from stringclasses 2
values | value stringlengths 16 6.77k |
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human | Suppose that all four of the numbers \[3 - 2\sqrt{2}, \; -3-2\sqrt{2}, \; 1+\sqrt{7}, \; 1-\sqrt{7}\]are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial? |
gpt | Because the polynomial has rational coefficients, the radical conjugate of each of the given roots must also be roots of the polynomial. However, $1+\sqrt{7}$ and $1-\sqrt{7}$ are each other's radical conjugates, so we only get $2$ more roots. (You might be tempted to think that $3-2\sqrt2$ and $-3-2\sqrt2$ are also a ... |
human | The function $f(x)$ satisfies
\[b^2 f(a) = a^2 f(b)\]for all real numbers $a$ and $b.$ If $f(2) \neq 0,$ find
\[\frac{f(5) - f(1)}{f(2)}.\] |
gpt | Setting $a = 5$ and $b = 2,$ we get
\[4f(5) = 25f(2),\]so $\frac{f(5)}{f(2)} = \frac{25}{4}.$
Setting $a = 1$ and $b = 2,$ we get
\[4f(1) = f(2),\]so $\frac{f(1)}{f(2)} = \frac{1}{4}.$ Hence,
\[\frac{f(5) - f(1)}{f(2)} = \frac{25}{4} - \frac{1}{4} = \boxed{6}.\] |
human | Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 32.$ Find the minimum value of
\[x^2 + 4xy + 4y^2 + 2z^2.\] |
gpt | First, $x^2 + 4xy + 4y^2 = (x + 2y)^2.$ By AM-GM,
\[x + 2y \ge 2 \sqrt{2xy},\]so $(x + 2y)^2 \ge 8xy.$ Hence,
\[x^2 + 4xy + 4y^2 + 2z^2 \ge 8xy + 2z^2.\]If we apply AM-GM directly to $8xy$ and $2z^2,$ then ignoring constants, we will get the term $\sqrt{xyz^2}.$ But the condition is $xyz = 32.$ So instead, we write... |
human | Find the sum of all real solutions to the equation \[\sqrt{x} + \sqrt{\frac{4}{x}} + \sqrt{x + \frac{4}{x}} = 6.\] |
gpt | We want to square the equation in order to eliminate the radicals. To do so, we first move the $\sqrt{x+\frac4x}$ term to the right-hand side, giving \[\sqrt{x} + \sqrt{\frac{4}{x}} = 6 - \sqrt{x+\frac{4}{x}}.\]Now we see that squaring will produce lots of common terms on the left-hand and right-hand sides, which cance... |
human | Find all real numbers $x$ which satisfy \[\frac{x-1}{x-3} \ge 2.\](Give your answer in interval notation.) |
gpt | Subtracting $2$ from both sides, we get \[\frac{x-1}{x-3} - 2 \ge 0,\]or \[\frac{x-1 - 2(x-3)}{x-3} = \frac{-x+5}{x-3} \ge 0.\]Negating both sides, we have \[\frac{x-5}{x-3} \le 0.\]Letting $f(x) = \frac{x-5}{x-3},$ we make a sign table with the two factors $x-5$ and $x-3$: \begin{tabular}{c|cc|c} &$x-5$ &$x-3$ &$f(x)$... |
human | Find the product of the nonreal roots of $x^4-4x^3+6x^2-4x=2005.$ |
gpt | We recognize part of the expansion of $(x-1)^4$ on the left-hand side. Adding $1$ to both sides, we have \[x^4-4x^3+6x^2-4x+1=2006,\]which means $(x-1)^4 = 2006.$ Therefore, \[x-1 = \sqrt[4]{2006}, i\sqrt[4]{2006}, -\sqrt[4]{2006}, -i\sqrt[4]{2006}.\]Since we want the nonreal roots, we only consider the roots \[ x = 1 ... |
human | Find the area of the region described by $x \ge 0,$ $y \ge 0,$ and
\[100 \{x\} \ge \lfloor x \rfloor + \lfloor y \rfloor.\]Note: For a real number $x,$ $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x.$ For example, $\{2.7\} = 0.7.$ |
gpt | Let $n = \lfloor x \rfloor,$ and let $\{x\} = (0.x_1 x_2 x_3 x_4 \dots)_{10},$ so the $x_i$ are the decimal digits. Then the given condition becomes
