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Let \[f(x) = \left\{
\begin{array}{cl} ax+3, &\text{ if }x>2, \\
x-5 &\text{ if } -2 \le x \le 2, \\
2x-b &\text{ if } x <-2.
\end{array}
\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper). | 0 | For the piecewise function to be continuous, the cases must "meet" at $2$ and $-2$. For example, $ax+3$ and $x-5$ must be equal when $x=2$. This implies $a(2)+3=2-5$, which we solve to get $2a=-6 \Rightarrow a=-3$. Similarly, $x-5$ and $2x-b$ must be equal when $x=-2$. Substituting, we get $-2-5=2(-2)-b$, which implies... | Algebra | Level 5 | MaybeKnown | [
"<|begin_of_text|><|start_header_id|>system<|end_header_id|>\n\nCutting Knowledge Date: December 2023\nToday Date: 26 Jul 2024\n\nSolve the following math problem step by step. Show your work, then give the final answer starting with 'Answer:'<|eot_id|><|start_header_id|>user<|end_header_id|>\n\nProblem: What is th... | [
"For the function to be continuous, the value of $f(x)$ must be the same at $x=2$ and $x=-2$. \n\nAt $x=2$, $f(x) = ax+3$. \nAt $x=-2$, $f(x) = x-5$. \n\nSetting these equal to each other, we get $a(2)+3 = -2-5$, so $2a = -9$, and $a = -\\frac{9}{2}$.\n\nAt $x=2$, $f(x) = ax+3 = -\\frac{9}{2}(2)+3 = -\\frac{9}{2}+3... | [
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A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | 98 | Let $x$ be the number of band members in each row for the original formation, when two are left over. Then we can write two equations from the given information: $$rx+2=m$$ $$(r-2)(x+1)=m$$ Setting these equal, we find: $$rx+2=(r-2)(x+1)=rx-2x+r-2$$ $$2=-2x+r-2$$ $$4=r-2x$$ We know that the band has less than 100 memb... | Algebra | Level 5 | MaybeKnown | [
"<|begin_of_text|><|start_header_id|>system<|end_header_id|>\n\nCutting Knowledge Date: December 2023\nToday Date: 26 Jul 2024\n\nSolve the following math problem step by step. Show your work, then give the final answer starting with 'Answer:'<|eot_id|><|start_header_id|>user<|end_header_id|>\n\nProblem: What is th... | [
"Let $m$ be the number of members in each row and $r$ be the number of rows. Then $mr + 2$ is the total number of band members. When the number of members in each row is increased by 1 and the number of rows is reduced by 2, the new formation has $(m+1)(r-2)$ members. Since this is equal to the total number of band... | [
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What is the degree of the polynomial $(4 +5x^3 +100 +2\pi x^4 + \sqrt{10}x^4 +9)$? | 4 | This polynomial is not written in standard form. However, we don't need to write it in standard form, nor do we need to pay attention to the coefficients. We just look for the exponents on $x$. We have an $x^4$ term and no other term of higher degree, so $\boxed{4}$ is the degree of the polynomial. | Algebra | Level 3 | HighlyKnown | [
"<|begin_of_text|><|start_header_id|>system<|end_header_id|>\n\nCutting Knowledge Date: December 2023\nToday Date: 26 Jul 2024\n\nSolve the following math problem step by step. Show your work, then give the final answer starting with 'Answer:'<|eot_id|><|start_header_id|>user<|end_header_id|>\n\nProblem: What is th... | [
"The degree of a polynomial is the highest power of the variable in the polynomial. In this case, the highest power of $x$ is $4$, so the degree of the polynomial is $4$.\n\nAnswer: 4",
"The degree of a polynomial is the highest power of the variable (in this case, $x$) in the polynomial. In the given polynomial,... | [
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Evaluate $\left\lceil3\left(6-\frac12\right)\right\rceil$. | 17 | Firstly, $3\left(6-\frac12\right)=18-1-\frac12=17-\frac12$. Because $0\le\frac12<1$, we have $\left\lceil17-\frac12\right\rceil=\boxed{17}$. | Algebra | Level 3 | HighlyKnown | [
"<|begin_of_text|><|start_header_id|>system<|end_header_id|>\n\nCutting Knowledge Date: December 2023\nToday Date: 26 Jul 2024\n\nSolve the following math problem step by step. Show your work, then give the final answer starting with 'Answer:'<|eot_id|><|start_header_id|>user<|end_header_id|>\n\nProblem: What is th... | [
"First, evaluate the expression inside the parentheses: $6 - \\frac{1}{2} = \\frac{11}{2}$. Then $3 \\cdot \\frac{11}{2} = \\frac{33}{2}$. Finally, $\\left\\lceil\\frac{33}{2}\\right\\rceil = \\left\\lceil16.5\\right\\rceil = 17$.\n\nAnswer: 17",
"First, evaluate the expression inside the parentheses: $6 - \\frac... | [
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Sam is hired for a 20-day period. On days that he works, he earns $\$$60. For each day that he does not work, $\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\$$660. How many days did he not work? | 6 | Call $x$ the number of days Sam works and $y$ the number of days he does not. We can set up the following system of equations to represent the given information: \begin{align*}
