conversations listlengths 2 2 | source stringclasses 1
value | score int64 5 5 |
|---|---|---|
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The parabolas $y = (x + 1)^2$ and $x + 4 = (y - 3)^2$ intersect at four points $(x_1,y_1),$ $(x_2,y_2),$ $(x_3,y_3),$ and $(x_4,y_4).$ Find\n\\[x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4.\\] [/INST] To find $x_1 + x_2 + x_... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the largest constant $m,$ so that for any positive real numbers $a,$ $b,$ $c,$ and $d,$\n\\[\\sqrt{\\frac{a}{b + c + d}} + \\sqrt{\\frac{b}{a + c + d}} + \\sqrt{\\frac{c}{a + b + d}} + \\sqrt{\\frac{d}{a + b + c}} > m.\\]... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: An ellipse has foci $(2, 2)$ and $(2, 6)$, and it passes through the point $(14, -3).$ Given this, we can write the equation of the ellipse in standard form as \\[\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1,\\]where $a, b,... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: How many of the first $1000$ positive integers can be expressed in the form\n\\[\\lfloor 2x \\rfloor + \\lfloor 4x \\rfloor + \\lfloor 6x \\rfloor + \\lfloor 8x \\rfloor\\]where $x$ is a real number, and $\\lfloor z \\rfloor$ ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: For a positive integer $n,$ let\n\\[a_n = \\sum_{k = 0}^n \\frac{1}{\\binom{n}{k}} \\quad \\text{and} \\quad b_n = \\sum_{k = 0}^n \\frac{k}{\\binom{n}{k}}.\\]Simplify $\\frac{a_n}{b_n}.$ [/INST] For the sum $b_n,$ let $j = n ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: 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 $ax^5+by^5.$ [/INST] For $n = 1, 2, 3, 4, 5,$ define $s_n = ax^n + by^n.$ We are given the values of $s_1,... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The numbers $a_1,$ $a_2,$ $a_3,$ $b_1,$ $b_2,$ $b_3,$ $c_1,$ $c_2,$ $c_3$ are equal to the numbers $1,$ $2,$ $3,$ $\\dots,$ $9$ in some order. Find the smallest possible value of\n\\[a_1 a_2 a_3 + b_1 b_2 b_3 + c_1 c_2 c_3.\\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the maximum value of\n\\[f(x) = 3x - x^3\\]for $0 \\le x \\le \\sqrt{3}.$ [/INST] Graphing the function, or trying different values of $x,$ we may think that the function is maximized at $x = 1,$ which would make the maxi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the equation of the directrix of the parabola $x = -\\frac{1}{6} y^2.$ [/INST] Recall that a parabola is defined as the set of all points that are equidistant to the focus $F$ and the directrix.\n\nSince the parabola $x =... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: An ellipse has its foci at $(-1, -1)$ and $(-1, -3).$ Given that it passes through the point $(4, -2),$ its equation can be written in the form \\[\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1\\]where $a, b, h, k$ are consta... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the quadratic function $f(x) = x^2 + ax + b$ such that\n\\[\\frac{f(f(x) + x)}{f(x)} = x^2 + 1776x + 2010.\\] [/INST] We have that\n\\begin{align*}\nf(f(x) + x) &= f(x^2 + (a + 1) x + b) \\\\\n&= (x^2 + (a + 1)x + b)^2 + ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the value of the sum\n\\[\n \\sum_z \\frac{1}{{\\left|1 - z\\right|}^2} \\, ,\n\\]where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$? [/INST] Since $z^7 = -1,$ $|z^7| = 1.$ Then $|z|^... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find a quadratic with rational coefficients and quadratic term $x^2$ that has $\\sqrt{3}-2$ as a root. [/INST] Since the root $\\sqrt{3}-2$ is irrational but the coefficients of the quadratic are rational, from the quadratic f... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A parabola has vertex $V = (0,0)$ and focus $F = (0,1).$ Let $P$ be a point in the first quadrant, lying on the parabola, so that $PF = 101.