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stringlengths 18
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A matrix reflects points over a line $\ell,$ passing through the origin, represented by the following matrix:
\[\begin{pmatrix} \frac{3}{5} & \frac{4}{5} \\ \frac{4}{5} & -\frac{3}{5} \end{pmatrix}.\]
Find the direction vector of line $\ell.$ Enter your answer in the form $\begin{pmatrix} x \\ y \end{pmatrix},$ where $x,$ and $y$ are integers, $x > 0,$ and $\gcd(|x|,|y|) = 1.$
|
\begin{pmatrix} 2 \\ 1 \end{pmatrix}
| 0.666667 |
Let
\[f(x) = \left\{
\begin{array}{cl}
x + 5 & \text{if $x < 10$}, \\
3x - 1 & \text{if $x \ge 10$}.
\end{array}
\right.\]
Find $f^{-1}(8) + f^{-1}(50)$.
|
20
| 0.916667 |
Calculate the area of a rhombus $EFGH$ where the circumradii for triangles $EFG$ and $EHG$ are $15$ and $30$ respectively.
|
576
| 0.083333 |
Is there a number that is congruent to 0 modulo every positive integer less than 10?
|
2520
| 0.916667 |
Expand $(x^{15} - 4x^8 + 2x^{-3} - 9) \cdot (3x^3)$.
|
3x^{18} - 12x^{11} - 27x^3 + 6
| 0.333333 |
Consider the equation \[\sqrt{(x-2)^2 + (y+3)^2} - \sqrt{(x-8)^2 + (y+3)^2} = 4.\] Find the positive value for the slope of an asymptote of this hyperbola.
|
\frac{\sqrt{5}}{2}
| 0.916667 |
Let \[g(x) = \left\{
\begin{array}{cl}
\frac{x}{28} & \text{ if }x\text{ is a multiple of 4 and 7}, \\
4x & \text{ if }x\text{ is only a multiple of 7}, \\
7x & \text{ if }x\text{ is only a multiple of 4}, \\
x+4 & \text{ if }x\text{ is not a multiple of 4 or 7}.
\end{array}
\right.\]If $g^a(x)$ denotes the function nested $a$ times, find the smallest value of $a$ greater than 1 such that $g(2)=g^a(2)$.
|
6
| 0.083333 |
Given that $b$ is an even multiple of $97$, find the greatest common divisor of $3b^2 + 41b + 74$ and $b + 19$.
|
1
| 0.666667 |
Determine $\cos B$ in the following right triangle:
[asy]
pair A, B, C;
A = (0,0);
B = (8,0);
C = (0,6);
draw(A--B--C--A);
draw(rightanglemark(C,A,B,10));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
label("$10$", (A+C)/2, NW);
label("$8$", (A+B)/2, S);
[/asy]
|
\frac{4}{5}
| 0.416667 |
Consider the functions:
$$\begin{array}{ccc}
f(x) & = & 4x^2 - 3x + 5\\
g(x) & = & x^2 - mx - 8
\end{array}$$
If $f(5) - g(5) = 20,$ what is the value of $m$?
|
-\frac{53}{5}
| 0.833333 |
If $128^7 = 16^x$, what is the value of $2^{-x}$? Express your answer as a common fraction.
|
2^{-x} = \frac{1}{2^{49/4}}
| 0.25 |
A $8\times 1$ board is completely covered by $m\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $8\times 1$ board in which at least two different colors are used. Find the remainder when $N$ is divided by $1000$.
|
768
| 0.083333 |
Three standard dice are rolled, yielding numbers $a, b, c$. Find the probability that the sum of these numbers is $a+b+c=15$.
|
\frac{5}{108}
| 0.666667 |
Find the vector $\mathbf{u}$ such that
\[\operatorname{proj}_{\begin{pmatrix} 3 \\ 1 \end{pmatrix}} \mathbf{u} = \begin{pmatrix} \frac{45}{10} \\ \frac{15}{10} \end{pmatrix}\]
and
\[\operatorname{proj}_{\begin{pmatrix} 1 \\ 2 \end{pmatrix}} \mathbf{u} = \begin{pmatrix} \frac{36}{5} \\ \frac{72}{5} \end{pmatrix}.\]
|
\begin{pmatrix} -\frac{6}{5} \\ \frac{93}{5} \end{pmatrix}
| 0.666667 |
Let $f(x) = \frac{4x^2 + 6x + 9}{x^2 - 2x + 5}$ and $g(x) = x + 2$. Calculate $f(g(x)) + g(f(x))$, evaluated when $x=0$.