\[\lfloor y \rfloor \le 100 \{x\} - \lfloor x \rfloor = (x_1 x_2.x_3 x_4 \dots)_{10} - n.\]Since $\lfloor y \rfloor$ is an integer, this is equivalent to
\[\lfloor y \rfl... |
human | Find the maximum value of $10^x - 100^x,$ over all real numbers $x.$ |
gpt | Let $y = 10^x.$ Then
\[10^x - 100^x = y - y^2 = \frac{1}{4} - \left( y - \frac{1}{2} \right)^2.\]Thus, the maximum value is $\boxed{\frac{1}{4}},$ which occurs when $y = \frac{1}{2},$ or $x = \log_{10} \left( \frac{1}{2} \right).$ |
human | Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 6.$ Find the minimum value of
\[\frac{x^2 + y^2}{x + y} + \frac{x^2 + z^2}{x + z} + \frac{y^2 + z^2}{y + z}.\] |
gpt | By QM-AM,
\[\sqrt{\frac{x^2 + y^2}{2}} \ge \frac{x + y}{2}.\]Then
\[\frac{x^2 + y^2}{2} \ge \left( \frac{x + y}{2} \right)^2,\]which we can re-arrange as
\[\frac{x^2 + y^2}{x + y} \ge \frac{x + y}{2}.\]Similarly,
\begin{align*}
\frac{x^2 + y^2}{x + y} &\ge \frac{x + y}{2}, \\
\frac{y^2 + z^2}{y + z} &\ge \frac{y + z}{... |
human | Solve the inequality
\[\left| \frac{2x - 1}{x - 1} \right| > 2.\] |
gpt | From the given inequality, either $\frac{2x - 1}{x - 1} > 2$ or $\frac{2x - 1}{x - 1} < -2.$
The inequality $\frac{2x - 1}{x - 1} > 2$ becomes
\[\frac{2x - 1}{x - 1} - 2 > 0,\]or
\[\frac{1}{x - 1} > 0.\]This is satisfied when $x > 1.$
The inequality $\frac{2x - 1}{x - 1} < -2$ becomes
\[\frac{2x - 1}{x - 1} + 2 < 0,\... |
human | Find all the solutions to
\[\frac{1}{x^2 + 11x - 8} + \frac{1}{x^2 + 2x - 8} + \frac{1}{x^2 - 13x - 8} = 0.\]Enter all the solutions, separated by commas. |
gpt | Let $y = x^2 - 13x - 8.$ Then we can write the given equation as
\[\frac{1}{y + 24x} + \frac{1}{y + 15x} + \frac{1}{y} = 0.\]Multiplying everything by $(y + 24x)(y + 15x)y,$ we get
\[(y + 15x)y + y(y + 24x) + (y + 24x)(y + 15x) = 0.\]This simplifies to $360x^2 + 78xy + 3y^2 = 0,$ which factors as $3(20x + y)(6x + y) =... |
human | Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x^2 + yf(z)) = xf(x) + zf(y)\]for all real numbers $x,$ $y,$ and $z.$
Let $n$ be the number of possible values of $f(5),$ and let $s$ be the sum of all possible values of $f(5).$ Find $n \times s.$ |
gpt | Setting $x = y = 0,$ we get
\[f(0) = zf(0)\]for all $z,$ so $f(0) = 0.$
Setting $y = 0,$ we get
\[f(x^2) = xf(x)\]for all $x.$
Setting $x = 0,$ we get
\[f(yf(z)) = zf(y).\]In particular, for $y = 1,$ $f(f(z)) = zf(1).$
Since $f(x^2) = xf(x),$
\[f(f(x^2)) = f(xf(x)).\]But $f(f(x^2)) = x^2 f(1)$ and $f(xf(x)) = xf(x),... |
human | A circle is tangent to the lines $4x - 3y = 30$ and $4x - 3y = -10.$ The center of the circle lies on the line $2x + y = 0.$ Find the center of the circle. |
gpt | Note that the lines $4x - 3y = 30$ and $4x - 3y = -10$ are parallel, so the center of the circle lies on the line which is exactly halfway between these lines, which is $4x - 3y = 10.$
[asy]
unitsize(2 cm);
pair A, B;
A = dir(-20);
B = dir(160);
draw(Circle((0,0),1));
draw((A + 1.5*dir(70))--(A - 1.5*dir(70)));
dra... |
human | An ellipse has foci at $(9, 20)$ and $(49, 55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis? |
gpt | Denote the ellipse by $\mathcal{E}.$ Let $F_1=(9,20)$ and $F_2=(49,55)$ be its foci, and let $X$ be the point where it touches the $x$-axis.