x+y &= 20 \\
60x - 30y &= 660 \\
\end{align*} The first equation represents the total number of days Sam works, and the second equation represe... | Algebra | Level 3 | HighlyKnown | [
"<|begin_of_text|><|start_header_id|>system<|end_header_id|>\n\nCutting Knowledge Date: December 2023\nToday Date: 26 Jul 2024\n\nSolve the following math problem step by step. Show your work, then give the final answer starting with 'Answer:'<|eot_id|><|start_header_id|>user<|end_header_id|>\n\nProblem: What is th... | [
"Let $x$ be the number of days Sam worked. Then $20-x$ is the number of days he did not work. His earnings are $60x - 30(20-x) = 60x - 600 + 30x = 90x - 600$. This is equal to $\\$$660, so $90x - 600 = 660$. Solving for $x$, we get $90x = 1260$, so $x = 14$. Therefore, Sam did not work for $20 - 14 = 6$ days.\n\nAn... | [
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Find the center of the circle with equation $x^2 - 6x + y^2 + 2y = 9$. | (3, -1) | Completing the square, we get $(x - 3)^2 + (y + 1)^2 = 19$. Therefore, the center of the circle is $\boxed{(3, -1)}$. | Algebra | Level 4 | HighlyKnown | [
"<|begin_of_text|><|start_header_id|>system<|end_header_id|>\n\nCutting Knowledge Date: December 2023\nToday Date: 26 Jul 2024\n\nSolve the following math problem step by step. Show your work, then give the final answer starting with 'Answer:'<|eot_id|><|start_header_id|>user<|end_header_id|>\n\nProblem: What is th... | [
"To find the center of the circle, we need to rewrite the equation in the standard form $(x-h)^2 + (y-k)^2 = r^2$. Completing the square for $x$ and $y$, we have $(x^2 - 6x + 9) + (y^2 + 2y + 1) = 9 + 9 + 1$, which simplifies to $(x-3)^2 + (y+1)^2 = 19$. Therefore, the center of the circle is $(3, -1)$.\n\nAnswer: ... | [
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What are all values of $p$ such that for every $q>0$, we have $$\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q?$$ Express your answer in interval notation in decimal form. | [0,3) | First we'll simplify that complicated expression. We attempt to factor the numerator of the left side: \begin{align*}
pq^2+p^2q+3q^2+3pq &= q(pq + p^2 + 3q + 3p) \\
&= q[ p(q+p) + 3(q+p) ] \\
&= q(p+3)(q+p).
\end{align*}Substituting this in for the numerator in our inequality gives $$\frac{3q(p+3)(p+q)}{p+q}>2p^2q.$$We... | Algebra | Level 5 | Unknown | [
"<|begin_of_text|><|start_header_id|>system<|end_header_id|>\n\nCutting Knowledge Date: December 2023\nToday Date: 26 Jul 2024\n\nSolve the following math problem step by step. Show your work, then give the final answer starting with 'Answer:'<|eot_id|><|start_header_id|>user<|end_header_id|>\n\nProblem: What is th... | [
"We can simplify the given inequality as follows:\n\n$\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q} > 2p^2q$\n\n$\\frac{3q(p+q+p^2)}{p+q} > 2p^2q$\n\n$3q(p+q+p^2) > 2p^2q(p+q)$\n\n$3(p+q+p^2) > 2p^2(p+q)$\n\n$3p+3q+3p^2 > 2p^3+2pq^2$\n\n$3p^2-2p^3+2pq^2+3q-2pq > 0$\n\n$-2p^3+3p^2+2pq^2+3q-2pq > 0$\n\n$-2p^3+3p^2+2pq^2-2pq+3q ... | [
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If $x = 2$ and $y = 5$, then what is the value of $\frac{x^4+2y^2}{6}$ ? | 11 | "We have \\[\\frac{x^4 + 2y^2}{6} = \\frac{2^4 + 2(5^2)}{6} = \\frac{16+2(25)}{6} = \\frac{16+50}{6(...TRUNCATED) | Algebra | Level 1 | HighlyKnown | ["<|begin_of_text|><|start_header_id|>system<|end_header_id|>\n\nCutting Knowledge Date: December 20(...TRUNCATED) | ["Substitute $x = 2$ and $y = 5$ into the expression: $\\frac{(2)^4 + 2(5)^2}{6} = \\frac{16 + 50}{6(...TRUNCATED) | [
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"The sequence of integers in the row of squares and in each of the two columns of squares form three(...TRUNCATED) | -7 | "Since $18 - 14 = 4$, the common difference in the first column of squares is 4, so the number above(...TRUNCATED) | Algebra | Level 2 | Unknown | ["<|begin_of_text|><|start_header_id|>system<|end_header_id|>\n\nCutting Knowledge Date: December 20(...TRUNCATED) | ["The common difference in the first column is $18-14 = 4$. The common difference in the second colu(...TRUNCATED) | [
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"Tim wants to invest some money in a bank which compounds quarterly with an annual interest rate of (...TRUNCATED) | 42409 | "Recall the formula $A=P\\left(1+\\frac{r}{n}\\right)^{nt}$, where $A$ is the end balance, $P$ is th(...TRUNCATED) | Algebra | Level 5 | MaybeKnown | ["<|begin_of_text|><|start_header_id|>system<|end_header_id|>\n\nCutting Knowledge Date: December 20(...TRUNCATED) | ["The formula for compound interest is $A = P(1 + \\frac{r}{n})^{nt}$, where $A$ is the amount of mo(...TRUNCATED) | [
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