|
\frac{56}{5}
| 0.75 |
Gage skated 1 hour each day for 6 days and 2 hours each day for 2 days. How many minutes would he have to skate the ninth day in order to average 100 minutes of skating each day for the entire time?
|
300 \text{ minutes}
| 0.916667 |
Piravena must make a trip from city $X$ to city $Y$, then from $Y$ to city $Z$, and finally from $Z$ back to $X$. The cities are arranged in a right-angled triangle, with $XZ = 4000$ km and $XY = 5000$ km. Travel costs are as follows: by bus, the cost is $\$0.20$ per kilometer; by airplane, there is a $\$150$ booking fee plus $\$0.15$ per kilometer. Determine the least expensive travel configurations and calculate the total minimum cost for the trip.
|
\$2250
| 0.666667 |
Suppose I have 8 shirts, 5 ties, 4 pairs of pants, and the choice of either wearing or not wearing one of 2 different jackets. If an outfit requires a shirt and pants, can optionally include a tie, and can also include a jacket or not, how many different outfits can I create?
|
576
| 0.833333 |
The values of a function \( g(x) \) are given in the table below.
\[
\begin{tabular}{|r||c|c|c|c|c|c|}
\hline
\( x \) & 1 & 2 & 3 & 5 & 8 & 13 \\
\hline
\( g(x) \) & 4 & 12 & 7 & 2 & 1 & 6 \\
\hline
\end{tabular}
\]
If \( g^{-1} \) exists, what is \( g^{-1}\left(\frac{g^{-1}(6) + g^{-1}(12)}{g^{-1}(2)}\right) \)?
|
3
| 0.75 |
Let \( x \) and \( y \) be two positive real numbers such that \( x + y = 28 \). Find the ordered pair \( (x, y) \) for which \( x^5 y^3 \) is maximized.
|
(17.5, 10.5)
| 0.666667 |
A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,3), (3,3), and (3,0). What is the probability that $x+y < 4$?
|
\frac{7}{9}
| 0.166667 |
At a school award ceremony, a class won 23 out of a possible 150 awards. Which one of the following fractions is closest to the fraction of awards that they won?
$$ \frac{1}{5} \quad \frac{1}{6} \quad \frac{1}{7} \quad \frac{1}{8} \quad \frac{1}{9} $$
|
\frac{1}{7}
| 0.75 |
Compute
\[
\sin^6 0^\circ + \sin^6 1^\circ + \sin^6 2^\circ + \dots + \sin^6 90^\circ.
\]
|
\frac{229}{8}
| 0.166667 |
A line is parameterized by a parameter $t$, such that the vector on the line at $t = 1$ is $\begin{pmatrix} 2 \\ 5 \end{pmatrix},$ and the vector on the line at $t = 4$ is $\begin{pmatrix} 8 \\ -7 \end{pmatrix}.$ Find the vector on the line at $t = -2.$
|
\begin{pmatrix} -4 \\ 17 \end{pmatrix}
| 0.666667 |
What is the greatest common divisor of $12345$ and $6789$?
|
3
| 0.75 |
Add 34.789, 15.2467, and 9.0056. Then, round to the nearest hundredth.
|
59.04
| 0.583333 |
What is the $205$th digit after the decimal point when $\frac{14}{360}$ is expressed as a decimal?
|
8
| 0.916667 |
How many positive three-digit integers less than 600 have at least two digits that are the same?
|
140
| 0.833333 |
Consider the function $$g(x) = \frac{2}{1+\frac{1}{2+\frac{1}{x}}}.$$ Determine the sum of the values of $x$ that make $g(x)$ undefined.
|
-\frac{5}{6}
| 0.833333 |
In how many ways can the digits of $47,\!520$ be arranged to form a 5-digit number, where numbers cannot begin with 0?
|
96
| 0.916667 |
An isosceles triangle $PQR$ has an area of 144 square inches. A horizontal line $\overline{ST}$ cuts $PQR$ into an isosceles trapezoid and a smaller isosceles triangle at the top. The area of the trapezoid is 108 square inches. If the altitude of triangle $PQR$ from $P$ is 24 inches, find the length of $\overline{ST}$.
