutting off the top of a larger cone. The frustum has a height of $18$ cm, the area of its larger base is $324\pi$ sq cm, and the area of its smaller base is $36\pi$ sq cm. Find the height of the smaller cone that was removed.
|
9 \text{ cm}
| 0.75 |
A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has a diameter of 16 and an altitude of 24. Both cylinder and cone share the same central axis. Determine the radius of the inscribed cylinder. Express your answer as a common fraction.
|
\frac{24}{5}
| 0.25 |
Let \( z = e^{i\theta} \), where \( \theta \) is a real number, and \( |z| = 1 \). Find the real part of \( \frac{1}{2 - z} \).
|
\frac{2 - \cos\theta}{5 - 4\cos\theta}
| 0.916667 |
Calculate the remainder of $11^{2023}$ when divided by 33.
|
11
| 0.916667 |
How many positive integers $N$ less than $500$ are there such that the equation $x^{\lfloor x\rfloor} = N$ has a solution for $x$?
|
287
| 0.416667 |
Let \(\mathbf{v}\) be a vector \((a, b)\) in \(\mathbb{R}^2\), and let \(\mathbf{u}(\theta)\) be a unit vector \((\cos\theta, \sin\theta)\). Find the minimum and maximum values of the dot product \(\mathbf{v} \cdot \mathbf{u}(\theta)\) as \(\theta\) varies from 0 to \(2\pi\).
|
-\sqrt{a^2+b^2}
| 0.25 |
Let \(c\) be a real number randomly selected from the interval \([-10,10]\). Then, \(p\) and \(q\) are two relatively prime positive integers such that \(p/q\) is the probability that the equation \(x^4 + 16c^2 = (3c^2 - 8c)x^2\) has \(\textit{at least}\) two distinct real solutions. Find the value of \(p+q\).
|
26
| 0.583333 |
What is the area, in square units, of a trapezoid bounded by the lines $y = x + 2$, $y = 12$, $y = 3$, and the $y$-axis? Express your answer as a decimal to the nearest tenth.
|
49.5
| 0.5 |
The Cookie Monster now encounters a different cookie, which is bounded by the equation $(x-2)^2 + (y+1)^2 = 5$. He wonders if this cookie is big enough to share. Calculate the radius of this cookie and determine the area it covers.
|
5\pi
| 0.916667 |
In Townville, vehicle license plates each contain three letters. The first letter is chosen from the set $\{B, F, J, N, Z\}$, the second letter from $\{E, U, Y\}$, and the third letter from $\{K, Q, X, S\}$. Townville decided to expand their license plate capacity by adding four new letters to these sets. Each new letter could be added to any set, but at least one letter must be added to each of the second and third sets. What is the largest possible number of ADDITIONAL vehicle license plates that can be made by optimally placing these four new letters?
|
90
| 0.583333 |
What is the least 3-digit base 8 positive integer that is divisible by 7? (Express your answer in base 8.)
|
106_8
| 0.75 |
Evaluate the expression $\log_{y^3}{x^2}\cdot\log_{x^4}{y^3}\cdot\log_{y^5}{x^4}\cdot\log_{x^2}{y^5}$ and express it as $b\log_y{x}$ for some constant $b$.
|
1
| 0.916667 |
The cards of a standard 40-card deck (20 black and 20 red) are dealt out in a circle. What is the expected number of pairs of adjacent cards that are both red? Express your answer as a common fraction.
|
\frac{380}{39}
| 0.916667 |
Compute the diameter, in inches, of a sphere with three times the volume of a sphere of radius 12 inches. Express the diameter in the form \( a\sqrt[3]{b} \) where \( a \) and \( b \) are positive integers and \( b \) contains no perfect cube factors. Compute \( a+b \).
|
27
| 0.75 |
In triangle $PQR$, the measure of $\angle P$ is $88^\circ$. The measure of $\angle Q$ is $18^\circ$ more than twice the measure of $\angle R$. Determine the measure, in degrees, of $\angle R$.