$ Find $P.$ [/INST] Using the vertex and focus, we can see that the equation of the ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Is\n\\[f(x) = \\log (x + \\sqrt{1 + x^2})\\]an even function, odd function, or neither?\n\nEnter \"odd\", \"even\", or \"neither\". [/INST] Note that\n\\begin{align*}\n-x + \\sqrt{1 + (-x)^2} &= -x + \\sqrt{1 + x^2} \\\\\n&= \... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A positive real number $x$ is such that \\[\n\\sqrt[3]{1-x^3} + \\sqrt[3]{1+x^3} = 1.\n\\]Find $x^6.$ [/INST] Cubing the given equation yields \\[\n1 = (1-x^3) + 3\\sqrt[3]{(1-x^3)(1+x^3)}\\left(\\sqrt[3]{1-x^3} + \\sqrt[3]{1+... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Express the following sum as a simple fraction in lowest terms.\n\n$$\\frac{1}{1\\times2} + \\frac{1}{2\\times3} + \\frac{1}{3\\times4} + \\frac{1}{4\\times5} + \\frac{1}{5\\times6}$$ [/INST] Any unit fraction whose denominato... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: For certain real values of $a, b, c,$ and $d_{},$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i^2 = -1.$ Find $b... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Given that $x"
},
{
"from": "gpt",
"value": "Solution placeholder"
}
] | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $a,$ $b,$ and $c$ be the roots of $x^3 - 7x^2 + 5x + 2 = 0.$ Find\n\\[\\frac{a}{bc + 1} + \\frac{b}{ac + 1} + \\frac{c}{ab + 1}.\\] [/INST] By Vieta's formulas, $a + b + c = 7,$ $ab + ac + bc = 5,$ and $abc = -2.$\n\nWe c... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find all real values of $x$ which satisfy\n\\[\\frac{1}{x + 1} + \\frac{6}{x + 5} \\ge 1.\\] [/INST] Subtracting 1 from both sides and putting everything over a common denominator, we get\n\\[\\frac{-x^2 + x + 6}{(x + 1)(x + 5... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Given that $a-b=5$ and $a^2+b^2=35$, find $a^3-b^3$. [/INST] We know that $(a-b)^2=a^2-2ab+b^2$. Therefore, we plug in the given values to get $5^2=35-2ab$. Solving, we get that $ab=5$. We also have the difference of cubes fac... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: For integers $a$ and $T,$ $T \\neq 0,$ a parabola whose general equation is $y = ax^2 + bx + c$ passes through the points $A = (0,0),$ $B = (2T,0),$ and $C = (2T + 1,28).$ Let $N$ be the sum of the coordinates of the vertex p... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Given that $x$ and $y$ are nonzero real numbers such that $x+\\frac{1}{y}=10$ and $y+\\frac{1}{x}=\\frac{5}{12},$ find all possible values for $x.$\n\n(Enter your answer as a comma-separated list.) [/INST] Multiplying the firs... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find all roots of the polynomial $x^3+x^2-4x-4$. Enter your answer as a list of numbers separated by commas. [/INST] By the Rational Root Theorem, any root of the polynomial must divide $4$. Therefore the roots are among the n... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of\n\\[\\frac{1}{a} + \\frac{2}{b}.\\] [/INST] By AM-HM,\n\\[\\frac{a + b + b}{3} \\ge \\frac{3}{\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Compute\n\\[\\frac{\\lfloor \\sqrt[4]{1} \\rfloor \\cdot \\lfloor \\sqrt[4]{3} \\rfloor \\cdot \\lfloor \\sqrt[4]{5} \\rfloor \\dotsm \\lfloor \\sqrt[4]{2015} \\rfloor}{\\lfloor \\sqrt[4]{2} \\rfloor \\cdot \\lfloor \\sqrt[4]{... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find all real numbers $p$ so that\n\\[x^4 + 2px^3 + x^2 + 2px + 1 = 0\\]has at least two distinct negative real roots. [/INST] We see that $x = 0$ cannot be a root of the polynomial. Dividing both sides by $x^2,$ we get\n\\[x... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the minimum of the function\n\\[\\frac{xy}{x^2 + y^2}\\]in the domain $\\frac{2}{5} \\le x \\le \\frac{1}{2}$ and $\\frac{1}{3} \\le y \\le \\frac{3}{8}.