[asy]
size(6cm);
draw(shift(((9, 20) + (49, 55))/2)*rotate(41.186)*scale(85/2,10*11^.5)*unitcircle); draw((-20,0)--(80,0),EndArrow); draw((0,-20)--(0,85),EndArrow);
dot("$F_1 (9, ... |
human | A cubic polynomial $p(x)$ satisfies
\[p(n) = \frac{1}{n^2}\]for $n = 1, 2, 3,$ and $4.$ Find $p(5).$ |
gpt | Let $q(x) = x^2 p(x) - 1.$ Then $q(x)$ is a polynomial of degree 5, and $q(n) = 0$ for $n = 1,$ 2, 3, and 4, so
\[q(x) = (ax + b)(x - 1)(x - 2)(x - 3)(x - 4)\]for some constants $a$ and $b.$
We know that $q(0) = 0^2 \cdot p(0) - 1 = -1.$ But setting $x = 0$ in the equation above, we get
\[q(0) = 24b,\]so $b = -\frac... |
human | The graph of \[y^4 - 4x^4 = 2y^2 - 1\]is the union of the graphs of two different conic sections. Which two types of conic sections are they?
(Write your answer as a list, with "C" for circle, "E" for ellipse, "H" for hyperbola, and "P" for parabola. For example, "C, H" if you think the graph consists of a circle and ... |
gpt | We can rewrite the given equation as \[y^4 - 2y^2 + 1 = 4x^4.\]The left-hand side is the perfect square of a binomial: \[(y^2-1)^2 = 4x^4.\]Therefore, either $y^2-1=2x^2$ or $y^2-1=-2x^2.$ That is, either $y^2-2x^2=1$ or $y^2+2x^2=1.$ These are the equations for a hyperbola and an ellipse, respectively, so the answer i... |
human | Find the value of $k$ so that the line $3x + 5y + k = 0$ is tangent to the parabola $y^2 = 24x.$ |
gpt | Solving for $x$ in $3x + 5y + k = 0,$ we get
\[x = -\frac{5y + k}{3}.\]Substituting into $y^2 = 24x,$ we get
\[y^2 = -40y - 8k,\]or $y^2 + 40y + 8k = 0.$ Since we have a tangent, this quadratic will have a double root, meaning that its discriminant will be 0. This give us $40^2 - 4(8k) = 0,$ so $k = \boxed{50}.$ |
human | Find the $\emph{positive}$ real number(s) $x$ such that $\frac{1}{2}\left( 3x^2-1\right) = \left( x^2-50x-10\right)\left( x^2+25x+5\right)$. |
gpt | Write $a = x^2-50x-10$ and $b = x^2+25x+5$. Then the equation given becomes
\[\frac{a+2b-1}{2} = ab,\]so $0=2ab-a-2b+1=(a-1)(2b-1)$. Then $a-1=x^2-50x-11=0$ or $2b-1=2x^2+50x+9=0$. The former has a positive root, $x=\boxed{25 + 2\sqrt{159}}$, while the latter does not. |
human | Suppose that $a$ and $b$ are positive integers such that $(a+bi)^2 = 3+4i$. What is $a+bi$? |
gpt | We have $(a+bi)^2 = a^2 + 2abi + (bi)^2 = (a^2 - b^2) + 2abi = 3 + 4i$. Equating real and imaginary parts, we get $a^2 - b^2 = 3$ and $2ab = 4$. The second equation implies $ab = 2$. Since $a$ and $b$ are positive integers and $ab=2$, we know one of them is 2 and the other is 1. Since $a^2-b^2 = 3$, we have $a=2$, $b=1... |
human | The vertices of an equilateral triangle lie on the hyperbola $xy=1$, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle? |
gpt | Without loss of generality, suppose that the centroid of the triangle is at the vertex $(-1,-1)$. In an equilateral triangle, the centroid and the circumcenter coincide, so the three vertices of the triangle are among the intersection points of the hyperbola $xy = 1$ and a circle centered at $(-1,-1)$.