|
\frac{74}{3} \text{ degrees}
| 0.416667 |
A point $(x, y)$ is randomly selected such that $0 \leq x \leq 4$ and $0 \leq y \leq 7$. What is the probability that $x+y \leq 5$? Express your answer as a common fraction.
|
\frac{3}{7}
| 0.25 |
How many positive three-digit integers, where each digit is odd and greater than 4, are divisible by 6?
|
0
| 0.833333 |
A line segment has endpoints at $A(3, 1)$ and $B(9, 7)$. The segment is extended through $B$ to a point $C$ such that $BC$ is half the length of $AB$. Determine the coordinates of point $C$.
|
(12, 10)
| 0.75 |
Given that the diagonals of a rhombus are always perpendicular bisectors of each other, what is the area of a rhombus with side length $\sqrt{117}$ units and diagonals that differ by 8 units?
|
101
| 0.916667 |
Let $\mathbf{A}$ and $\mathbf{B}$ be matrices such that
\[\mathbf{A} + \mathbf{B} = \mathbf{A} \mathbf{B}.\]
If $\mathbf{A} \mathbf{B} = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix},$ find $\mathbf{B} \mathbf{A}$.
|
\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}
| 0.916667 |
For certain real numbers \(a\), \(b\), and \(c\), the polynomial \[g(x) = x^3 + ax^2 + x + 8\] has three distinct roots, and each root of \(g(x)\) is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 50x + c.\] What is \(f(1)\)?
|
-1333
| 0.25 |
Suppose functions $g$ and $f$ are such that $g(x) = 4f^{-1}(x)$ and $f(x) = \frac{30}{x+2}$. Determine the value of $x$ for which $g(x) = 20$.
|
\frac{30}{7}
| 0.833333 |
Find the $2 \times 2$ matrix $\mathbf{M}$ such that for a $2 \times 2$ matrix $\mathbf{N},$ $\mathbf{M} \mathbf{N}$ results in swapping the first row and second row of $\mathbf{N},$ and tripling the second row of $\mathbf{N}.$ In other words,
\[
\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} c & d \\ 3a & 3b \end{pmatrix}.
\]
|
\begin{pmatrix} 0 & 1 \\ 3 & 0 \end{pmatrix}
| 0.75 |
How many real numbers \( y \) are solutions to the equation \[ |y-2| = |y-1| + |y-4| \]?
|
0
| 0.25 |
The third and sixth terms of a geometric sequence of real numbers are $6!$ and $7!$ respectively. What is the first term?
|
\frac{720}{7^{2/3}}
| 0.916667 |
In triangle $ABC$, the lengths of the sides are $AB = 4$, $AC = 7$, and $BC = 5$. The medians $AD$, $BE$, and $CF$ intersect at the centroid $G$. Let the projections of $G$ onto $BC$, $AC$, and $AB$ be $P$, $Q$, and $R$, respectively. Find $GP + GQ + GR$.
|
\frac{166\sqrt{6}}{105}
| 0.083333 |
Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \ne 1$. Compute
\[\omega^{10} + \omega^{12} + \omega^{14} + \dots + \omega^{50}.\]
|
1
| 0.166667 |
Find the matrix \(\mathbf{P}\) such that for any vector \(\mathbf{v},\) \(\mathbf{P} \mathbf{v}\) is the projection of \(\mathbf{v}\) onto the vector \(\begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}\).
|
\begin{pmatrix} \frac{1}{6} & \frac{1}{6} & \frac{1}{3} \\ \frac{1}{6} & \frac{1}{6} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{2}{3} \end{pmatrix}
| 0.916667 |
After finishing a meeting, Jay, Paul, and Sam start walking from the same point. Jay walks north at a rate of 1 mile every 20 minutes, Paul walks south at a rate of 3 miles every 40 minutes, and Sam walks north at a rate of 1.5 miles every 30 minutes. Calculate the total distance between Paul and the farthest among Jay and Sam after 2 hours.
|
15 \text{ miles}
| 0.916667 |
Find the remainder when $3x^7 - 2x^5 + 5x^3 - 8$ is divided by $x^2 + 3x + 2$.