$ [/INST] We can write\n\\[\\frac{xy}{x^2 + y^2} = \\frac{1}{\\frac{... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Three of the four endpoints of the axes of an ellipse are, in some order, \\[(-2, 4), \\; (3, -2), \\; (8, 4).\\]Find the distance between the foci of the ellipse. [/INST] The two axes of the ellipse are perpendicular bisector... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the remainder when $x^{2015} + 1$ is divided by $x^8 - x^6 + x^4 - x^2 + 1.$ [/INST] Note that\n\\[(x^2 + 1)(x^8 - x^6 + x^4 - x^2 + 1) = x^{10} + 1.\\]Also, $x^{10} + 1$ is a factor of $x^{2010} + 1$ via the factorizatio... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first d... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $f(x) = ax^6 + bx^4 - cx^2 + 3.$ If $f(91) = 1$, find $f(91) + f(-91)$. [/INST] Since only the even exponents have non-zero coefficients, $f$ is an even function, and we know that $f(-x) = f(x)$. Hence $f(-91) = f(91) = 1... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $f(x) = x^2 + 6x + c$ for all real numbers $x$, where $c$ is some real number. For what values of $c$ does $f(f(x))$ have exactly $3$ distinct real roots? [/INST] Suppose the function $f(x) = 0$ has only one distinct root... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the minimum value of\n\\[x^2 + 2xy + 3y^2 - 6x - 2y,\\]over all real numbers $x$ and $y.$ [/INST] Suppose that $y$ is a fixed number, and $x$ can vary. If we try to complete the square in $x,$ we would write\n\\[x^2 + (2... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $f(x) = x^2-2x$. How many distinct real numbers $c$ satisfy $f(f(f(f(c)))) = 3$? [/INST] We want the size of the set $f^{-1}(f^{-1}(f^{-1}(f^{-1}(3)))).$ Note that $f(x) = (x-1)^2-1 = 3$ has two solutions: $x=3$ and $x=-1$... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $ f(x) = x^3 + x + 1$. Suppose $ g$ is a cubic polynomial such that $ g(0) = - 1$, and the roots of $ g$ are the squares of the roots of $ f$. Find $ g(9)$. [/INST] Let $r,$ $s,$ and $t$ be the roots of $f(x),$ so that $f(... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the maximum value of\n\\[\\cos \\theta_1 \\sin \\theta_2 + \\cos \\theta_2 \\sin \\theta_3 + \\cos \\theta_3 \\sin \\theta_4 + \\cos \\theta_4 \\sin \\theta_5 + \\cos \\theta_5 \\sin \\theta_1,\\]over all real numbers $\\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: If\n\\begin{align*}\na + b + c &= 1, \\\\\na^2 + b^2 + c^2 &= 2, \\\\\na^3 + b^3 + c^3 &= 3,\n\\end{align*}find $a^4 + b^4 + c^4.$ [/INST] Squaring the equation $a + b + c = 1,$ we get\n\\[a^2 + b^2 + c^2 + 2ab + 2ac + 2bc = 1... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The terms of the sequence $(a_i)$ defined by $a_{n + 2} = \\frac {a_n + 2009} {1 + a_{n + 1}}$ for $n \\ge 1$ are positive integers. Find the minimum possible value of $a_1 + a_2$. [/INST] The definition gives $$a_3(a_2+1) = a... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $z$ be a complex number with $|z| = \\sqrt{2}.$ Find the maximum value of\n\\[|(z - 1)^2 (z + 1)|.\\] [/INST] Let $z = x + yi,$ where $x$ and $y$ are real numbers. Since $|z| = \\sqrt{2},$ $x^2 + y^2 = 2.$ Then\n\\begin... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: For $-25 \\le x \\le 25,$ find the maximum value of $\\sqrt{25 + x} + \\sqrt{25 - x}.$ [/INST] By QM-AM,\n\\[\\frac{\\sqrt{25 + x} + \\sqrt{25 - x}}{2} \\le \\sqrt{\\frac{25 + x + 25 - x}{2}} = 5,\\]so $\\sqrt{25 + x} + \\sqrt... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $f$ be a function satisfying $f(xy) = f(x)/y$ for all positive real numbers $x$ and $y$. If $f(500) = 3$, what is the value of $f(600)$? [/INST] Note that $$f(600) = f \\left( 500 \\cdot \\frac{6}{5} \\right) = \\frac{f(50... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Compute $$\\sum_{n=1}^{\\infty} \\frac{3n-1}{2^n}.$$ [/INST] Let\n$$S = \\sum_{n=1}^{\\infty} \\frac{3n-1}{2^n} = \\frac{2}{2} + \\frac{5}{4} + \\frac{8}{8} + \\frac{11}{16} + \\dotsb.$$Then\n$$2S = \\sum_{n=1}^{\\infty} \\fra... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $x,$ $y,$ and $z$ be nonnegative numbers such that $x^2 + y^2 + z^2 = 1.$ Find the maximum value of\n\\[2xy \\sqrt{6} + 8yz.\\] [/INST] Our strategy is to take $x^2 + y^2 + z^2$ and divide into several expression, apply A... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $Q(x)=a_0+a_1x+\\dots+a_nx^n$ be a polynomial with integer coefficients, and $0\\le a_i<3$ for all $0\\le i\\le n$.\n\nGiven that $Q(\\sqrt{3})=20+17\\sqrt{3}$, compute $Q(2)$. [/INST] We have that\n\\[Q(\\sqrt{3}) = a_0 +... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the focus of the parabola $y = -3x^2 - 6x.$ [/INST] Recall that a parabola is defined as the set of all points that are equidistant to the focus $F$ and the directrix. Completing the square on $x,$ we get\n\\[y = -3(x + ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: If $x$ and $y$ are positive real numbers such that $5x^2 + 10xy = x^3 + 2x^2 y,$ what is the value of $x$? [/INST] Notice that we can factor out a $5x$ from each term on the left-hand side to give $5x(x+2y)$. Similarly, we can... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.5 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Compute $\\sqrt{(31)(30)(29)(28)+1}.$ [/INST] Let $x = 29.$ Then we can write \\[\\begin{aligned} (31)(30)(29)(28) + 1 &= (x+2)(x+1)(x)(x-1) + 1 \\\\ &= [(x+2)(x-1)][(x+1)x] - 1 \\\\& = (x^2+x-2)(x^2+x) + 1 \\\\&= (x^2+x)^2 - ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The polynomial\n$$g(x) = x^3 - x^2 - (m^2 + m) x + 2m^2 + 4m + 2$$is divisible by $x-4$ and all of its zeroes are integers. Find all possible values of $m$. [/INST] Since $g(x)$ is divisible by $x-4$, we have $g(4)=0$. We also... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The function $f$ takes nonnegative integers to real numbers, such that $f(1) = 1,$ and\n\\[f(m + n) + f(m - n) = \\frac{f(2m) + f(2n)}{2}\\]for all nonnnegative integers $m \\ge n.$ Find the sum of all possible values of $f(1... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Real numbers $x,$ $y,$ and $z$ satisfy the following equality:\n\\[4(x + y + z) = x^2 + y^2 + z^2.\\]Let $M$ be the maximum value of $xy + xz + yz,$ and let $m$ be the minimum value of $xy + xz + yz.$ Find $M + 10m.$ [/INST] ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the range of the function \\[f(x) = \\frac{x}{x^2-x+1},\\]where $x$ can be any real number. (Give your answer in interval notation.) [/INST] Let $y$ be a number in the range of $f.$ This means that there is a real number ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2010$ and let $M$ be the largest of the sum $a+b$, $b+c$, $c+d$ and $d+e$. What is the smallest possible value of $M$? [/INST] We have that\n\\[M = \\max \\{... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the ordered pair $(a,b)$ of positive integers, with $a < b,$ for which\n\\[\\sqrt{1 + \\sqrt{21 + 12 \\sqrt{3}}} = \\sqrt{a} + \\sqrt{b}.\\] [/INST] First, we simplify $\\sqrt{21 + 12 \\sqrt{3}}.$ Let\n\\[\\sqrt{21 + 12 ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: If $x$ is real, compute the maximum integer value of\n\\[\\frac{3x^2 + 9x + 17}{3x^2 + 9x + 7}.\\] [/INST] First, we can write\n\\[\\frac{3x^2 + 9x + 17}{3x^2 + 9x + 7} = \\frac{(3x^2 + 9x + 7) + 10}{3x^2 + 9x + 7} = 1 + \\fra... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: There are two values of $a$ for which the equation $4x^2 + ax + 8x + 9 = 0$ has only one solution for $x$. What is the sum of these values of $a$? [/INST] We can write the quadratic as\n\\[4x^2 + (a + 8)x + 9 = 0.