Suppose the hy... |
human | If $x$ is a number between 0 and 1, which of the following represents the smallest value?
A). $x$
B). $x^2$
C). $2x$
D). $\sqrt{x}$
E). $\frac{1}{x}$
Express your answer as A, B, C, D, or E. |
gpt | Since $0 < x < 1,$
\[x^2 < x < 2x,\]and $x^2 < x < \sqrt{x}$ and $x < 1 < \frac{1}{x}.$ Therefore, the smallest number is always $x^2,$ and the answer is $\boxed{\text{B}}.$ |
human | Let $a$, $b$, $c$, and $d$ be real numbers with $|a-b|=2$, $|b-c|=3$, and $|c-d|=4$. What is the sum of all possible values of $|a-d|$? |
gpt | We use the result that if $x$ and $y$ are real numbers, then the distance between them on the real number line is $|x - y|.$
First, we place $a$:
[asy]
unitsize(0.5 cm);
int i;
draw((-11,0)--(11,0));
for (i = -10; i <= 10; ++i) {
draw((i,-0.2)--(i,0.2));
}
label("$a$", (0,-0.2), S);
[/asy]
We then place a labe... |
human | Let $a,$ $b,$ $c,$ $d,$ $e,$ $f,$ $g,$ and $h$ be real numbers such that $abcd = 4$ and $efgh = 9.$ Find the minimum value of
\[(ae)^2 + (bf)^2 + (cg)^2 + (dh)^2.\] |
gpt | By AM-GM,
\begin{align*}
(ae)^2 + (bf)^2 + (cg)^2 + (dh)^2 &\ge 4 \sqrt[4]{(ae)^2 (bf)^2 (cg)^2 (dh)^2} \\
&= 4 \sqrt[4]{(abcdefgh)^2} \\
&= 24.
\end{align*}Equality occurs when $(ae)^2 = (bf)^2 = (cg)^2 = (dh)^2,$ $abcd = 4,$ and $efgh = 9.$ For example, we can take $a = b = c = d = \sqrt{2}$ and $e = f = g = h = \sq... |
human | Find the real roots of
\[x^4 - 2x^3 - x + 2 = 0.\] |
gpt | We can factor the polynomial as
\begin{align*}
x^4 - 2x^3 - x + 2 &= (x - 2) x^3 - (x - 2) \\
&= (x - 2)(x^3 - 1) \\
&= (x - 2)(x - 1)(x^2 + x + 1).