|
6
| 0.75 |
When the graph of $y = 2x^2 - x + 7$ is shifted three units to the right and five units up, we get the graph of $y = ax^2 + bx + c$. Find the value of $a+b+c$.
|
22
| 0.833333 |
Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 12$, $BC = 6$, $\overline{PA} \perp \overline{AB}$, $\overline{PA} \perp \overline{AD}$, and $PA = 10$, find the volume of $PABCD$. Additionally, compute the area of triangle $ABC$.
|
36 \text{ square units}
| 0.5 |
Let $\alpha$ and $\beta$ be real numbers. Find the minimum value of:
\[
(3 \cos \alpha + 4 \sin \beta - 5)^2 + (3 \sin \alpha + 4 \cos \beta - 12)^2.
\]
|
36
| 0.083333 |
What is the smallest positive integer that must always divide the sum of the first twelve terms of any arithmetic sequence where the terms are positive integers?
|
6
| 0.833333 |
Let $a,$ $b,$ and $c$ be positive real numbers. Find the minimum value of
\[
\frac{a^2}{kb} + \frac{b^2}{kc} + \frac{c^2}{ka}.
\]
for some positive constant \( k \).
|
\frac{3}{k}
| 0.166667 |
Five faucets fill a 125-gallon tub in 8 minutes. How long, in minutes, does it take ten faucets to fill a 50-gallon tub? Assume that all faucets dispense water at the same rate.
|
1.6
| 0.75 |
The Photography club has 24 members: 12 boys and 12 girls. A 5-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl?
|
\frac{1705}{1771}
| 0.833333 |
The line \( y = \frac{3}{4}x - 15 \) is parameterized by \( (x,y) = (f(t),20t - 10) \). Determine the function \( f(t) \).
|
f(t) = \frac{80}{3}t + \frac{20}{3}
| 0.166667 |
In the diagram below, we have $AB = 30$ and $\angle ADB = 90^\circ$. If $\sin A = \frac{4}{5}$ and $\sin C = \frac{1}{4}$, then what is $DC$?
[asy]
pair A,B,C,D;
A = (0,0);
B = (12*sqrt(26/21),24);
D = (12*sqrt(26/21),0);
C = (12*sqrt(26/21) + 32*sqrt(26/5),0);
draw(D--B--A--C--B);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label("$D$",D,S);
draw(rightanglemark(B,D,A,63));
[/asy]
|
24\sqrt{15}
| 0.166667 |
In spherical coordinates, the point $\left( 4, \frac{\pi}{3}, \frac{7 \pi}{4} \right)$ is equivalent to what other point, in the standard spherical coordinate representation? Enter your answer in the form $(\rho,\theta,\phi),$ where $\rho > 0,$ $0 \le \theta < 2 \pi,$ and $0 \le \phi \le \pi.$
|
\left( 4, \frac{4 \pi}{3}, \frac{\pi}{4} \right)
| 0.083333 |
Rationalize the denominator of $\frac{\sqrt{8}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}$ and express your answer in simplest form.
|
\sqrt{10} - \sqrt{6} + \frac{\sqrt{15}}{2} - \frac{3}{2}
| 0.583333 |
If \(y^2 - 6y + 9 = 0\) and \(y + z = 11\), what are the values of \(y\) and \(z\)?
|
8
| 0.916667 |
There exist constants $b_1,$ $b_2,$ $b_3$ such that
\[\cos^3 \theta = b_1 \cos \theta + b_2 \cos 2 \theta + b_3 \cos 3 \theta\] for all angles $\theta$. Find \(b_1^2 + b_2^2 + b_3^2.\)
|
\frac{5}{8}
| 0.833333 |
We flip a fair coin 12 times. What is the probability that we get exactly 9 heads and all heads occur consecutively?
|
\dfrac{1}{1024}
| 0.666667 |
Find the number of complex numbers $z$ satisfying $|z| = 1$ and
\[\left| \frac{z}{\overline{z}} - \frac{\overline{z}}{z} \right| = 1.\]
|
8
| 0.5 |
An amoeba is placed in a controlled environment on day 1. Starting from the initial day, each amoeba splits into two amoebas every day, except that every third day, only half of the amoebas are capable of splitting due to environmental stress. After 10 days, how many amoebas will there be, assuming the controlled environment starts with one amoeba?
|
432
| 0.083333 |
For what value of $k$ does the equation $(x+5)(x+2) = k + 3x$ have exactly one real solution? Express your answer as a common fraction.