\\]If the qua... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Calculate $\\frac{1}{4} \\cdot \\frac{2}{5} \\cdot \\frac{3}{6} \\cdot \\frac{4}{7} \\cdots \\frac{49}{52} \\cdot \\frac{50}{53}$. Express your answer as a common fraction. [/INST] Note that from $\\frac{4}{7}$ to $\\frac{50}{... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Determine the largest positive integer $n$ such that there exist positive integers $x, y, z$ so that \\[\nn^2 = x^2+y^2+z^2+2xy+2yz+2zx+3x+3y+3z-6\n\\] [/INST] The given equation rewrites as $n^2 = (x+y+z+1)^2+(x+y+z+1)-8$. Wr... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The function $f$ satisfies \\[\nf(x) + f(2x+y) + 5xy = f(3x - y) + 2x^2 + 1\n\\]for all real numbers $x,y$. Determine the value of $f(10)$. [/INST] Setting $x = 10$ and $y=5$ gives $f(10) + f(25) + 250 = f(25) + 200 + 1$, from... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $1000x^3+27= (ax^2 + bx +c )(d x^2 +ex + f)$ for all $x$, then what is $a^2+b^2+c^2+d^2+e^2+f^2$? [/INST] Apply the sum of cubes factorization to the expression $1000... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: For positive integers $n$, define $S_n$ to be the minimum value of the sum\n\\[\\sum_{k=1}^n \\sqrt{(2k-1)^2+a_k^2},\\]where $a_1,a_2,\\ldots,a_n$ are positive real numbers whose sum is $17$. Find the unique positive integer $... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $x,$ $y,$ $z$ be real numbers, all greater than 3, so that\n\\[\\frac{(x + 2)^2}{y + z - 2} + \\frac{(y + 4)^2}{z + x - 4} + \\frac{(z + 6)^2}{x + y - 6} = 36.\\]Enter the ordered triple $(x,y,z).$ [/INST] By Cauchy-Schwar... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The set of points $(x,y)$ such that $|x - 3| \\le y \\le 4 - |x - 1|$ defines a region in the $xy$-plane. Compute the area of this region. [/INST] Plotting $y = |x - 3|$ and $y = 4 - |x - 1|,$ we find that the two graphs inte... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The planet Xavier follows an elliptical orbit with its sun at one focus. At its nearest point (perigee), it is 2 astronomical units (AU) from the sun, while at its furthest point (apogee) it is 12 AU away. When Xavier is mid... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: For some integer $m$, the polynomial $x^3 - 2011x + m$ has the three integer roots $a$, $b$, and $c$. Find $|a| + |b| + |c|.$ [/INST] By Vieta's formulas, \\[\\left\\{ \\begin{aligned} a + b + c &= 0 \\\\ ab+bc+ac&=-2011. \\e... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $x,$ $y,$ $z$ be real numbers such that\n\\begin{align*}\nx + y + z &= 4, \\\\\nx^2 + y^2 + z^2 &= 6.\n\\end{align*}Let $m$ and $M$ be the smallest and largest possible values of $x,$ respectively. Find $m + M.$ [/INST] F... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the distance between the foci of the ellipse\n\\[\\frac{x^2}{20} + \\frac{y^2}{4} = 7.\\] [/INST] First, we divide both sides by 7, to get\n\\[\\frac{x^2}{140} + \\frac{y^2}{28} = 1.\\]Thus, $a^2 = 140$ and $b^2 = 28,$ so... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: One focus of the ellipse $\\frac{x^2}{2} + y^2 = 1$ is at $F = (1,0).$ There exists a point $P = (p,0),$ where $p > 0,$ such that for any chord $\\overline{AB}$ that passes through $F,$ angles $\\angle APF$ and $\\angle BPF$ ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\\cdots a_k$ is an integer? [/INST]... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $x,$ $y,$ $z$ be real numbers such that $4x^2 + y^2 + 16z^2 = 1.$ Find the maximum value of\n\\[7x + 2y + 8z.\\] [/INST] By Cauchy-Schwarz\n\\[\\left( \\frac{49}{4} + 4 + 4 \\right) (4x^2 + y^2 + 16z^2) \\ge (7x + 2y + 8z... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: For what values of $x$ is $\\frac{\\log{(3-x)}}{\\sqrt{x-1}}$ defined? [/INST] The expression inside the square root must be greater than 0 because the denominator cannot be equal to 0. Therefore, $x-1>0$, so $x>1$. The expres... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $a,$ $b,$ $c$ be positive real numbers. Find the smallest possible value of\n\\[6a^3 + 9b^3 + 32c^3 + \\frac{1}{4abc}.\\] [/INST] By AM-GM,\n\\[6a^3 + 9b^3 + 32c^3 \\ge 3 \\sqrt[3]{6a^3 \\cdot 9b^3 \\cdot 32c^3} = 36abc.\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that\n\\[P(1) = P(3) = P(5) = P(7) = a\\]and\n\\[P(2) = P(4) = P(6) = P(8) = -a.\\]What is the smallest possible value of $a$? [/INST] There must be so... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $a,$ $b,$ $c$ be complex numbers such that\n\\begin{align*}\nab + 4b &= -16, \\\\\nbc + 4c &= -16, \\\\\nca + 4a &= -16.\n\\end{align*}Enter all possible values of $abc,$ separated by commas. [/INST] Adding the equations, ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The polynomial equation \\[x^3 + bx + c = 0,\\]where $b$ and $c$ are rational numbers, has $5-\\sqrt{2}$ as a root. It also has an integer root. What is it? [/INST] Because the coefficients of the polynomial are rational, the ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The function $y=\\frac{x^3+8x^2+21x+18}{x+2}$ can be simplified into the function $y=Ax^2+Bx+C$, defined everywhere except at $x=D$. What is the sum of the values of $A$, $B$, $C$, and $D$? [/INST] The fact that the function c... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The function $f(x)$ satisfies\n\\[f(x - y) = f(x) f(y)\\]for all real numbers $x$ and $y,$ and $f(x) \\neq 0$ for all real numbers $x.$ Find $f(3).$ [/INST] Setting $x = 3$ and $y = \\frac{3}{2},$ we get\n\\[f \\left( \\frac{... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find all solutions to the equation\\[ \\sqrt[4]{x} = \\frac{12}{7 - \\sqrt[4]{x}}.\\] [/INST] Let $y = \\sqrt[4]{x}.$ Then we have $y = \\frac{12}{7-y},$ or $y(7-y) = 12.$ Rearranging and factoring, we get \\[(y-3)(y-4) = 0.\\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the number of ordered quadruples $(a,b,c,d)$ of nonnegative real numbers such that\n\\begin{align*}\na^2 + b^2 + c^2 + d^2 &= 4, \\\\\n(a + b + c + d)(a^3 + b^3 + c^3 + d^3) &= 16.\n\\end{align*} [/INST] Note that\n\\[(a^... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $x,$ $y,$ $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of\n\\[\\frac{1}{x + y} + \\frac{1}{x + z} + \\frac{1}{y + z}.\\] [/INST] By Cauchy-Schwarz,\n\\[[(x + y) + (x + z) + (y + z)] \\lef... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: An ellipse has foci at $F_1 = (0,2)$ and $F_2 = (3,0).$ The ellipse intersects the $x$-axis at the origin, and one other point. What is the other point of intersection? [/INST] The distance between the origin and $F_1$ is 2,... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: If\n\\[\\frac{x}{a} + \\frac{y}{b} + \\frac{z}{c} = 3 \\quad \\text{and} \\quad \\frac{a}{x} + \\frac{b}{y} + \\frac{c}{z} = 0,\\]find $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} + \\frac{z^2}{c^2}.$ [/INST] Let $p = \\frac{x}{a},$ $... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The partial fraction decomposition of\n\\[\\frac{x^2 - 19}{x^3 - 2x^2 - 5x + 6}\\]is\n\\[\\frac{A}{x - 1} + \\frac{B}{x + 2} + \\frac{C}{x - 3}.\\]Find the product $ABC.$ [/INST] We have that\n\\[\\frac{x^2 - 19}{x^3 - 2x^2 - ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Suppose that all four of the numbers \\[2 - \\sqrt{5}, \\;4+\\sqrt{10}, \\;14 - 2\\sqrt{7}, \\;-\\sqrt{2}\\]are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polyn... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $x$ and $y$ be real numbers such that $x + y = 3.