\end{align*}The quadratic factor $x^2 + x + 1$ has no real roots, so the real roots are $\boxed{1,2}.$ |
human | Let $G$ be the set of polynomials of the form $$ P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50, $$where $ c_1,c_2,\dots, c_{n-1} $ are integers and $P(z)$ has distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$? |
gpt | Since the coefficients of the polynomial are real numbers, any nonreal roots must come in conjugate pairs. Thus, when we factor $P(z)$ over the integers, each factor is either of the form $z - c,$ where $c$ is an integer, or
\[(z - a - bi)(z - a + bi) = z^2 - 2az + a^2 + b^2,\]where $a$ and $b$ are integers, and $b \n... |
human | Let $a_1,$ $a_2,$ $\dots,$ $a_{12}$ be positive real numbers such that $a_1 + a_2 + \dots + a_{12} = 1.$ Find the minimum value of
\[\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_{12}}.\] |
gpt | By Cauchy-Schwarz,
\[(a_1 + a_2 + \dots + a_{12}) \left( \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_{12}} \right) \ge (1 + 1 + \dots + 1)^2 = 12^2 = 144,\]so
\[\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_{12}} \ge 144.\]Equality occurs when $a_i = \frac{1}{12}$ for all $i,$ so the minimum value is $\boxe... |
human | Let $\alpha \neq 1$ be a complex number such that the distance from $\alpha^2$ to 1 is twice the distance from $\alpha$ to 1, while the distance from $\alpha^4$ to 1 is four times the distance from $\alpha$ to 1. Enter all possible values of $\alpha,$ separated by commas. |
gpt | From the given conditions, $|\alpha^2 - 1| = 2 |\alpha - 1|$ and $|\alpha^4 - 1| = 4 |\alpha - 1|.$ From the first equation,
\[|\alpha + 1||\alpha - 1| = 2 |\alpha - 1|.\]Since $\alpha \neq 1,$ $|\alpha - 1| \neq 0.$ Thus, we can safely cancel the factors of $|\alpha - 1|,$ to get
\[|\alpha + 1| = 2.\]From the second... |
human | For what values of $x$ is \[\frac{x-10x^2+25x^3}{8-x^3}\]nonnegative? Answer as an interval. |
gpt | First we factor $x$ from the numerator, \[\frac{x(1-10x+25x^2)}{8-x^3}.\]Now we see the square of a binomial in the numerator, so our expression is equal to \[\frac{x(1-5x)^2}{8-x^3}.\]The denominator only has the single (real) root $x=2$, and we can make more sense of that by applying the difference of cubes factoriz... |
human | If $1+2x+3x^2 + \dotsb=9$, find $x$. |
gpt | Let $S = 1 + 2x + 3x^2 + \dotsb.$ Then
\[xS = x + 2x^2 + 3x^3 + \dotsb.\]Subtracting these equations, we get
\[(1 - x) S = 1 + x + x^2 + \dotsb = \frac{1}{1 - x},\]so $S = \frac{1}{(1 - x)^2}.$ Thus, we want to solve
\[\frac{1}{(1 - x)^2} = 9.\]then $(1 - x)^2 = \frac{1}{9},$ so $1 - x = \pm \frac{1}{3}.$ Since $x$... |
human | The parabolas $y = (x + 1)^2$ and $x + 4 = (y - 3)^2$ intersect at four points. All four points lie on a circle of radius $r.$ Find $r^2.$ |
gpt | Add the equations $y = (x + 1)^2$ and $x + 4 = (y - 3)^2$ to get
\[x + y + 4 = (x + 1)^2 + (y - 3)^2.\](Any point that satisfies both equations must satisfy this equation as well.)
Completing the square in $x$ and $y$, we get
\[\left( x + \frac{1}{2} \right)^2 + \left( y - \frac{7}{2} \right)^2 = \frac{13}{2}.\]Thus, ... |
human | Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 1$ and
\[f(x^2 - y^2) = (x - y) (f(x) + f(y))\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times s.$ |
gpt | Setting $x = y,$ we get $f(0) = 0.$
Setting $x = -1$ and $y = 0,$ we get
\[f(1) = -f(-1),\]so $f(-1) = -1.$
Setting $y = 1$ and $y = -1,$ we get
\begin{align*}
f(x^2 - 1) &= (x - 1) (f(x) + 1), \\
f(x^2 - 1) &= (x + 1) (f(x) - 1),
\end{align*}respectively. Hence, $(x - 1) (f(x) + 1) = (x + 1) (f(x) - 1),$ which simp... |
human | Factor
\[\frac{(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3}{(a - b)^3 + (b - c)^3 + (c - a)^3}.\] |
gpt | We will use the identity
\[x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz).\]Setting $x = a^2 - b^2,$ $y = b^2 - c^2,$ $z = c^2 - a^2,$ we get
\[(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3 - 3(a^2 - b^2)(b^2 - c^2)(c^2 - a^2) = 0.\]Setting $x = a - b,$ $y = b - c,$ $z = c - a,$ we get
\[(a - b)^3... |
human | Let $x$ and $y$ be nonzero real numbers. Let $m$ and $M$ be the minimium value and maximum value of
\[\frac{|x + y|}{|x| + |y|},\]respectively. Find $M - m.$ |
gpt | Both $|x + y|$ and $|x| + |y|$ are nonnegative, so $\frac{|x + y|}{|x| + |y|}$ must be nonnegative. When $x = 1$ and $y = -1,$
\[\frac{|x + y|}{|x| + |y|} = \frac{0}{2} = 0,\]so this is clearly the minimum.