|
6
| 0.916667 |
A function $g(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have:
\[3g\left(x\right) + g\left(\frac{1}{x}\right) = 7x + 5\]
Let $T$ denote the sum of all of the values of $x$ for which $g(x) = 3005$. Compute the integer nearest to $T$.
|
1144
| 0.916667 |
A sphere is inscribed in a right cone with base radius $15$ cm and height $30$ cm. The radius of the sphere can be expressed as $b\sqrt{d} - b$ cm. What is the value of $b + d$?
|
12.5
| 0.083333 |
Find how many positive integers \( n \), \( 1 \leq n \leq 1500 \), make the polynomial \( x^2 - 2x - n \) factorizable into two linear factors with integer coefficients.
|
37
| 0.5 |
A square pyramid has a base edge of 40 inches and an altitude of 15 inches. A smaller square pyramid, whose altitude is one-third the altitude of the original pyramid, is cut away from the apex. What is the volume of the remaining frustum as a fraction of the volume of the original pyramid?
|
\frac{26}{27}
| 0.833333 |
Let $\mathbf{p}$ and $\mathbf{q}$ be two three-dimensional unit vectors such that the angle between them is $45^\circ$. Calculate the area of the parallelogram whose diagonals are represented by the vectors $\mathbf{p} + 2\mathbf{q}$ and $2\mathbf{p} + \mathbf{q}$.
|
\frac{3\sqrt{2}}{4}
| 0.333333 |
Laura has modified her training routine. Now, she bikes 25 miles at a rate of $3x + 1$ miles per hour, spends ten minutes transitioning to her running gear, and then runs 8 miles at $x$ miles per hour. The entire workout lasts 130 minutes. Determine Laura’s running speed, rounding to the nearest hundredth of a mile per hour. A calculator may be used.
|
8.00 \text{ mph}
| 0.75 |
A circle with a radius of 5 units has its center at $(0, 0)$. A circle with a radius of 3 units has its center at $(10, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction.
|
\frac{25}{4}
| 0.416667 |
In a right triangle $PQR$ with $\angle R = 90^\circ$, it is known that $3\sin Q = 4\cos Q$. Determine $\sin Q$ and $\tan Q$.
|
\tan Q = \frac{4}{3}
| 0.916667 |
Automobile license plates for a state consist of four letters followed by a dash and two single digits. How many different license plate combinations are possible if two distinct letters each are repeated exactly once, but digits cannot be repeated and the letter 'A' cannot be used?
|
162,\!000
| 0.916667 |
In a local chess club, there are 5 teams each with 6 members. Teams rotate hosting monthly meetings. At each meeting, each team selects three members to be on the organizing committee, except the host team, which selects four members. Calculate the number of possible 16-member organizing committees.
|
12,000,000
| 0.5 |
Let $a$, $b$, $c$, and $d$ be positive real numbers. What is the smallest possible value of $(a+b+c+d)\left(\frac{1}{a+b+c}+\frac{1}{a+b+d}+\frac{1}{a+c+d}+\frac{1}{b+c+d}\right)$?
|
\frac{16}{3}
| 0.916667 |
How many prime numbers between 30 and 80 have a prime remainder when divided by 12?
|
9
| 0.416667 |
Let $a, b,$ and $t$ be real numbers such that $2a + b = 2t$. Find, in terms of $t$, the minimum value of $a^2 + b^2$.
|
\frac{4t^2}{5}
| 0.833333 |
How many even perfect square factors does $2^6 \cdot 7^{10}$ have?
|
18
| 0.75 |
Calculate the remainder when the sum $1! + 2! + 3! + \cdots + 9! + 10!$ is divided by 7.
|
5
| 0.833333 |
Each of the numbers $b_1, b_2, \dots, b_{100}$ is $\pm 1.$ Find the smallest possible positive value of
\[\sum_{1 \le i < j \le 100} b_i b_j.\]
|
22
| 0.583333 |
Find the ordered triplet $(x, y, z)$ that solves the following system of equations:
\begin{align*}
3x + 2y &= z - 1,\\
2x - y &= 4z + 2,\\
x + 4y &= 3z + 9
\end{align*}
|
\left(-\frac{24}{13}, \frac{18}{13}, -\frac{23}{13}\right)
| 0.916667 |
What is the greatest common divisor of $123^2 + 235^2 - 347^2$ and $122^2 + 234^2 - 348^2$?
|
1
| 0.583333 |
In the diagram below, we have $AB = 30$ and $\angle ADB = 90^\circ$. If $\cos A = \frac{4}{5}$ and $\sin C = \frac{2}{5}$, then what is $AD$?