|
354x + 340
| 0.916667 |
Given that \( x \) is real and \( x^3 + \frac{1}{x^3} = -52 \), find \( x + \frac{1}{x} \).
|
-4
| 0.916667 |
Find $n$ such that $2^6 \cdot 3^3 \cdot 5^1 \cdot n = 10!$.
|
420
| 0.75 |
What is the sum of all three-digit and four-digit positive integers up to 2000?
|
1996050
| 0.75 |
The area of the floor in a rectangular room is 320 square feet. The room is 16 feet long. The homeowners plan to cover the floor with tiles that are each 1 foot by 1 foot. How many tiles will be in each row?
|
20
| 0.666667 |
Let $g(x) = \left\lceil\dfrac{2}{x+3}\right\rceil$ for $x > -3$, and $g(x) = \left\lfloor\dfrac{2}{x+3}\right\rfloor$ for $x < -3$. ($g(x)$ is not defined at $x = -3$.) Which integer is not in the range of $g(x)$?
|
0
| 0.916667 |
For the ellipse $25x^2 - 100x + 4y^2 + 8y + 16 = 0,$ find the distance between the foci.
|
\frac{2\sqrt{462}}{5}
| 0.833333 |
Determine in how many different ways five students can stand in a line if three of those students insist on standing together.
|
36
| 0.666667 |
Suppose that $a$, $b$, and $c$ are digits, and the repeating decimal $0.\overline{abc}$ is expressed as a fraction in lowest terms. Assume $a$, $b$, and $c$ are not all nine and not all zero. How many different denominators are possible?
|
7
| 0.416667 |
How many different 8-digit positive integers are there?
|
90,\!000,\!000
| 0.75 |
Let $a,$ $b,$ and $c$ be the roots of
\[ x^3 - 5x + 7 = 0. \]
Find the monic polynomial, in $x,$ whose roots are $a + 3,$ $b + 3,$ and $c + 3.$
|
x^3 - 9x^2 + 22x - 5
| 0.416667 |
Let $g(x)$ be a polynomial of degree 2011 with real coefficients, and let its roots be $s_1, s_2, \dots, s_{2011}$. There are exactly 1010 distinct values among
\[ |s_1|, |s_2|, \dots, |s_{2011}|. \]
What is the minimum number of real roots that $g(x)$ can have?
|
9
| 0.5 |
Let $\mathcal{T}$ be the set $\{1, 2, 3, \ldots, 12\}$. Let $m$ be the number of sets of two non-empty disjoint subsets of $\mathcal{T}$. Find the remainder obtained when $m$ is divided by $1000$.
|
625
| 0.25 |
Evaluate $3(a^3+b^3)\div(a^2-ab+b^2)$ where $a=7$ and $b=5$.
|
36
| 0.833333 |
Let $a$, $b$, $c$ be the sides of a triangle, and let $\alpha$, $\beta$, $\gamma$ be the respective angles opposite these sides. If $a^2 + b^2 = 2001c^2$, find the value of
\[
\frac{\cot \gamma}{\cot \alpha + \cot \beta}.
\]
|
1000
| 0.583333 |
What is the sum of all integer values $n$ such that $\binom{30}{15} + \binom{30}{n} = \binom{31}{16}$?
|
30
| 0.916667 |
Calculate how many five-character license plates can be made if they consist of two consonants followed by two vowels and end with a digit? (Consider Y as a vowel for this problem.)
|
144{,}000
| 0.083333 |
Rationalize the denominator of $\frac{3}{4\sqrt{7} + 3\sqrt{13}}$ and write your answer in the form $\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is $A+B+C+D+E$?
|
22
| 0.416667 |
Find the quotient when $x^3 + 5x^2 + 3x + 9$ is divided by $x - 2$.
|
x^2 + 7x + 17
| 0.916667 |
Find the units digit of the product, \( 1! \times 2! \times 3! \times \cdots \times 15! \).
|
0
| 0.916667 |
Calculate the distance between the points $(2,6)$ and $(5,2)$.
|
5
| 0.083333 |
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