On the other hand, by the Triangle Inequality, $|x| + |y| \ge |x + y|,$ so
\[\frac{|x + y|}{|x| + |y|} \le 1.\]... |
human | Find \[\left|\left(1 + \sqrt{3}i\right)^4\right|\] |
gpt | First of all, we know that $|ab|=|a|\cdot |b|$, so \[\left|\left(1 + \sqrt{3}i\right)^4\right|=\left|1 + \sqrt{3} i\right|^4\]We also find that \[\left|1 +\sqrt{3}i\right|=\sqrt{\left(1\right)^2+\left(\sqrt{3}\right)^2}=\sqrt{4}=2\]Therefore, our answer is $2^4=\boxed{16}$. |
human | Let $x$, $y$, and $z$ be distinct real numbers that sum to $0$. Compute \[ \dfrac {xy+yz+zx}{x^2+y^2+z^2}. \] |
gpt | We have $x+y+z=0,$ and squaring this equation gives \[(x^2+y^2+z^2) + 2(xy+yz+zx) = 0.\]Thus, $x^2+y^2+z^2=-2(xy+yz+zx).$ Since $x, y, z$ are distinct, it is not possible that $x^2+y^2+z^2=0,$ so we have \[\frac{xy+yz+zx}{x^2+y^2+z^2} = \boxed{-\frac12}.\] |
human | The equations of the asymptotes of a hyperbola are $y = 2x+5$ and $y = -2x+1.$ Given that the hyperbola passes through the point $(0, 7),$ the standard form for the equation of the hyperbola is \[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1,\]where $a,$ $b$, $h,$ and $k$ are constants with $a, b > 0.$ Find $a + h.$ |
gpt | Solving the system $y=2x+5$ and $y=-2x+1,$ we get $(x, y) = (-1, 3).$ Therefore, the asymptotes of the hyperbola intersect at $(-1, 3),$ which must be the center of the hyperbola. Therefore, $(h, k) = (-1, 3),$ so the equation of the hyperbola is \[\frac{(y-3)^2}{a^2} - \frac{(x+1)^2}{b^2} = 1\]for some $a$ and $b.$ Th... |
human | Let $p(x)$ be a monic quartic polynomial such that $p(1) = 2,$ $p(2) = 5,$ $p(3) = 10,$ and $p(4) = 17.$ Find $p(5).$ |
gpt | Note that $p(x)$ takes on the same values as $x^2 + 1$ for $x = 1,$ 2, 3, and 4. So, let
\[q(x) = p(x) - x^2 - 1.\]Then $q(x)$ is also a monic quartic polynomial. Also, $q(1) = q(2) = q(3) = q(4) = 0,$ so
\[q(x) = (x - 1)(x - 2)(x - 3)(x - 4).\]Hence, $p(x) = (x - 1)(x - 2)(x - 3)(x - 4) + x^2 + 1.$ We can set $x = ... |
human | Let $x$ and $y$ be positive real numbers such that $x + y = 10.$ Find the minimum value of $\frac{1}{x} + \frac{1}{y}.$ |
gpt | By AM-HM,
\[\frac{x + y}{2} \ge \frac{2}{\frac{1}{x} + \frac{1}{y}}.\]Hence,
\[\frac{1}{x} + \frac{1}{y} \ge \frac{4}{x + y} = \frac{4}{10} = \frac{2}{5}.\]Equality occurs when $x = y = 5,$ so the minimum value is $\boxed{\frac{2}{5}}.$ |
human | Let $P= \left( 1 - \frac{1}{2} \right) \left( 1 - \frac{1}{3} \right) \left( 1 - \frac{1}{4} \right) \dotsm \left( 1 - \frac{1}{n} \right)$. What is the value of $P$ if $n=2007$? Express your answer as a common fraction. |
gpt | Simplifying each term in $P,$ \[P=\left( \frac{1}{2} \right) \left( \frac{2}{3} \right) \left( \frac{3}{4} \right) \dotsm \left( \frac{n-1}{n} \right) . \]The denominator of each fraction cancels with the numerator of the next fraction, so $P=\frac{1}{n}.$ When $n=2007,$ $P=\boxed{\frac{1}{2007}}.$ |
human | The graph below shows a portion of the curve defined by the quartic polynomial $P(x)=x^4+ax^3+bx^2+cx+d$.