[asy]
pair A,B,C,D;
A = (0,0);
B = (12*sqrt(5),18);
D = (12*sqrt(5),0);
C = (12*sqrt(5) + 45,0);
draw(D--B--A--C--B);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label("$D$",D,S);
draw(rightanglemark(B,D,A,63));
[/asy]
|
AD = 24
| 0.916667 |
Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers that satisfy the three equations
\begin{align*}
17x + by + cz &= 0 \\
ax + 29y + cz &= 0 \\
ax + by + 37z &= 0.
\end{align*}
Suppose that $ a \ne 17$ and $ x \ne 0$. What is the value of
\[ \frac{a}{a - 17} + \frac{b}{b - 29} + \frac{c}{c - 37} \, ?\]
|
1
| 0.833333 |
A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has a diameter of 8 and an altitude of 10, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. Express your answer as a common fraction.
|
\frac{20}{9}
| 0.333333 |
$\triangle XYZ$ is inscribed inside $\triangle PQR$ such that points $X, Y, Z$ lie on $RQ, QP, PR$, respectively. The circumcircles of $\triangle PXZ, \triangle QYX, \triangle RZY$ have centers $O_4, O_5, O_6$, respectively. $PQ = 26, QR = 29, RP = 25$, and $\stackrel{\frown}{PY} = \stackrel{\frown}{RZ},\ \stackrel{\frown}{QZ} = \stackrel{\frown}{PX},\ \stackrel{\frown}{QY} = \stackrel{\frown}{RX}$. The length of $RQ$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$.
|
30
| 0.75 |
Winnie has 15 red balloons, 42 blue balloons, 54 yellow balloons, and 92 purple balloons. She decides to distribute these balloons equally among her 11 closest friends. How many balloons does Winnie end up keeping for herself?
|
5
| 0.666667 |
What is the smallest multiple of 7 that is greater than -50?
|
-49
| 0.666667 |
I have a drawer with 6 shirts, 7 pairs of shorts, and 6 pairs of socks, and 3 hats. If I reach in and randomly remove three articles of clothing, what is the probability that I get one shirt, one pair of shorts, and one hat?
|
\frac{63}{770}
| 0.083333 |
What is the sum of the squares of the lengths of the **medians** of a triangle whose side lengths are $13,$ $14,$ and $15$?
|
442.5
| 0.916667 |
Let $p,$ $q,$ and $r$ be real numbers, and let $P,$ $Q,$ $R$ be points such that the midpoint of $\overline{QR}$ is $(p,0,0),$ the midpoint of $\overline{PR}$ is $(0,q,0),$ and the midpoint of $\overline{PQ}$ is $(0,0,r).$ Determine the value of
\[\frac{PQ^2 + PR^2 + QR^2}{p^2 + q^2 + r^2}.\]
|
8
| 0.916667 |
What is the number of square units in the area of trapezoid EFGH with vertices E(0,0), F(0,-3), G(5,0), and H(5,8)?
|
27.5
| 0.666667 |
If 5 daps are equivalent to 4 dops, and 3 dops are equivalent to 8 dips, how many daps are equivalent to 24 dips?
|
11.25 \text{ daps}
| 0.916667 |
Find the largest real number $x$ such that
\[\frac{\lfloor x \rfloor}{x} = \frac{8}{9}.\]
|
\frac{63}{8}
| 0.916667 |
Let $n$ be a positive integer and $x, y$ be invertible integers modulo $n$ such that $x \equiv 2y \pmod{n}$ and $y \equiv 3x^{-1} \pmod{n}$. What is the remainder when $xy$ is divided by $n$?
|
3
| 0.916667 |
How many five-character license plates can be created if the plate must consist of two consonants (from a set of 21 consonants), followed by two vowels (from a set of 5 vowels), and ending with a digit? (Assume an alphabet like Spanish where Y is not considered a vowel.)
|
110,250
| 0.916667 |
Convert $253_{10}$ to base 2. Let $x$ be the number of zeros and $y$ be the number of ones in base 2. What is the value of $y-x?$
|
6
| 0.75 |
Let $m$ be the smallest integer whose cube root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/10000$. Find $n$.