$ Find the maximum value of\n\\[x^4 y + x^3 y + x^2 y + xy + xy^2 + xy^3 + xy^4.\\] [/INST] First, we can factor out $xy,$ to get\n\\[xy (x^3 + x^2 + x + 1 + y + y^2 + y^3)... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Determine the value of the expression\n\\[\\log_2 (27 + \\log_2 (27 + \\log_2 (27 + \\cdots))),\\]assuming it is positive. [/INST] Let\n\\[x = \\log_2 (27 + \\log_2 (27 + \\log_2 (27 + \\dotsb))).\\]Then\n\\[x = \\log_2 (27 + ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the largest positive integer $n$ such that\n\\[\\sin^n x + \\cos^n x \\ge \\frac{1}{n}\\]for all real numbers $x.$ [/INST] Setting $x = \\pi,$ we get\n\\[(-1)^n \\ge \\frac{1}{n},\\]so $n$ must be even. Let $n = 2m.$\n\n... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $a$ and $b$ be real numbers. Consider the following five statements:\n\n$\\frac{1}{a} < \\frac{1}{b}$\n$a^2 > b^2$\n$a < b$\n$a < 0$\n$b < 0$\n\nWhat is the maximum number of these statements that can be true for any valu... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=80$. What is $x$? [/INST] Applying Simon's Favorite Factoring Trick, we add 1 to both sides to get $xy + x + y + 1 = 81,$ so\n\\[(x + 1)(y + 1) = 81.\\]The only possibility is ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the roots of $z^2 - z = 5 - 5i.$\n\nEnter the roots, separated by commas. [/INST] We can write $z^2 - z - (5 - 5i) = 0.$ By the quadratic formula,\n\\[z = \\frac{1 \\pm \\sqrt{1 + 4(5 - 5i)}}{2} = \\frac{1 \\pm \\sqrt{21... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Suppose $f(x) = 6x - 9$ and $g(x) = \\frac{x}{3} + 2$. Find $f(g(x)) - g(f(x))$. [/INST] We have that\n$$\\begin{aligned} f(g(x)) &= f\\left(\\frac{x}{3} + 2\\right) = 6\\left(\\frac{x}{3} + 2\\right) - 9 \\\\\n&= 2x + 12 - 9\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Which type of conic section is described by the equation \\[\\sqrt{x^2 + (y-1)^2} + \\sqrt{(x-5)^2 + (y+3)^2} = 10?\\]Enter \"C\" for circle, \"P\" for parabola, \"E\" for ellipse, \"H\" for hyperbola, and \"N\" for none of th... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find all $t$ such that $x-t$ is a factor of $6x^2+13x-5.$\n\nEnter your answer as a list separated by commas. [/INST] From the Factor theorem, if $x-t$ is a factor of $ 6x^2+13x-5$ we know that\n$$6t^2+13t - 5 = 0$$Factoring g... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: There are integers $b,c$ for which both roots of the polynomial $x^2-x-1$ are also roots of the polynomial $x^5-bx-c$. Determine the product $bc$. [/INST] Let $r$ be a root of $x^2-x-1$. Then, rearranging, we have\n$$r^2 = r+1... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: When the graph of $y = 2x^2 - x + 7$ is shifted four units to the right, we obtain the graph of $y = ax^2 + bx + c$. Find $a + b + c$. [/INST] When we shift the graph of $y = 2x^2 - x + 7$ four units to the right, we obtain t... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $a$ and $b$ be the roots of $k(x^2 - x) + x + 5 = 0.$ Let $k_1$ and $k_2$ be the values of $k$ for which $a$ and $b$ satisfy\n\\[\\frac{a}{b} + \\frac{b}{a} = \\frac{4}{5}.\\]Find\n\\[\\frac{k_1}{k_2} + \\frac{k_2}{k_1}.\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The function $f$ is linear and satisfies $f(d+1)-f(d) = 3$ for all real numbers $d$. What is $f(3)-f(5)$? [/INST] Taking $d = 3,$ we get\n\\[f(4) - f(3) = 3.\\]Taking $d = 4,$ we get\n\\[f(5) - f(4) = 3.\\]Adding these equatio... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A function $f$ is defined by $f(z) = (4 + i) z^2 + \\alpha z + \\gamma$ for all complex numbers $z$, where $\\alpha$ and $\\gamma$ are complex numbers and $i^2 = - 1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the ... | custom_source | 5 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.