[asy]
unitsize(0.8 cm);
int i;
real func (real x) {
return(0.5*(x^4/4 - 2*x^3/3 - 3/2*x^2) + 2.7);
}
draw(graph(func,-4.5,4.5));
draw((-4.5,0)--(4.5,0));
draw((0,-5.5)--(0,5.5));
for (i = -4; i <= 4; ++i) {
... |
gpt | We claim that the product of the non-real zeros is the smallest.
(A) The value of $P(-1)$ is greater than 4.
(B) Since the leading coefficient in $P(x)$ is 1, the product of the zeros of $P$ is $d = P(0),$ which is greater than 4.
(D) The sum of the coefficient of $P(x)$ is $P(1),$ which is greater than 2.
(E) The ... |
human | Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x - y)) = f(x) f(y) - f(x) + f(y) - xy\]for all $x,$ $y.$ Find the sum of all possible values of $f(1).$ |
gpt | Let $a = f(0)$ and $b = f(f(0))$. Setting $y = x$ in the given equation, we get
\[[f(x)]^2 - x^2 = b \quad (1)\]for all $x$. In particular, for $x = 0$, $a^2 = b$.
Setting $y = 0$ in the given equation, we get
\[f(f(x)) = (a - 1) f(x) + a \quad (2)\]for all $x$.
Substituting $f(x)$ for $x$ in equation (1), we get
\... |
human | If $y=\frac{12x^4+4x^3+9x^2+5x+3}{3x^4+2x^3+8x^2+3x+1}$, at what value of $y$ will there be a horizontal asymptote? |
gpt | For a rational function, if the degree of the polynomial in the numerator is the same as the degree of the polynomial in the denominator, there is a horizontal asymptote at the $y$-value that equals the ratio of the leading coefficient of the numerator to the leading coefficient of the denominator. For this function th... |
human | Suppose that $a$ and $b$ are positive integers such that $(a-bi)^2 = 8-6i$. What is $a-bi$? |
gpt | We have $(a-bi)^2 = a^2 - 2abi + (bi)^2 = (a^2 - b^2) - 2abi = 8-6i$. Equating real and imaginary parts, we get $a^2 - b^2 = 8$ and $-2ab = -6$, or $ab = 3$. Since $a$ and $b$ are positive integers and $ab=3$, we know one of them is 3 and the other is 1. Since $a^2-b^2 = 8$, trial and error gives $a=3$, $b=1$. So $a-bi... |
human | Find the four roots of
\[2x^4 + x^3 - 6x^2 + x + 2 = 0.\]Enter the four roots (counting multiplicity), separated by commas. |
gpt | Dividing the equation by $x^2,$ we get
\[2x^2 + x - 6 + \frac{1}{x} + \frac{2}{x^2} = 0.\]Let $y = x + \frac{1}{x}.$ Then
\[y^2 = x^2 + 2 + \frac{1}{x^2},\]so $x^2 + \frac{1}{x^2} = y^2 - 2.$ Thus, we can re-write the equation above as
\[2(y^2 - 2) + y - 6 = 0.\]This simplifies to $2y^2 + y - 10 = 0.$ The roots are ... |
human | Let $a,$ $b,$ and $c$ be distinct real numbers. Simplify the expression
\[\frac{(x + a)^3}{(a - b)(a - c)} + \frac{(x + b)^3}{(b - a)(b - c)} + \frac{(x + c)^3}{(c - a)(c - b)}.\] |
gpt | Let
\[p(x) = \frac{(x + a)^3}{(a - b)(a - c)} + \frac{(x + b)^3}{(b - a)(b - c)} + \frac{(x + c)^3}{(c - a)(c - b)}.\]Then