|
58
| 0.083333 |
Let $x$, $y$, and $z$ be real numbers such that
\[\cos 2x + \cos 2y + \cos 2z = \sin 2x + \sin 2y + \sin 2z = 0.\]
Find the sum of all possible values of $\cos 4x + \cos 4y + \cos 4z.$
|
0
| 0.833333 |
Find the slope and the $x$-intercept of the line $4x + 7y = 28$.
|
7
| 0.166667 |
Find the integer $m,$ $-90 < m < 90,$ such that $\tan m^\circ = \tan 1230^\circ.$
|
-30
| 0.916667 |
I have four distinct mystery novels, four distinct science fiction novels, and four distinct historical novels. I'm packing for a trip and want to take two books from different genres. How many different pairs of books can I take?
|
48
| 0.916667 |
Calculate $9 \cdot 7\frac{2}{5}$.
|
66\frac{3}{5}
| 0.75 |
When a water tank is $30\%$ full, it contains 54 gallons less than when it is $10\%$ empty. How many gallons of water does the tank hold when it is full?
|
90 \text{ gallons}
| 0.833333 |
Isosceles triangle $PQR$ has an area of 180 square inches and is divided by line $\overline{XY}$ into a smaller isosceles triangle and an isosceles trapezoid. The area of the trapezoid is 135 square inches. If the altitude of triangle $PQR$ from $P$ is 30 inches, determine the length of $\overline{XY}$.
|
6 \text{ inches}
| 0.916667 |
If $a$, $b$, and $c$ are positive numbers such that $ab=45\sqrt[3]{3}$, $ac=75\sqrt[3]{3}$, and $bc=27\sqrt[3]{3}$, then find the value of $abc$.
|
135\sqrt{15}
| 0.166667 |
What is \[3 - 5x - 7x^2 + 9 + 11x - 13x^2 - 15 + 17x + 19x^2\] in terms of $x$?
|
-x^2 + 23x - 3
| 0.833333 |
There are 12 sprinters in an international track event final, including 5 Americans. Gold, silver, and bronze medals are awarded to the first, second, and third place finishers respectively. Determine the number of ways the medals can be awarded if no more than two Americans receive medals.
|
1260
| 0.916667 |
The terms of the sequence $(b_i)$ are defined by $b_{n + 2} = \frac {b_n + 3001} {1 + b_{n + 1}}$ for $n \ge 1$, where all terms are positive integers. Find the minimum possible value of $b_1 + b_2$.
|
3002
| 0.166667 |
The Johnson family has 5 sons and 4 daughters. In how many ways can they be seated in a row of 9 chairs such that at least 2 boys are next to each other?
|
360000
| 0.666667 |
Pirate Pete shares his treasure with Pirate Paul using a modified pattern. Pete initiates by saying, "One for me, one for you," giving himself one coin and starting Paul's pile with one coin. He then says, "Two for me, and two for you," giving himself two more coins and making Paul’s pile two coins in total. This pattern continues with Pete saying, "Three for me, three for you," and so on, until Pete gives himself $x$ more coins but makes Paul’s pile $x$ coins in total. When all coins are distributed, Pirate Pete has exactly five times as many coins as Pirate Paul. How many gold coins do they have in total?
|
54
| 0.166667 |
Find $c$ if $\log_{c}125=-\frac{5}{3}$.
|
c = \frac{1}{5^{\frac{9}{5}}}
| 0.166667 |
The sequence \(b_1, b_2, \ldots\) is geometric with \(b_1 = b\) and a common ratio \(s\), where \(b\) and \(s\) are positive integers. Suppose that \(\log_4 b_1 + \log_4 b_2 + \cdots + \log_4 b_7 = 343\). Determine the number of possible ordered pairs \((b,s)\).
|
33
| 0.5 |
Angle $EAB$ is a right angle, and $BE = 12$ units. If the length $AB$ is $5$ units, what is the sum of the square units in the areas of the two squares $ABCD$ and $AEFG$?
|
144
| 0.916667 |
What is the greatest number of points of intersection that can occur when $2$ different circles and $3$ different straight lines are drawn on the same piece of paper?
|
17
| 0.833333 |
If $4$ lunks can be traded for $2$ kunks, and $3$ kunks will buy $6$ oranges, how many lunks are needed to purchase two dozen oranges?
|
24
| 0.916667 |
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