\begin{align*}
p(-a) &= \frac{(-a + a)^3}{(a - b)(a - c)} + \frac{(-a + b)^3}{(b - a)(b - c)} + \frac{(-a + c)^3}{(c - a)(c - b)} \\
&= \frac{(b - a)^3}{(b - a)(b - c)} + \frac{(c - a)^3}{(c - a)(c... |
human | Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(xf(y) + x) = xy + f(x)\]for all $x,$ $y.$
Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times s.$ |
gpt | Setting $x = 1$ and $y = -1 - f(1),$ we get
\[f(f(-1 - f(1)) + 1) = -1 - f(1) + f(1) = -1.\]Let $a = f(-1 - f(1)) + 1,$ so $f(a) = -1.$
Setting $y = a,$ we get
\[f(0) = ax + f(x).\]Let $b = f(0),$ so $f(x) = -ax + b.$ Substituting into the given functional equation, we get
\[-a(x(-ay + b) + x) + b = xy - ax + b.\]Thi... |
human | When $p(x) = Ax^5 + Bx^3 + Cx + 4$ is divided by $x - 3,$ the remainder is 11. Find the remainder when $p(x)$ is divided by $x + 3.$ |
gpt | By the Remainder Theorem, $p(3) = 11,$ so
\[A \cdot 3^5 + B \cdot 3^3 + C \cdot 3 + 4 = 11.\]Then $A \cdot 3^5 + B \cdot 3^3 + C \cdot 3 = 7.$
Again by the Remainder Theorem, when $p(x)$ is divided by $x + 3,$ the remainder is
\begin{align*}
p(-3) &= A \cdot (-3)^5 + B \cdot (-3)^3 + C \cdot (-3) + 4 \\
&= -A \cdot 3^... |
human | Let $F_1 = \left( -3, 1 - \frac{\sqrt{5}}{4} \right)$ and $F_ 2= \left( -3, 1 + \frac{\sqrt{5}}{4} \right).$ Then the set of points $P$ such that
\[|PF_1 - PF_2| = 1\]form a hyperbola. The equation of this hyperbola can be written as
\[\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1,\]where $a, b > 0.$ Find $h + k ... |
gpt | The center of the hyperbola is the midpoint of $\overline{F_1 F_2},$ which is $(-3,1).$ Thus, $h = -3$ and $k = 1.$
Also, $2a = 1,$ so $a = \frac{1}{2}.$ The distance between the foci is $2c = \frac{\sqrt{5}}{2},$ so $c = \frac{\sqrt{5}}{4}.$ Then $b^2 = c^2 - a^2 = \frac{5}{16} - \frac{1}{4} = \frac{1}{16},$ so $b... |
human | Determine the value of
\[2002 + \frac{1}{2} \left( 2001 + \frac{1}{2} \left( 2000 + \dots + \frac{1}{2} \left( 3 + \frac{1}{2} \cdot 2 \right) \right) \dotsb \right).\] |
gpt | Let
\begin{align*}
S &= 2002 + \frac{1}{2} \left( 2001 + \frac{1}{2} \left( 2000 + \dots + \frac{1}{2} \left( 3 + \frac{1}{2} \cdot 2 \right) \right) \dotsb \right) \\
&= 2002 + \frac{2001}{2} + \frac{2000}{2^2} + \dots + \frac{3}{2^{1999}} + \frac{2}{2^{2000}}.
\end{align*}Then
\[2S = 2 \cdot 2002 + 2001 + \frac{2000}... |
human | If $\log (xy^3)= 1$ and $\log (x^2y) = 1$, what is $\log (xy)$? |
gpt | We have \[1 = \log(xy^{3}) = \log x + 3\log y \quad\text{and}\quad 1 = \log(x^{2}y) = 2\log x + \log y.\]Solving yields $\log x = \frac{2}{5}$ and $\log y = \frac{1}{5}$. Thus \[\log(xy) = \log x + \log y = \boxed{\frac{3}{5}}.\] |
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